| Title: | Estimate Gaussian and Student's t Mixture Vector Autoregressive Models |
|---|---|
| Description: | Unconstrained and constrained maximum likelihood estimation of structural and reduced form Gaussian mixture vector autoregressive, Student's t mixture vector autoregressive, and Gaussian and Student's t mixture vector autoregressive models, quantile residual tests, graphical diagnostics, simulations, forecasting, and estimation of generalized impulse response function and generalized forecast error variance decomposition. Leena Kalliovirta, Mika Meitz, Pentti Saikkonen (2016) <doi:10.1016/j.jeconom.2016.02.012>, Savi Virolainen (forthcoming) <doi:10.1080/07350015.2024.2322090>, Savi Virolainen (2022) <doi:10.48550/arXiv.2109.13648>. |
| Authors: | Savi Virolainen [aut, cre]
|
| Maintainer: | Savi Virolainen <[email protected]> |
| License: | GPL-3 |
| Version: | 2.1.3 |
| Built: | 2024-12-04 15:48:25 UTC |
| Source: | CRAN |
gmvarkit is a package for reduced form and structural Gaussian mixture vector
autoregressive (GMVAR), Student's t Mixture Vector Autoregressive (StMVAR),
or Gaussian and Student's t Mixture Vector Autoregressive (G-StMVAR) model analysis.
It provides functions for unconstrained and constrained maximum likelihood estimation of the model parameters,
quantile residuals tests, graphical diagnostics, estimation of generalized impulse response function,
estimation of generalized forecast error variance decomposition, simulation from GMVAR processes,
forecasting, and more.
The readme file is a good place to start and the vignette might be useful too.
Maintainer: Savi Virolainen [email protected] (ORCID)
Useful links:
Report bugs at https://github.com/saviviro/gmvarkit/issues
add_data adds or updates data to object of class 'gsmvar' that defines
a GMVAR, StMVAR, or G-StMVAR model. Also calculates mixing weights and quantile residuals accordingly.
add_data(data, gsmvar, calc_cond_moments = TRUE, calc_std_errors = FALSE)add_data(data, gsmvar, calc_cond_moments = TRUE, calc_std_errors = FALSE)
data |
a matrix or class |
gsmvar |
an object of class |
calc_cond_moments |
should conditional means and covariance matrices should be calculated?
Default is |
calc_std_errors |
should approximate standard errors be calculated? |
Returns an object of class 'gsmvar' defining the specified GSMVAR, StMVAR, or G-StMVAR model with the data added to the model. If the object already contained data, the data will be updated.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
fitGSMVAR, GSMVAR, iterate_more, update_numtols
# GMVAR(1, 2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(p=1, M=2, d=2, params=params12) mod12 mod12_2 <- add_data(gdpdef, mod12) mod12_2 # StMVAR(1, 2), d=2 model: mod12t <- GSMVAR(p=1, M=2, d=2, params=c(params12, 10, 12), model="StMVAR") mod12t mod12t_2 <- add_data(gdpdef, mod12t) mod12t_2 # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(p=2, M=2, d=2, params=params22s, structural_pars=list(W=W_22)) mod22s mod22s_2 <- add_data(gdpdef, mod22s) mod22s_2# GMVAR(1, 2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(p=1, M=2, d=2, params=params12) mod12 mod12_2 <- add_data(gdpdef, mod12) mod12_2 # StMVAR(1, 2), d=2 model: mod12t <- GSMVAR(p=1, M=2, d=2, params=c(params12, 10, 12), model="StMVAR") mod12t mod12t_2 <- add_data(gdpdef, mod12t) mod12t_2 # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(p=2, M=2, d=2, params=params22s, structural_pars=list(W=W_22)) mod22s mod22s_2 <- add_data(gdpdef, mod22s) mod22s_2
fitGSMVAR
DEPRECATED! USE THE FUNCTION alt_gsmvar INSTEAD! alt_gsmvar constructs
a GMVAR model based on results from an arbitrary estimation round of fitGSMVAR.
alt_gmvar( gmvar, which_round = 1, which_largest, calc_cond_moments = TRUE, calc_std_errors = TRUE )alt_gmvar( gmvar, which_round = 1, which_largest, calc_cond_moments = TRUE, calc_std_errors = TRUE )
gmvar |
object of class 'gmvar' |
which_round |
based on which estimation round should the model be constructed? An integer value in 1,..., |
which_largest |
based on estimation round with which largest log-likelihood should the model be constructed?
An integer value in 1,..., |
calc_cond_moments |
should conditional means and covariance matrices should be calculated?
Default is |
calc_std_errors |
should approximate standard errors be calculated? |
It's sometimes useful to examine other estimates than the one with the highest log-likelihood. This function
is wrapper around GSMVAR that picks the correct estimates from an object returned by fitGSMVAR.
Returns an object of class 'gsmvar' defining the specified reduced form or structural GMVAR,
StMVAR, or G-StMVAR model. Can be used to work with other functions provided in gmvarkit.
Note that the first autocovariance/correlation matrix in $uncond_moments is for the lag zero,
the second one for the lag one, etc.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Kalliovirta L. and Saikkonen P. 2010. Reliable Residuals for Multivariate Nonlinear Time Series Models. Unpublished Revision of HECER Discussion Paper No. 247.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
fitGSMVAR
alt_gsmvar constructs a GMVAR, StMVAR, or G-StMVAR model based on results from
an arbitrary estimation round of fitGSMVAR.
alt_gsmvar( gsmvar, which_round = 1, which_largest, calc_cond_moments = TRUE, calc_std_errors = TRUE )alt_gsmvar( gsmvar, which_round = 1, which_largest, calc_cond_moments = TRUE, calc_std_errors = TRUE )
gsmvar |
an object of class |
which_round |
based on which estimation round should the model be constructed? An integer value in 1,..., |
which_largest |
based on estimation round with which largest log-likelihood should the model be constructed?
An integer value in 1,..., |
calc_cond_moments |
should conditional means and covariance matrices should be calculated?
Default is |
calc_std_errors |
should approximate standard errors be calculated? |
It's sometimes useful to examine other estimates than the one with the highest log-likelihood. This function
is wrapper around GSMVAR that picks the correct estimates from an object returned by fitGSMVAR.
Returns an object of class 'gsmvar' defining the specified reduced form or structural GMVAR,
StMVAR, or G-StMVAR model. Can be used to work with other functions provided in gmvarkit.
Note that the first autocovariance/correlation matrix in $uncond_moments is for the lag zero,
the second one for the lag one, etc.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Kalliovirta L. and Saikkonen P. 2010. Reliable Residuals for Multivariate Nonlinear Time Series Models. Unpublished Revision of HECER Discussion Paper No. 247.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
fitGSMVAR, GSMVAR, iterate_more, update_numtols
# GMVAR(1,2) model fit12 <- fitGSMVAR(gdpdef, p=1, M=2, ncalls=2, seeds=4:5) fit12 fit12_2 <- alt_gsmvar(fit12, which_largest=2) fit12_2# GMVAR(1,2) model fit12 <- fitGSMVAR(gdpdef, p=1, M=2, ncalls=2, seeds=4:5) fit12 fit12_2 <- alt_gsmvar(fit12, which_largest=2) fit12_2
calc_gradient or calc_hessian calculates the gradient or Hessian matrix
of the given function at the given point using central difference numerical approximation.
get_gradient or get_hessian calculates the gradient or Hessian matrix of the
log-likelihood function at the parameter estimates of a class 'gsmvar' object. get_soc
returns eigenvalues of the Hessian matrix, and get_foc is the same as get_gradient
but named conveniently.
calc_gradient(x, fn, h = 6e-06, varying_h = NULL, ...) calc_hessian(x, fn, h = 6e-06, varying_h = NULL, ...) get_gradient(gsmvar, custom_h = NULL) get_hessian(gsmvar, custom_h = NULL) get_foc(gsmvar, custom_h = NULL) get_soc(gsmvar, custom_h = NULL)calc_gradient(x, fn, h = 6e-06, varying_h = NULL, ...) calc_hessian(x, fn, h = 6e-06, varying_h = NULL, ...) get_gradient(gsmvar, custom_h = NULL) get_hessian(gsmvar, custom_h = NULL) get_foc(gsmvar, custom_h = NULL) get_soc(gsmvar, custom_h = NULL)
x |
a numeric vector specifying the point where the gradient or Hessian should be calculated. |
fn |
a function that takes in argument |
h |
difference used to approximate the derivatives. |
varying_h |
a numeric vector with the same length as |
... |
other arguments passed to |
gsmvar |
an object of class |
custom_h |
same as |
In particular, the functions get_foc and get_soc can be used to check whether
the found estimates denote a (local) maximum point, a saddle point, or something else. Note that
profile log-likelihood functions can be conveniently plotted with the function profile_logliks.
Gradient functions return numerical approximation of the gradient and Hessian functions return
numerical approximation of the Hessian. get_soc returns eigenvalues of the Hessian matrix.
No argument checks!
# Simple function foo <- function(x) x^2 + x calc_gradient(x=1, fn=foo) calc_gradient(x=-0.5, fn=foo) # More complicated function foo <- function(x, a, b) a*x[1]^2 - b*x[2]^2 calc_gradient(x=c(1, 2), fn=foo, a=0.3, b=0.1) # GMVAR(1,2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) get_gradient(mod12) get_hessian(mod12) get_soc(mod12)# Simple function foo <- function(x) x^2 + x calc_gradient(x=1, fn=foo) calc_gradient(x=-0.5, fn=foo) # More complicated function foo <- function(x, a, b) a*x[1]^2 - b*x[2]^2 calc_gradient(x=c(1, 2), fn=foo, a=0.3, b=0.1) # GMVAR(1,2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) get_gradient(mod12) get_hessian(mod12) get_soc(mod12)
check_parameters checks whether the given parameter vector satisfies
the model assumptions. Does NOT consider the identifiability condition!
check_parameters( p, M, d, params, model = c("GMVAR", "StMVAR", "G-StMVAR"), parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, weight_constraints = NULL, structural_pars = NULL, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 )check_parameters( p, M, d, params, model = c("GMVAR", "StMVAR", "G-StMVAR"), parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, weight_constraints = NULL, structural_pars = NULL, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 )
p |
a positive integer specifying the autoregressive order of the model. |
M |
|
d |
the number of time series in the system. |
params |
a real valued vector specifying the parameter values.
Above, In the GMVAR model, The notation is similar to the cited literature. |
model |
is "GMVAR", "StMVAR", or "G-StMVAR" model considered? In the G-StMVAR model, the first |
parametrization |
|
constraints |
a size |
same_means |
Restrict the mean parameters of some regimes to be the same? Provide a list of numeric vectors
such that each numeric vector contains the regimes that should share the common mean parameters. For instance, if
|
weight_constraints |
a numeric vector of length |
structural_pars |
If
See Virolainen (forthcoming) for the conditions required to identify the shocks and for the B-matrix as well (it is |
stat_tol |
numerical tolerance for stationarity of the AR parameters: if the "bold A" matrix of any regime
has eigenvalues larger that |
posdef_tol |
numerical tolerance for positive definiteness of the error term covariance matrices: if the error term covariance matrix of any regime has eigenvalues smaller than this, the model is classified as not satisfying positive definiteness assumption. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error. |
df_tol |
the parameter vector is considered to be outside the parameter space if all degrees of
freedom parameters are not larger than |
Throws an informative error if there is something wrong with the parameter vector.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
@keywords internal
## Not run: # These examples will cause an informative error # GMVAR(1, 1), d=2 model: params11 <- c(1.07, 127.71, 0.99, 0.00, -0.01, 1.00, 4.05, 2.22, 8.87) check_parameters(p=1, M=1, d=2, params=params11) # GMVAR(2, 2), d=2 model: params22 <- c(1.39, -0.77, 1.31, 0.14, 0.09, 1.29, -0.39, -0.07, -0.11, -0.28, 0.92, -0.03, 4.84, 1.01, 5.93, 1.25, 0.08, -0.04, 1.27, -0.27, -0.07, 0.03, -0.31, 5.85, 10.57, 9.84, 0.74) check_parameters(p=2, M=2, d=2, params=params22) # GMVAR(2, 2), d=2 model with AR-parameters restricted to be # the same for both regimes: C_mat <- rbind(diag(2*2^2), diag(2*2^2)) params222c <- c(1.03, 2.36, 1.79, 3.00, 1.25, 0.06,0.04, 1.34, -0.29, -0.08, -0.05, -0.36, 0.93, -0.15, 5.20, 5.88, 3.56, 9.80, 1.37) check_parameters(p=2, M=2, d=2, params=params22c, constraints=C_mat) # Structural GMVAR(2, 2), d=2 model identified with sign-constraints # (no error): params22s <- c(1.03, 2.36, 1.79, 3, 1.25, 0.06, 0.04, 1.34, -0.29, -0.08, -0.05, -0.36, 1.2, 0.05, 0.05, 1.3, -0.3, -0.1, -0.05, -0.4, 0.89, 0.72, -0.37, 2.16, 7.16, 1.3, 0.37) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) check_parameters(p=2, M=2, d=2, params=params22s, structural_pars=list(W=W_22)) ## End(Not run)## Not run: # These examples will cause an informative error # GMVAR(1, 1), d=2 model: params11 <- c(1.07, 127.71, 0.99, 0.00, -0.01, 1.00, 4.05, 2.22, 8.87) check_parameters(p=1, M=1, d=2, params=params11) # GMVAR(2, 2), d=2 model: params22 <- c(1.39, -0.77, 1.31, 0.14, 0.09, 1.29, -0.39, -0.07, -0.11, -0.28, 0.92, -0.03, 4.84, 1.01, 5.93, 1.25, 0.08, -0.04, 1.27, -0.27, -0.07, 0.03, -0.31, 5.85, 10.57, 9.84, 0.74) check_parameters(p=2, M=2, d=2, params=params22) # GMVAR(2, 2), d=2 model with AR-parameters restricted to be # the same for both regimes: C_mat <- rbind(diag(2*2^2), diag(2*2^2)) params222c <- c(1.03, 2.36, 1.79, 3.00, 1.25, 0.06,0.04, 1.34, -0.29, -0.08, -0.05, -0.36, 0.93, -0.15, 5.20, 5.88, 3.56, 9.80, 1.37) check_parameters(p=2, M=2, d=2, params=params22c, constraints=C_mat) # Structural GMVAR(2, 2), d=2 model identified with sign-constraints # (no error): params22s <- c(1.03, 2.36, 1.79, 3, 1.25, 0.06, 0.04, 1.34, -0.29, -0.08, -0.05, -0.36, 1.2, 0.05, 0.05, 1.3, -0.3, -0.1, -0.05, -0.4, 0.89, 0.72, -0.37, 2.16, 7.16, 1.3, 0.37) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) check_parameters(p=2, M=2, d=2, params=params22s, structural_pars=list(W=W_22)) ## End(Not run)
cond_moment_plot plots the one-step in-sample conditional means/variances of the model along with
the individual time series contained in the model (e.g. the time series the model was fitted to). Also plots
the regimewise conditional means/variances multiplied with mixing weights.
cond_moment_plot( gsmvar, which_moment = c("mean", "variance"), grid = FALSE, ... )cond_moment_plot( gsmvar, which_moment = c("mean", "variance"), grid = FALSE, ... )
gsmvar |
an object of class |
which_moment |
should conditional means or variances be plotted? |
grid |
add grid to the plots? |
... |
additional paramters passed to |
The conditional mean plot works best if the data contains positive values only.
acf from the package stats and the plot method for class 'acf' objects is employed.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Lütkepohl H. 2005. New Introduction to Multiple Time Series Analysis, Springer.
McElroy T. 2017. Computation of vector ARMA autocovariances. Statistics and Probability Letters, 124, 92-96.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
profile_logliks, fitGSMVAR, GSMVAR,
quantile_residual_tests, LR_test, Wald_test,
diagnostic_plot
# GMVAR(2, 2), d=2 model; params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22) cond_moment_plot(mod22, which_moment="mean") cond_moment_plot(mod22, which_moment="variance") cond_moment_plot(mod22, which_moment="mean", grid=TRUE, lty=3) # G-StMVAR(2, 1, 1), d=2 model: params22gs <- c(0.697, 0.154, 0.049, 0.374, 0.476, 0.318, -0.645, -0.302, -0.222, 0.193, 0.042, -0.013, 0.048, 0.554, 0.033, 0.184, 0.005, -0.186, 0.683, 0.256, 0.031, 0.026, 0.204, 0.583, -0.002, 0.048, 0.182, 4.334) mod22gs <- GSMVAR(gdpdef, p=2, M=c(1, 1), params=params22gs, model="G-StMVAR") cond_moment_plot(mod22gs, which_moment="mean") cond_moment_plot(mod22gs, which_moment="variance") #StMVAR(4, 1), d=2 model: params41t <- c(0.512, -0.002, 0.243, 0.024, -0.088, 0.452, 0.242, 0.011, 0.093, 0.162, -0.097, 0.033, -0.339, 0.19, 0.091, 0.006, 0.168, 0.101, 0.516, -0.005, 0.054, 4.417) mod41t <- GSMVAR(gdpdef, p=4, M=1, params=params41t, model="StMVAR") cond_moment_plot(mod41t, which_moment="mean") cond_moment_plot(mod41t, which_moment="variance")# GMVAR(2, 2), d=2 model; params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22) cond_moment_plot(mod22, which_moment="mean") cond_moment_plot(mod22, which_moment="variance") cond_moment_plot(mod22, which_moment="mean", grid=TRUE, lty=3) # G-StMVAR(2, 1, 1), d=2 model: params22gs <- c(0.697, 0.154, 0.049, 0.374, 0.476, 0.318, -0.645, -0.302, -0.222, 0.193, 0.042, -0.013, 0.048, 0.554, 0.033, 0.184, 0.005, -0.186, 0.683, 0.256, 0.031, 0.026, 0.204, 0.583, -0.002, 0.048, 0.182, 4.334) mod22gs <- GSMVAR(gdpdef, p=2, M=c(1, 1), params=params22gs, model="G-StMVAR") cond_moment_plot(mod22gs, which_moment="mean") cond_moment_plot(mod22gs, which_moment="variance") #StMVAR(4, 1), d=2 model: params41t <- c(0.512, -0.002, 0.243, 0.024, -0.088, 0.452, 0.242, 0.011, 0.093, 0.162, -0.097, 0.033, -0.339, 0.19, 0.091, 0.006, 0.168, 0.101, 0.516, -0.005, 0.054, 4.417) mod41t <- GSMVAR(gdpdef, p=4, M=1, params=params41t, model="StMVAR") cond_moment_plot(mod41t, which_moment="mean") cond_moment_plot(mod41t, which_moment="variance")
cond_moments compute conditional regimewise means, conditional means, and conditional covariance matrices
of a GMVAR, StMVAR, or G-StMVAR model.
cond_moments( data, p, M, params, model = c("GMVAR", "StMVAR", "G-StMVAR"), parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, weight_constraints = NULL, structural_pars = NULL, to_return = c("regime_cmeans", "regime_ccovs", "total_cmeans", "total_ccovs", "arch_scalars"), minval = NA, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 )cond_moments( data, p, M, params, model = c("GMVAR", "StMVAR", "G-StMVAR"), parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, weight_constraints = NULL, structural_pars = NULL, to_return = c("regime_cmeans", "regime_ccovs", "total_cmeans", "total_ccovs", "arch_scalars"), minval = NA, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 )
data |
a matrix or class |
p |
a positive integer specifying the autoregressive order of the model. |
M |
|
params |
a real valued vector specifying the parameter values.
Above, In the GMVAR model, The notation is similar to the cited literature. |
model |
is "GMVAR", "StMVAR", or "G-StMVAR" model considered? In the G-StMVAR model, the first |
parametrization |
|
constraints |
a size |
same_means |
Restrict the mean parameters of some regimes to be the same? Provide a list of numeric vectors
such that each numeric vector contains the regimes that should share the common mean parameters. For instance, if
|
weight_constraints |
a numeric vector of length |
structural_pars |
If
See Virolainen (forthcoming) for the conditions required to identify the shocks and for the B-matrix as well (it is |
to_return |
should the regimewise conditional means, total conditional means, or total conditional covariance matrices be returned? |
minval |
the value that will be returned if the parameter vector does not lie in the parameter space (excluding the identification condition). |
stat_tol |
numerical tolerance for stationarity of the AR parameters: if the "bold A" matrix of any regime
has eigenvalues larger that |
posdef_tol |
numerical tolerance for positive definiteness of the error term covariance matrices: if the error term covariance matrix of any regime has eigenvalues smaller than this, the model is classified as not satisfying positive definiteness assumption. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error. |
df_tol |
the parameter vector is considered to be outside the parameter space if all degrees of
freedom parameters are not larger than |
The first p values are used as the initial values, and by conditional we mean conditioning on the past. Formulas for the conditional means and covariance matrices are given in equations (3) and (4) of KMS (2016).
to_return=="regime_cmeans":an [T-p, d, M] array containing the regimewise conditional means
(the first p values are used as the initial values).
to_return=="regime_ccovs":an [d, d, T-p, M] array containing the regimewise conditional
covariance matrices (the first p values are used as the initial values). The index [ , , t, m] gives the time
t conditional covariance matrix for the regime m.
to_return=="total_cmeans":a [T-p, d] matrix containing the conditional means of the process
(the first p values are used as the initial values).
to_return=="total_ccov":an [d, d, T-p] array containing the conditional covariance matrices of the process
(the first p values are used as the initial values).
to_return=="arch_scalars":a [T-p, M] matrix containing the regimewise arch scalars
multiplying error term covariance matrix in the conditional covariance matrix of the regime. For GMVAR type regimes, these
are all ones (the first p values are used as the initial values).
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Lütkepohl H. 2005. New Introduction to Multiple Time Series Analysis, Springer.
McElroy T. 2017. Computation of vector ARMA autocovariances. Statistics and Probability Letters, 124, 92-96.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
Other moment functions:
get_regime_autocovs(),
get_regime_means(),
uncond_moments()
# GMVAR(2, 2), d=2 model; params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) cond_moments(data=gdpdef, p=2, M=2, params=params22, to_return="regime_cmeans") cond_moments(data=gdpdef, p=2, M=2, params=params22, to_return="total_cmeans") cond_moments(data=gdpdef, p=2, M=2, params=params22, to_return="total_ccovs")# GMVAR(2, 2), d=2 model; params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) cond_moments(data=gdpdef, p=2, M=2, params=params22, to_return="regime_cmeans") cond_moments(data=gdpdef, p=2, M=2, params=params22, to_return="total_cmeans") cond_moments(data=gdpdef, p=2, M=2, params=params22, to_return="total_ccovs")
diag_Omegas Simultaneously diagonalizes two covariance matrices using
eigenvalue decomposition.
diag_Omegas(Omega1, Omega2)diag_Omegas(Omega1, Omega2)
Omega1 |
a positive definite |
Omega2 |
another positive definite |
See the return value and Muirhead (1982), Theorem A9.9 for details.
Returns a length vector where the first elements
are with the columns of being (specific) eigenvectors of
the matrix and the rest elements are the
corresponding eigenvalues "lambdas". The result satisfies and
.
If Omega2 is not supplied, returns a vectorized symmetric (and pos. def.)
square root matrix of Omega1.
No argument checks! Does not work with dimension !
Muirhead R.J. 1982. Aspects of Multivariate Statistical Theory, Wiley.
d <- 2 W0 <- matrix(1:(d^2), nrow=2) lambdas0 <- 1:d (Omg1 <- W0%*%t(W0)) (Omg2 <- W0%*%diag(lambdas0)%*%t(W0)) res <- diag_Omegas(Omg1, Omg2) W <- matrix(res[1:(d^2)], nrow=d, byrow=FALSE) tcrossprod(W) # == Omg1 lambdas <- res[(d^2 + 1):(d^2 + d)] W%*%diag(lambdas)%*%t(W) # == Omg2d <- 2 W0 <- matrix(1:(d^2), nrow=2) lambdas0 <- 1:d (Omg1 <- W0%*%t(W0)) (Omg2 <- W0%*%diag(lambdas0)%*%t(W0)) res <- diag_Omegas(Omg1, Omg2) W <- matrix(res[1:(d^2)], nrow=d, byrow=FALSE) tcrossprod(W) # == Omg1 lambdas <- res[(d^2 + 1):(d^2 + d)] W%*%diag(lambdas)%*%t(W) # == Omg2
diagnostic_plot plots a multivariate quantile residual diagnostic plot
for either autocorrelation, conditional heteroskedasticity, or normality, or simply draws
the quantile residual time series.
diagnostic_plot( gsmvar, type = c("all", "series", "ac", "ch", "norm"), maxlag = 12, wait_time = 4 )diagnostic_plot( gsmvar, type = c("all", "series", "ac", "ch", "norm"), maxlag = 12, wait_time = 4 )
gsmvar |
an object of class |
type |
which type of diagnostic plot should be plotted?
|
maxlag |
the maximum lag considered in types |
wait_time |
if |
Auto- and cross-correlations (types "ac" and "ch") are calculated with the function
acf from the package stats and the plot method for class 'acf' objects is employed.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Kalliovirta L. and Saikkonen P. 2010. Reliable Residuals for Multivariate Nonlinear Time Series Models. Unpublished Revision of HECER Discussion Paper No. 247.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
profile_logliks, fitGSMVAR, GSMVAR, quantile_residual_tests,
LR_test, Wald_test, Rao_test, cond_moment_plot, acf,
density, predict.gsmvar
# GMVAR(1,2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) diagnostic_plot(mod12, type="series") diagnostic_plot(mod12, type="ac") # GMVAR(2,2), d=2 model: params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22) diagnostic_plot(mod22, type="ch") diagnostic_plot(mod22, type="norm") # G-StMVAR(2, 1, 1), d=2 model: params22gs <- c(0.697, 0.154, 0.049, 0.374, 0.476, 0.318, -0.645, -0.302, -0.222, 0.193, 0.042, -0.013, 0.048, 0.554, 0.033, 0.184, 0.005, -0.186, 0.683, 0.256, 0.031, 0.026, 0.204, 0.583, -0.002, 0.048, 0.182, 4.334) mod22gs <- GSMVAR(gdpdef, p=2, M=c(1, 1), params=params22gs, model="G-StMVAR") diagnostic_plot(mod22gs, wait_time=0)# GMVAR(1,2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) diagnostic_plot(mod12, type="series") diagnostic_plot(mod12, type="ac") # GMVAR(2,2), d=2 model: params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22) diagnostic_plot(mod22, type="ch") diagnostic_plot(mod22, type="norm") # G-StMVAR(2, 1, 1), d=2 model: params22gs <- c(0.697, 0.154, 0.049, 0.374, 0.476, 0.318, -0.645, -0.302, -0.222, 0.193, 0.042, -0.013, 0.048, 0.554, 0.033, 0.184, 0.005, -0.186, 0.683, 0.256, 0.031, 0.026, 0.204, 0.583, -0.002, 0.048, 0.182, 4.334) mod22gs <- GSMVAR(gdpdef, p=2, M=c(1, 1), params=params22gs, model="G-StMVAR") diagnostic_plot(mod22gs, wait_time=0)
estimate_gsmvar uses a genetic algorithm and variable metric algorithm to estimate the parameters
of a structural GMVAR, StMVAR, or G-StMVAR model with the method of maximum likelihood and preliminary
estimates from (typically identified) reduced form or structural GSMVAR model.
estimate_sgsmvar( gsmvar, new_W, ncalls = 16, ncores = 2, maxit = 1000, seeds = NULL )estimate_sgsmvar( gsmvar, new_W, ncalls = 16, ncores = 2, maxit = 1000, seeds = NULL )
gsmvar |
an object of class |
new_W |
What should be the constraints on the W-matrix (or equally B-matrix)? Provide a |
ncalls |
the number of estimation rounds that should be performed. |
ncores |
the number of CPU cores to be used in numerical differentiation. Multiple cores are not supported on Windows, though. |
maxit |
the maximum number of iterations in the variable metric algorithm. |
seeds |
a length |
The purpose of estimate_gsmvar is to provide a convenient tool to estimate (typically over)identified
structural GSMVAR models when preliminary estimates are available from a fitted reduced form or structural GMVAR
model. Often one estimates a two-regime reduced form model and then uses the function gsmvar_to_sgsmvar to
obtain the corresponding, statistically identified structural model. After obtaining the statistically identified
structural model, overidentifying constraints may be placed the W-matrix (or equally B-matrix). This function makes
imposing the overidentifying constraints and estimating the overidentified structural model convenient.
Reduced form models can be directly used as lower-triangular Cholesky identified SVARs without having
to estimate a structural model separately.
Note that the surface of the log-likelihood function is extremely multimodal, and this function is designed
to only explore the neighbourhood of the preliminary estimates, so it finds its way reliably to the correct MLE
only the preliminary estimates are close it in the first place. Use the function directly fitGSMVAR for a more thorough
search of the parameter space, if necessary. This function calls fitGSMVAR by construction an initial population to
the genetic algorithm from a preliminary guess of the new estimates. The smart mutations are set to begin from the
first generation.
In order to the impose the constraints you wish, it might be useful to first run the model through the functioons
reorder_W_columns and swap_W_signs. Particularly check that the sign constraints are readily
satisfied. If not, the estimated solution might not be correct MLE.
estimate_sgsmvar can also be used to estimate models that are not identified, i.e., one regime models. If it
supplied with a reduced form model, it will first apply the function gsmvar_to_sgsmvar, then impose the
constraints and finally estimate the model.
Returns an object of class 'gsmvar' defining the estimated GMVAR, StMVAR, or G-StMVAR model.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Kalliovirta L. and Saikkonen P. 2010. Reliable Residuals for Multivariate Nonlinear Time Series Models. Unpublished Revision of HECER Discussion Paper No. 247.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
fitGSMVAR, GSMVAR, optim,
profile_logliks, iterate_more gsmvar_to_sgsmvar
## These are long running examples that use parallel computing! ## Running the below examples takes 30 seconds # GMVAR(1,2) model fit12 <- fitGSMVAR(gdpdef, p=1, M=2, ncalls=2, seeds=1:2) # Reduced form fit12s <- gsmvar_to_sgsmvar(fit12) # Structural fit12s # Constrain the lower right element of W (or B-matrix) to zero, and for # global identification the first elements of each column to strictly positive. (new_W <- matrix(c(1, NA, 1, 0), nrow=2)) new_fit12s <- estimate_sgsmvar(fit12s, new_W, ncalls=2, ncores=2, seeds=1:2) new_fit12s # Overidentified model # Cholesky VAR(1) fit11 <- fitGSMVAR(gdpdef, p=1, M=1, ncalls=2, seeds=1:2) # Reduced form (new_W <- matrix(c(1, NA, 0, 1), nrow=2)) new_fit11s <- estimate_sgsmvar(fit11, new_W, ncalls=2, ncores=2, seeds=1:2) print(new_fit11s, digits=4) # Also: gsmvar_to_sgsmvar(fit11, cholesky=TRUE) gives Cholesky VAR## These are long running examples that use parallel computing! ## Running the below examples takes 30 seconds # GMVAR(1,2) model fit12 <- fitGSMVAR(gdpdef, p=1, M=2, ncalls=2, seeds=1:2) # Reduced form fit12s <- gsmvar_to_sgsmvar(fit12) # Structural fit12s # Constrain the lower right element of W (or B-matrix) to zero, and for # global identification the first elements of each column to strictly positive. (new_W <- matrix(c(1, NA, 1, 0), nrow=2)) new_fit12s <- estimate_sgsmvar(fit12s, new_W, ncalls=2, ncores=2, seeds=1:2) new_fit12s # Overidentified model # Cholesky VAR(1) fit11 <- fitGSMVAR(gdpdef, p=1, M=1, ncalls=2, seeds=1:2) # Reduced form (new_W <- matrix(c(1, NA, 0, 1), nrow=2)) new_fit11s <- estimate_sgsmvar(fit11, new_W, ncalls=2, ncores=2, seeds=1:2) print(new_fit11s, digits=4) # Also: gsmvar_to_sgsmvar(fit11, cholesky=TRUE) gives Cholesky VAR
The cyclical component of the log of industrial production index was obtained by applying the linear projection filter proposed by Hamilton (2018) using the parameter values h=24 and p=12. In order to obtain as accurate estimates as possible, we applied the filter to the full available sample from January 1991 to December 2021 before extracting our sample period from it. package lpirfs (Adämmer, 2021).
euromoneeuromone
A numeric matrix of class 'ts' with 276 rows and 4 columns with one time series in each column:
The cyclical component of the log of industrial production index, url is https://sdw.ecb.europa.eu/quickview.do?SERIES_KEY=132.STS.M.I8.Y.PROD.NS0010.4.000.
The log-difference of harmonized consumer price index, url is https://sdw.ecb.europa.eu/quickview.do?SERIES_KEY=122.ICP.M.U2.Y.000000.3.INX.
The log-difference of Brent crude oil price (Europe), https://fred.stlouisfed.org/series/MCOILBRENTEU.
The EONIA from January 1999 to October 2008 and after that the Wu and Xia (2016) shadow rate, urls are https://sdw.ecb.europa.eu/quickview.do?SERIES_KEY=143.FM.M.U2.EUR.4F.MM.EONIA.HSTA and https://sites.google.com/view/jingcynthiawu/shadow-rates.
The Federal Reserve Bank of St. Louis database and the Federal Reserve Bank of Atlanta's website
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
Wu J. and Xia F. 2016. Measuring the macroeconomic impact of monetary policy at the zero lower bound. Journal of Money, Credit and Banking, 48(2-3): 253-291.
DEPRECATED! USE THE FUNCTION fitGSMVAR INSTEAD!
fitGMVAR estimates a GMVAR model model in two phases:
in the first phase it uses a genetic algorithm to find starting values for a gradient based
variable metric algorithm, which it then uses to finalize the estimation in the second phase.
Parallel computing is utilized to perform multiple rounds of estimations in parallel.
fitGMVAR( data, p, M, conditional = TRUE, parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, structural_pars = NULL, ncalls = M^6, ncores = 2, maxit = 1000, seeds = NULL, print_res = TRUE, ... )fitGMVAR( data, p, M, conditional = TRUE, parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, structural_pars = NULL, ncalls = M^6, ncores = 2, maxit = 1000, seeds = NULL, print_res = TRUE, ... )
data |
a matrix or class |
p |
a positive integer specifying the autoregressive order of the model. |
M |
|
conditional |
a logical argument specifying whether the conditional or exact log-likelihood function |
parametrization |
|
constraints |
a size |
same_means |
Restrict the mean parameters of some regimes to be the same? Provide a list of numeric vectors
such that each numeric vector contains the regimes that should share the common mean parameters. For instance, if
|
structural_pars |
If
See Virolainen (forthcoming) for the conditions required to identify the shocks and for the B-matrix as well (it is |
ncalls |
the number of estimation rounds that should be performed. |
ncores |
the number CPU cores to be used in parallel computing. |
maxit |
the maximum number of iterations in the variable metric algorithm. |
seeds |
a length |
print_res |
should summaries of estimation results be printed? |
... |
additional settings passed to the function |
If you wish to estimate a structural model without overidentifying constraints that is identified statistically,
specify your W matrix is structural_pars to be such that it contains the same sign constraints in a single row
(e.g. a row of ones) and leave the other elements as NA. In this way, the genetic algorithm works the best.
The ordering and signs of the columns of the W matrix can be changed afterwards with the functions
reorder_W_columns and swap_W_signs.
Because of complexity and high multimodality of the log-likelihood function, it's not certain that the estimation algorithms will end up in the global maximum point. It's expected that most of the estimation rounds will end up in some local maximum or saddle point instead. Therefore, a (sometimes large) number of estimation rounds is required for reliable results. Because of the nature of the model, the estimation may fail especially in the cases where the number of mixture components is chosen too large. With two regimes and couple hundred observations in a two-dimensional time series, 50 rounds is usually enough. Several hundred estimation rounds often suffices for reliably fitting two-regimes models to 3 or 4 dimensional time series. With more than two regimes and more than couple hundred observations, thousands of estimation rounds (or more) are often required to obtain reliable results.
The estimation process is computationally heavy and it might take considerably long time for large models with
large number of observations. If the iteration limit maxit in the variable metric algorithm is reached,
one can continue the estimation by iterating more with the function iterate_more. Alternatively, one may
use the found estimates as starting values for the genetic algorithm and and employ another round of estimation
(see ?GAfit how to set up an initial population with the dot parameters).
If the estimation algorithm fails to create an initial population for the genetic algorithm, it usually helps to scale the individual series so that the AR coefficients (of a VAR model) will be relative small, preferably less than one. Even if one is able to create an initial population, it should be preferred to scale the series so that most of the AR coefficients will not be very large, as the estimation algorithm works better with relatively small AR coefficients. If needed, another package can be used to fit linear VARs to the series to see which scaling of the series results in relatively small AR coefficients.
The code of the genetic algorithm is mostly based on the description by Dorsey and Mayer (1995) but it includes some extra features that were found useful for this particular estimation problem. For instance, the genetic algorithm uses a slightly modified version of the individually adaptive crossover and mutation rates described by Patnaik and Srinivas (1994) and employs (50%) fitness inheritance discussed by Smith, Dike and Stegmann (1995).
The gradient based variable metric algorithm used in the second phase is implemented with function optim
from the package stats.
Note that the structural models are even more difficult to estimate than the reduced form models due to
the different parametrization of the covariance matrices, so larger number of estimation rounds should be considered.
Also, be aware that if the lambda parameters are constrained in any other way than by restricting some of them to be
identical, the parameter "lambda_scale" of the genetic algorithm (see ?GAfit) needs to be carefully adjusted accordingly.
When estimating a structural model that imposes overidentifiying constraints to a time series with ,
it is highly recommended to create an initial population based on the estimates of a statistically identified model
(when ). This is because currently obtaining the ML estimate reliably to such a structural model seems
difficult in many application.
Finally, the function fails to calculate approximate standard errors and the parameter estimates are near the border
of the parameter space, it might help to use smaller numerical tolerance for the stationarity and positive
definiteness conditions. The numerical tolerance of an existing model can be changed with the function
update_numtols.
Filtering inappropriate estimates: If filter_estimates == TRUE, the function will automatically filter
out estimates that it deems "inappropriate". That is, estimates that are not likely solutions of interest.
Specifically, solutions that incorporate a near-singular error term covariance matrix (any eigenvalue less than ),
mixing weights such that they are close to zero for almost all for at least one regime, or mixing weight parameter
estimate close to zero (or one). It also filters out estimates with any modulus "bold A" eigenvalues larger than 0.9985,
as the solution is near the boundary of the stationarity region and likely not a local maximum. You can also set
filter_estimates=FALSE and find the solutions of interest yourself by using the
function alt_gsmvar.
Returns an object of class 'gsmvar' defining the estimated (reduced form or structural) GMVAR, StMVAR, or G-StMVAR model.
Multivariate quantile residuals (Kalliovirta and Saikkonen 2010) are also computed and included in the returned object.
In addition, the returned object contains the estimates and log-likelihood values from all the estimation rounds performed.
The estimated parameter vector can be obtained at gsmvar$params (and corresponding approximate standard errors
at gsmvar$std_errors). See ?GSMVAR for the form of the parameter vector, if needed.
Remark that the first autocovariance/correlation matrix in $uncond_moments is for the lag zero,
the second one for the lag one, etc.
Dorsey R. E. and Mayer W. J. 1995. Genetic algorithms for estimation problems with multiple optima, nondifferentiability, and other irregular features. Journal of Business & Economic Statistics, 13, 53-66.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Patnaik L.M. and Srinivas M. 1994. Adaptive Probabilities of Crossover and Mutation in Genetic Algorithms. Transactions on Systems, Man and Cybernetics 24, 656-667.
Smith R.E., Dike B.A., Stegmann S.A. 1995. Fitness inheritance in genetic algorithms. Proceedings of the 1995 ACM Symposium on Applied Computing, 345-350.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
fitGSMVAR estimates a GMVAR, StMVAR, or G-StMVAR model model in two phases:
in the first phase it uses a genetic algorithm to find starting values for a gradient based
variable metric algorithm, which it then uses to finalize the estimation in the second phase.
Parallel computing is utilized to perform multiple rounds of estimations in parallel.
fitGSMVAR( data, p, M, model = c("GMVAR", "StMVAR", "G-StMVAR"), conditional = TRUE, parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, weight_constraints = NULL, structural_pars = NULL, ncalls = (M + 1)^5, ncores = 2, maxit = 1000, seeds = NULL, print_res = TRUE, use_parallel = TRUE, filter_estimates = TRUE, calc_std_errors = TRUE, ... )fitGSMVAR( data, p, M, model = c("GMVAR", "StMVAR", "G-StMVAR"), conditional = TRUE, parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, weight_constraints = NULL, structural_pars = NULL, ncalls = (M + 1)^5, ncores = 2, maxit = 1000, seeds = NULL, print_res = TRUE, use_parallel = TRUE, filter_estimates = TRUE, calc_std_errors = TRUE, ... )
data |
a matrix or class |
p |
a positive integer specifying the autoregressive order of the model. |
M |
|
model |
is "GMVAR", "StMVAR", or "G-StMVAR" model considered? In the G-StMVAR model, the first |
conditional |
a logical argument specifying whether the conditional or exact log-likelihood function |
parametrization |
|
constraints |
a size |
same_means |
Restrict the mean parameters of some regimes to be the same? Provide a list of numeric vectors
such that each numeric vector contains the regimes that should share the common mean parameters. For instance, if
|
weight_constraints |
a numeric vector of length |
structural_pars |
If
See Virolainen (forthcoming) for the conditions required to identify the shocks and for the B-matrix as well (it is |
ncalls |
the number of estimation rounds that should be performed. |
ncores |
the number CPU cores to be used in parallel computing. |
maxit |
the maximum number of iterations in the variable metric algorithm. |
seeds |
a length |
print_res |
should summaries of estimation results be printed? |
use_parallel |
employ parallel computing? If |
filter_estimates |
should the likely inappropriate estimates be filtered? See details. |
calc_std_errors |
calculate approximate standard errors for the estimates? |
... |
additional settings passed to the function |
If you wish to estimate a structural model without overidentifying constraints that is identified statistically,
specify your W matrix is structural_pars to be such that it contains the same sign constraints in a single row
(e.g. a row of ones) and leave the other elements as NA. In this way, the genetic algorithm works the best.
The ordering and signs of the columns of the W matrix can be changed afterwards with the functions
reorder_W_columns and swap_W_signs.
Because of complexity and high multimodality of the log-likelihood function, it's not certain that the estimation algorithms will end up in the global maximum point. It's expected that most of the estimation rounds will end up in some local maximum or saddle point instead. Therefore, a (sometimes large) number of estimation rounds is required for reliable results. Because of the nature of the model, the estimation may fail especially in the cases where the number of mixture components is chosen too large. With two regimes and couple hundred observations in a two-dimensional time series, 50 rounds is usually enough. Several hundred estimation rounds often suffices for reliably fitting two-regimes models to 3 or 4 dimensional time series. With more than two regimes and more than couple hundred observations, thousands of estimation rounds (or more) are often required to obtain reliable results.
The estimation process is computationally heavy and it might take considerably long time for large models with
large number of observations. If the iteration limit maxit in the variable metric algorithm is reached,
one can continue the estimation by iterating more with the function iterate_more. Alternatively, one may
use the found estimates as starting values for the genetic algorithm and and employ another round of estimation
(see ?GAfit how to set up an initial population with the dot parameters).
If the estimation algorithm fails to create an initial population for the genetic algorithm, it usually helps to scale the individual series so that the AR coefficients (of a VAR model) will be relative small, preferably less than one. Even if one is able to create an initial population, it should be preferred to scale the series so that most of the AR coefficients will not be very large, as the estimation algorithm works better with relatively small AR coefficients. If needed, another package can be used to fit linear VARs to the series to see which scaling of the series results in relatively small AR coefficients.
The code of the genetic algorithm is mostly based on the description by Dorsey and Mayer (1995) but it includes some extra features that were found useful for this particular estimation problem. For instance, the genetic algorithm uses a slightly modified version of the individually adaptive crossover and mutation rates described by Patnaik and Srinivas (1994) and employs (50%) fitness inheritance discussed by Smith, Dike and Stegmann (1995).
The gradient based variable metric algorithm used in the second phase is implemented with function optim
from the package stats.
Note that the structural models are even more difficult to estimate than the reduced form models due to
the different parametrization of the covariance matrices, so larger number of estimation rounds should be considered.
Also, be aware that if the lambda parameters are constrained in any other way than by restricting some of them to be
identical, the parameter "lambda_scale" of the genetic algorithm (see ?GAfit) needs to be carefully adjusted accordingly.
When estimating a structural model that imposes overidentifiying constraints to a time series with ,
it is highly recommended to create an initial population based on the estimates of a statistically identified model
(when ). This is because currently obtaining the ML estimate reliably to such a structural model seems
difficult in many application.
Finally, the function fails to calculate approximate standard errors and the parameter estimates are near the border
of the parameter space, it might help to use smaller numerical tolerance for the stationarity and positive
definiteness conditions. The numerical tolerance of an existing model can be changed with the function
update_numtols.
Filtering inappropriate estimates: If filter_estimates == TRUE, the function will automatically filter
out estimates that it deems "inappropriate". That is, estimates that are not likely solutions of interest.
Specifically, solutions that incorporate a near-singular error term covariance matrix (any eigenvalue less than ),
mixing weights such that they are close to zero for almost all for at least one regime, or mixing weight parameter
estimate close to zero (or one). It also filters out estimates with any modulus "bold A" eigenvalues larger than 0.9985,
as the solution is near the boundary of the stationarity region and likely not a local maximum. You can also set
filter_estimates=FALSE and find the solutions of interest yourself by using the
function alt_gsmvar.
Returns an object of class 'gsmvar' defining the estimated (reduced form or structural) GMVAR, StMVAR, or G-StMVAR model.
Multivariate quantile residuals (Kalliovirta and Saikkonen 2010) are also computed and included in the returned object.
In addition, the returned object contains the estimates and log-likelihood values from all the estimation rounds performed.
The estimated parameter vector can be obtained at gsmvar$params (and corresponding approximate standard errors
at gsmvar$std_errors). See ?GSMVAR for the form of the parameter vector, if needed.
Remark that the first autocovariance/correlation matrix in $uncond_moments is for the lag zero,
the second one for the lag one, etc.
The following S3 methods are supported for class 'gsmvar': logLik, residuals, print, summary,
predict, simulate, and plot.
Dorsey R. E. and Mayer W. J. 1995. Genetic algorithms for estimation problems with multiple optima, nondifferentiability, and other irregular features. Journal of Business & Economic Statistics, 13, 53-66.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Patnaik L.M. and Srinivas M. 1994. Adaptive Probabilities of Crossover and Mutation in Genetic Algorithms. Transactions on Systems, Man and Cybernetics 24, 656-667.
Smith R.E., Dike B.A., Stegmann S.A. 1995. Fitness inheritance in genetic algorithms. Proceedings of the 1995 ACM Symposium on Applied Computing, 345-350.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
GSMVAR, iterate_more, stmvar_to_gstmvar, predict.gsmvar,
profile_logliks, simulate.gsmvar, quantile_residual_tests, print_std_errors,
swap_parametrization, get_gradient, GIRF, GFEVD, LR_test,
Wald_test, gsmvar_to_sgsmvar, stmvar_to_gstmvar, reorder_W_columns,
swap_W_signs, cond_moment_plot, update_numtols
## These are long running examples that use parallel computing! # Running all the below examples will take approximately 3-4 minutes. # GMVAR(1,2) model: 10 estimation rounds with seeds set # for reproducibility fit12 <- fitGSMVAR(gdpdef, p=1, M=2, ncalls=10, seeds=1:10) fit12 plot(fit12) summary(fit12) print_std_errors(fit12) profile_logliks(fit12) # The rest of the examples only use a single estimation round with a given # seed that produces the MLE to reduce running time of the examples. When # estimating models for empirical applications, a large number of estimation # rounds (ncalls = a large number) should be performed to ensure reliability # of the estimates (see the section details). # StMVAR(2, 2) model fit22t <- fitGSMVAR(gdpdef, p=2, M=2, model="StMVAR", ncalls=1, seeds=1) fit22t # Overly large degrees of freedom estimate in the 2nd regime! fit22gs <- stmvar_to_gstmvar(fit22t) # So switch it to GMVAR type! fit22gs # This is the appropriate G-StMVAR model based on the above StMVAR model. fit22gss <- gsmvar_to_sgsmvar(fit22gs) # Switch to structural model fit22gss # This is the implied statistically identified structural model. # Structural GMVAR(1,2) model identified with sign # constraints. W_122 <- matrix(c(1, 1, -1, 1), nrow=2) fit12s <- fitGSMVAR(gdpdef, p=1, M=2, structural_pars=list(W=W_122), ncalls=1, seeds=1) fit12s # A statistically identified structural model can also be obtained with # gsmvar_to_sgsmvar(fit12) # GMVAR(2,2) model with autoregressive parameters restricted # to be the same for both regimes C_mat <- rbind(diag(2*2^2), diag(2*2^2)) fit22c <- fitGSMVAR(gdpdef, p=2, M=2, constraints=C_mat, ncalls=1, seeds=1) fit22c # G-StMVAR(2, 1, 1) model with autoregressive parameters and unconditional means restricted # to be the same for both regimes: fit22gscm <- fitGSMVAR(gdpdef, p=2, M=c(1, 1), model="G-StMVAR", constraints=C_mat, parametrization="mean", same_means=list(1:2), ncalls=1, seeds=1) # GMVAR(2,2) model with autoregressive parameters restricted # to be the same for both regimes and non-diagonal elements # the coefficient matrices constrained to zero. tmp <- matrix(c(1, rep(0, 10), 1, rep(0, 8), 1, rep(0, 10), 1), nrow=2*2^2, byrow=FALSE) C_mat2 <- rbind(tmp, tmp) fit22c2 <- fitGSMVAR(gdpdef, p=2, M=2, constraints=C_mat2, ncalls=1, seeds=1) fit22c2## These are long running examples that use parallel computing! # Running all the below examples will take approximately 3-4 minutes. # GMVAR(1,2) model: 10 estimation rounds with seeds set # for reproducibility fit12 <- fitGSMVAR(gdpdef, p=1, M=2, ncalls=10, seeds=1:10) fit12 plot(fit12) summary(fit12) print_std_errors(fit12) profile_logliks(fit12) # The rest of the examples only use a single estimation round with a given # seed that produces the MLE to reduce running time of the examples. When # estimating models for empirical applications, a large number of estimation # rounds (ncalls = a large number) should be performed to ensure reliability # of the estimates (see the section details). # StMVAR(2, 2) model fit22t <- fitGSMVAR(gdpdef, p=2, M=2, model="StMVAR", ncalls=1, seeds=1) fit22t # Overly large degrees of freedom estimate in the 2nd regime! fit22gs <- stmvar_to_gstmvar(fit22t) # So switch it to GMVAR type! fit22gs # This is the appropriate G-StMVAR model based on the above StMVAR model. fit22gss <- gsmvar_to_sgsmvar(fit22gs) # Switch to structural model fit22gss # This is the implied statistically identified structural model. # Structural GMVAR(1,2) model identified with sign # constraints. W_122 <- matrix(c(1, 1, -1, 1), nrow=2) fit12s <- fitGSMVAR(gdpdef, p=1, M=2, structural_pars=list(W=W_122), ncalls=1, seeds=1) fit12s # A statistically identified structural model can also be obtained with # gsmvar_to_sgsmvar(fit12) # GMVAR(2,2) model with autoregressive parameters restricted # to be the same for both regimes C_mat <- rbind(diag(2*2^2), diag(2*2^2)) fit22c <- fitGSMVAR(gdpdef, p=2, M=2, constraints=C_mat, ncalls=1, seeds=1) fit22c # G-StMVAR(2, 1, 1) model with autoregressive parameters and unconditional means restricted # to be the same for both regimes: fit22gscm <- fitGSMVAR(gdpdef, p=2, M=c(1, 1), model="G-StMVAR", constraints=C_mat, parametrization="mean", same_means=list(1:2), ncalls=1, seeds=1) # GMVAR(2,2) model with autoregressive parameters restricted # to be the same for both regimes and non-diagonal elements # the coefficient matrices constrained to zero. tmp <- matrix(c(1, rep(0, 10), 1, rep(0, 8), 1, rep(0, 10), 1), nrow=2*2^2, byrow=FALSE) C_mat2 <- rbind(tmp, tmp) fit22c2 <- fitGSMVAR(gdpdef, p=2, M=2, constraints=C_mat2, ncalls=1, seeds=1) fit22c2
GAfit estimates the specified GMVAR, StMVAR, or G-StMVAR model using a genetic algorithm.
It's designed to find starting values for gradient based methods.
GAfit( data, p, M, model = c("GMVAR", "StMVAR", "G-StMVAR"), conditional = TRUE, parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, weight_constraints = NULL, structural_pars = NULL, ngen = 200, popsize, smart_mu = min(100, ceiling(0.5 * ngen)), initpop = NULL, mu_scale, mu_scale2, omega_scale, W_scale, lambda_scale, ar_scale = 0.2, upper_ar_scale = 1, ar_scale2 = 1, regime_force_scale = 1, red_criteria = c(0.05, 0.01), pre_smart_mu_prob = 0, to_return = c("alt_ind", "best_ind"), minval, seed = NULL )GAfit( data, p, M, model = c("GMVAR", "StMVAR", "G-StMVAR"), conditional = TRUE, parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, weight_constraints = NULL, structural_pars = NULL, ngen = 200, popsize, smart_mu = min(100, ceiling(0.5 * ngen)), initpop = NULL, mu_scale, mu_scale2, omega_scale, W_scale, lambda_scale, ar_scale = 0.2, upper_ar_scale = 1, ar_scale2 = 1, regime_force_scale = 1, red_criteria = c(0.05, 0.01), pre_smart_mu_prob = 0, to_return = c("alt_ind", "best_ind"), minval, seed = NULL )
data |
a matrix or class |
p |
a positive integer specifying the autoregressive order of the model. |
M |
|
model |
is "GMVAR", "StMVAR", or "G-StMVAR" model considered? In the G-StMVAR model, the first |
conditional |
a logical argument specifying whether the conditional or exact log-likelihood function |
parametrization |
|
constraints |
a size |
same_means |
Restrict the mean parameters of some regimes to be the same? Provide a list of numeric vectors
such that each numeric vector contains the regimes that should share the common mean parameters. For instance, if
|
weight_constraints |
a numeric vector of length |
structural_pars |
If
See Virolainen (forthcoming) for the conditions required to identify the shocks and for the B-matrix as well (it is |
ngen |
a positive integer specifying the number of generations to be ran through in the genetic algorithm. |
popsize |
a positive even integer specifying the population size in the genetic algorithm.
Default is |
smart_mu |
a positive integer specifying the generation after which the random mutations in the genetic algorithm are "smart". This means that mutating individuals will mostly mutate fairly close (or partially close) to the best fitting individual (which has the least regimes with time varying mixing weights practically at zero) so far. |
initpop |
a list of parameter vectors from which the initial population of the genetic algorithm will be generated from. The parameter vectors should be...
Above, In the GMVAR model, The notation is similar to the cited literature. |
mu_scale |
a size |
mu_scale2 |
a size |
omega_scale |
a size |
W_scale |
a size |
lambda_scale |
a length If the lambda parameters are constrained with the This argument is ignored if As with omega_scale and W_scale, this argument should be adjusted carefully if specified by hand. NOTE that if lambdas are constrained in some other way than restricting some of them to be identical, this parameter should be adjusted accordingly in order to the estimation succeed! |
ar_scale |
a positive real number between zero and one, adjusting how large AR parameter values are typically
proposed in construction of the initial population: larger value implies larger coefficients (in absolute value).
After construction of the initial population, a new scale is drawn from |
upper_ar_scale |
the upper bound for |
ar_scale2 |
a positive real number adjusting how large AR parameter values are typically proposed in some random mutations (if AR constraints are employed, in all random mutations): larger value implies smaller coefficients (in absolute value). Values larger than 1 can be used if the AR coefficients are expected to be very small. If set smaller than 1, be careful as it might lead to failure in the creation of stationary parameter candidates |
regime_force_scale |
a non-negative real number specifying how much should natural selection favor individuals
with less regimes that have almost all mixing weights (practically) at zero. Set to zero for no favoring or large
number for heavy favoring. Without any favoring the genetic algorithm gets more often stuck in an area of the
parameter space where some regimes are wasted, but with too much favouring the best genes might never mix into
the population and the algorithm might converge poorly. Default is |
red_criteria |
a length 2 numeric vector specifying the criteria that is used to determine whether a regime is
redundant (or "wasted") or not.
Any regime |
pre_smart_mu_prob |
A number in |
to_return |
should the genetic algorithm return the best fitting individual which has "positive enough" mixing
weights for as many regimes as possible ( |
minval |
a real number defining the minimum value of the log-likelihood function that will be considered.
Values smaller than this will be treated as they were |
seed |
a single value, interpreted as an integer, or NULL, that sets seed for the random number generator in the beginning of
the function call. If calling |
The core of the genetic algorithm is mostly based on the description by Dorsey and Mayer (1995). It utilizes a slightly modified version of the individually adaptive crossover and mutation rates described by Patnaik and Srinivas (1994) and employs (50%) fitness inheritance discussed by Smith, Dike and Stegmann (1995).
By "redundant" or "wasted" regimes we mean regimes that have the time varying mixing weights practically at zero for almost all t. A model including redundant regimes would have about the same log-likelihood value without the redundant regimes and there is no purpose to have redundant regimes in a model.
Some of the AR coefficients are drawn with the algorithm by Ansley and Kohn (1986). However,
when using large ar_scale with large p or d, numerical inaccuracies caused
by the imprecision of the float-point presentation may result in errors or nonstationary AR-matrices.
Using smaller ar_scale facilitates the usage of larger p or d. Therefore, we bound
upper_ar_scale from above by when p*d>40 and by otherwise.
Returns the estimated parameter vector which has the form described in initpop.
Ansley C.F., Kohn R. 1986. A note on reparameterizing a vector autoregressive moving average model to enforce stationarity. Journal of statistical computation and simulation, 24:2, 99-106.
Dorsey R. E. and Mayer W. J. 1995. Genetic algorithms for estimation problems with multiple optima, nondifferentiability, and other irregular features. Journal of Business & Economic Statistics, 13, 53-66.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Patnaik L.M. and Srinivas M. 1994. Adaptive Probabilities of Crossover and Mutation in Genetic Algorithms. Transactions on Systems, Man and Cybernetics 24, 656-667.
Smith R.E., Dike B.A., Stegmann S.A. 1995. Fitness inheritance in genetic algorithms. Proceedings of the 1995 ACM Symposium on Applied Computing, 345-350.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
# Preliminary estimation of a G-StMVAR(1, 1, 1) model with 50 generations. GA_estimates <- GAfit(gdpdef, p=1, M=c(1, 1), model="G-StMVAR", ngen=50, seed=1) GA_estimates# Preliminary estimation of a G-StMVAR(1, 1, 1) model with 50 generations. GA_estimates <- GAfit(gdpdef, p=1, M=c(1, 1), model="G-StMVAR", ngen=50, seed=1) GA_estimates
A dataset containing a quarterly U.S. time series with two components: the percentage change of real GDP and the percentage change of GDP implicit price deflator, covering the period from 1959Q1 - 2019Q4.
gdpdefgdpdef
A numeric matrix of class 'ts' with 244 rows and 2 columns with one time series in each column:
The quarterly percent change of real U.S. GDP, from 1959Q1 to 2019Q4, https://fred.stlouisfed.org/series/GDPC1.
The quarterly percent change of U.S. GDP implicit price deflator, from 1959Q1 to 2019Q4, https://fred.stlouisfed.org/series/GDPDEF.
The Federal Reserve Bank of St. Louis database
get_boldA_eigens calculates absolute values of the eigenvalues of
the "bold A" matrices containing the AR coefficients for each mixture component.
get_boldA_eigens(gsmvar)get_boldA_eigens(gsmvar)
gsmvar |
an object of class |
Returns a matrix with rows and columns - one column for each regime.
The th column contains the absolute values (or modulus) of the eigenvalues of the "bold A" matrix containing
the AR coefficients correspinding to regime .
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
@keywords internal
# GMVAR(2, 2), d=2 model params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(p=2, M=2, d=2, params=params22) get_boldA_eigens(mod22)# GMVAR(2, 2), d=2 model params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(p=2, M=2, d=2, params=params22) get_boldA_eigens(mod22)
get_omega_eigens calculates the eigenvalues of the "Omega" error
term covariance matrices for each mixture component.
get_omega_eigens(gsmvar)get_omega_eigens(gsmvar)
gsmvar |
an object of class |
Returns a matrix with rows and columns - one column for each regime.
The th column contains the eigenvalues of the "Omega" error term covariance matrix
of the th regime.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
@keywords internal
# GMVAR(2, 2), d=2 model params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(p=2, M=2, d=2, params=params22) get_omega_eigens(mod22)# GMVAR(2, 2), d=2 model params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(p=2, M=2, d=2, params=params22) get_omega_eigens(mod22)
get_regime_autocovs calculates the first p regimewise autocovariance
matrices for the given GMVAR, StMVAR, or G-StMVAR model.
get_regime_autocovs(gsmvar)get_regime_autocovs(gsmvar)
gsmvar |
an object of class |
Returns an array containing the first p regimewise autocovariance matrices.
The subset [, , j, m] contains the j-1:th lag autocovariance matrix of the m:th regime.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Lütkepohl H. 2005. New Introduction to Multiple Time Series Analysis, Springer.
McElroy T. 2017. Computation of vector ARMA autocovariances. Statistics and Probability Letters, 124, 92-96.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
Other moment functions:
cond_moments(),
get_regime_means(),
uncond_moments()
# GMVAR(1,2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) get_regime_autocovs(mod12) # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) mod22s get_regime_autocovs(mod22s)# GMVAR(1,2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) get_regime_autocovs(mod12) # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) mod22s get_regime_autocovs(mod22s)
get_regime_means calculates regime means
for the given GMVAR, StMVAR, or G-StMVAR model.
get_regime_means(gsmvar)get_regime_means(gsmvar)
gsmvar |
an object of class |
Returns a matrix containing regime mean in the m:th column, .
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
@keywords internal
uncond_moments, get_regime_autocovs, cond_moments
Other moment functions:
cond_moments(),
get_regime_autocovs(),
uncond_moments()
# GMVAR(1,2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) mod12 get_regime_means(mod12) # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) mod22s get_regime_means(mod22s)# GMVAR(1,2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) mod12 get_regime_means(mod12) # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) mod22s get_regime_means(mod22s)
GFEVD estimates generalized forecast error variance decomposition for structural
(and reduced form) GMVAR, StMVAR, and G-StMVAR models.
GFEVD( gsmvar, shock_size = 1, N = 30, initval_type = c("data", "random", "fixed"), R1 = 250, R2 = 250, init_regimes = NULL, init_values = NULL, which_cumulative = numeric(0), include_mixweights = FALSE, ncores = 2, seeds = NULL ) ## S3 method for class 'gfevd' plot(x, ...) ## S3 method for class 'gfevd' print(x, ..., digits = 2, N_to_print)GFEVD( gsmvar, shock_size = 1, N = 30, initval_type = c("data", "random", "fixed"), R1 = 250, R2 = 250, init_regimes = NULL, init_values = NULL, which_cumulative = numeric(0), include_mixweights = FALSE, ncores = 2, seeds = NULL ) ## S3 method for class 'gfevd' plot(x, ...) ## S3 method for class 'gfevd' print(x, ..., digits = 2, N_to_print)
gsmvar |
an object of class |
shock_size |
What shocks size should be used for all shocks? By the definition of the SGMVAR, SStMVAR, and SG-StMVAR model, the conditional covariance matrix of the structural shock is identity matrix. |
N |
a positive integer specifying the horizon how far ahead should the GFEVD be calculated. |
initval_type |
What type initial values are used for estimating the GIRFs that the GFEVD is based on?
|
R1 |
the number of repetitions used to estimate GIRF for each initial value. |
R2 |
the number of initial values to be drawn if |
init_regimes |
a numeric vector of length at most |
init_values |
a size |
which_cumulative |
a numeric vector with values in |
include_mixweights |
should the GFEVD be estimated for the mixing weights as well? Note that this is
ignored if |
ncores |
the number CPU cores to be used in parallel computing. Only single core computing is supported if an initial value is specified (and the GIRF won't thus be estimated multiple times). |
seeds |
a numeric vector containing the random number generator seed for estimation of each GIRF. Should have the length...
Set to |
x |
object of class |
... |
currently not used. |
digits |
the number of decimals to print |
N_to_print |
an integer specifying the horizon how far to print the estimates. The default is that all the values are printed. |
The model DOES NOT need to be structural in order for this function to be
applicable. When an identified structural GMVAR, StMVAR, or G-StMVAR model is
provided in the argument gsmvar, the identification imposed by the model
is used. When a reduced form model is provided in the argument gsmvar,
lower triangular Cholesky identification is used to identify the shocks.
The GFEVD is a forecast error variance decomposition calculated with the generalized impulse response function (GIRF). Lanne and Nyberg (2016) for details. Note, however, that the related GIRFs are calculated using the algorithm given in Virolainen (2022).
Returns and object of class 'gfevd' containing the GFEVD for all the variables and if
include_mixweights=TRUE also to the mixing weights. Note that the decomposition does not
exist at horizon zero for mixing weights because the related GIRFs are always zero at impact.
plot(gfevd): plot method
print(gfevd): print method
Lanne M. and Nyberg H. 2016. Generalized Forecast Error Variance Decomposition for Linear and Nonlineae Multivariate Models. Oxford Bulletin of Economics and Statistics, 78, 4, 595-603.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
GIRF, linear_IRF, fitGSMVAR, GSMVAR,
gsmvar_to_sgsmvar, reorder_W_columns, swap_W_signs, simulate.gsmvar
# These are long-running examples that use parallel computing. # It takes approximately 30 seconds to run all the below examples. ## StMVAR(1, 2), d=2 model identified recursively by lower-triangular ## Cholesky decomposition (i.e., reduced form model is specified): params12t <- c(0.55, 0.11, 0.34, 0.05, -0.01, 0.72, 0.58, 0.01, 0.06, 0.17, 0.25, 0.34, 0.05, -0.01, 0.72, 0.50, -0.01, 0.20, 0.60, 3.00, 12.00) mod12t <- GSMVAR(gdpdef, p=1, M=2, params=params12t, model="StMVAR") # Estimating the GFEVD using all possible histories in the data as the # initial values: gfevd0 <- GFEVD(mod12t, N=24, R1=10, initval_type="data") gfevd0 plot(gfevd0) ## NOTE: Use larger R1 is empirical applications! Small R1 is used ## here only to fasten the execution time of the examples. ## Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) mod22s # Alternatively, use: #fit22s <- fitGSMVAR(gdpdef, p=2, M=2, structural_pars=list(W=W_22), # ncalls=20, seeds=1:20) # To obtain an estimated version of the same model. # Estimating the GFEVD using all possible histories in the data as the # initial values: gfevd1 <- GFEVD(mod22s, N=24, R1=10, initval_type="data") gfevd1 plot(gfevd1) # Estimate GFEVD with the initial values generated from the stationary # distribution of the process: gfevd2 <- GFEVD(mod22s, N=24, R1=10, R2=100, initval_type="random") gfevd2 plot(gfevd2) # Estimate GFEVD with fixed hand specified initial values. We use the # unconditional mean of the process: myvals <- rbind(mod22s$uncond_moments$uncond_mean, mod22s$uncond_moments$uncond_mean) gfevd3 <- GFEVD(mod22s, N=36, R1=50, initval_type="fixed", init_values=myvals, include_mixweights=TRUE) gfevd3 plot(gfevd3)# These are long-running examples that use parallel computing. # It takes approximately 30 seconds to run all the below examples. ## StMVAR(1, 2), d=2 model identified recursively by lower-triangular ## Cholesky decomposition (i.e., reduced form model is specified): params12t <- c(0.55, 0.11, 0.34, 0.05, -0.01, 0.72, 0.58, 0.01, 0.06, 0.17, 0.25, 0.34, 0.05, -0.01, 0.72, 0.50, -0.01, 0.20, 0.60, 3.00, 12.00) mod12t <- GSMVAR(gdpdef, p=1, M=2, params=params12t, model="StMVAR") # Estimating the GFEVD using all possible histories in the data as the # initial values: gfevd0 <- GFEVD(mod12t, N=24, R1=10, initval_type="data") gfevd0 plot(gfevd0) ## NOTE: Use larger R1 is empirical applications! Small R1 is used ## here only to fasten the execution time of the examples. ## Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) mod22s # Alternatively, use: #fit22s <- fitGSMVAR(gdpdef, p=2, M=2, structural_pars=list(W=W_22), # ncalls=20, seeds=1:20) # To obtain an estimated version of the same model. # Estimating the GFEVD using all possible histories in the data as the # initial values: gfevd1 <- GFEVD(mod22s, N=24, R1=10, initval_type="data") gfevd1 plot(gfevd1) # Estimate GFEVD with the initial values generated from the stationary # distribution of the process: gfevd2 <- GFEVD(mod22s, N=24, R1=10, R2=100, initval_type="random") gfevd2 plot(gfevd2) # Estimate GFEVD with fixed hand specified initial values. We use the # unconditional mean of the process: myvals <- rbind(mod22s$uncond_moments$uncond_mean, mod22s$uncond_moments$uncond_mean) gfevd3 <- GFEVD(mod22s, N=36, R1=50, initval_type="fixed", init_values=myvals, include_mixweights=TRUE) gfevd3 plot(gfevd3)
GIRF estimates generalized impulse response function for
structural (and reduced form) GMVAR, StMVAR, and G-StMVAR models.
GIRF( gsmvar, which_shocks, shock_size = 1, N = 30, R1 = 250, R2 = 250, init_regimes = 1:sum(gsmvar$model$M), init_values = NULL, which_cumulative = numeric(0), scale = NULL, scale_type = c("instant", "peak"), scale_horizon = N, ci = c(0.95, 0.8), include_mixweights = TRUE, ncores = 2, plot_res = TRUE, seeds = NULL, ... ) ## S3 method for class 'girf' plot(x, add_grid = FALSE, margs, ...) ## S3 method for class 'girf' print(x, ..., digits = 2, N_to_print)GIRF( gsmvar, which_shocks, shock_size = 1, N = 30, R1 = 250, R2 = 250, init_regimes = 1:sum(gsmvar$model$M), init_values = NULL, which_cumulative = numeric(0), scale = NULL, scale_type = c("instant", "peak"), scale_horizon = N, ci = c(0.95, 0.8), include_mixweights = TRUE, ncores = 2, plot_res = TRUE, seeds = NULL, ... ) ## S3 method for class 'girf' plot(x, add_grid = FALSE, margs, ...) ## S3 method for class 'girf' print(x, ..., digits = 2, N_to_print)
gsmvar |
an object of class |
which_shocks |
a numeric vector of length at most |
shock_size |
a non-zero scalar value specifying the common size for all scalar components of the structural shock. Note that the conditional covariance matrix of the structural shock is an identity matrix and that the (generalized) impulse responses may not be symmetric to the sign and size of the shock. |
N |
a positive integer specifying the horizon how far ahead should the generalized impulse responses be calculated. |
R1 |
the number of repetitions used to estimate GIRF for each initial value. |
R2 |
the number of initial values to be drawn from a stationary
distribution of the process or of a specific regime? The confidence bounds
will be sample quantiles of the GIRFs based on different initial values.
Ignored if the argument |
init_regimes |
a numeric vector of length at most |
init_values |
a size |
which_cumulative |
a numeric vector with values in |
scale |
should the GIRFs to some of the shocks be scaled so that they
correspond to a specific magnitude of instantaneous or peak response
of some specific variable (see the argument |
scale_type |
If argument |
scale_horizon |
If |
ci |
a numeric vector with elements in |
include_mixweights |
should the generalized impulse response be
calculated for the mixing weights as well? |
ncores |
the number CPU cores to be used in parallel computing. Only single core computing is supported if an initial value is specified (and the GIRF won't thus be estimated multiple times). |
plot_res |
|
seeds |
a length |
... |
arguments passed to |
x |
object of class |
add_grid |
should grid be added to the plots? |
margs |
numeric vector of length four that adjusts the
|
digits |
the number of decimals to print |
N_to_print |
an integer specifying the horizon how far to print the estimates and confidence intervals. The default is that all the values are printed. |
The model DOES NOT need to be structural in order for this function to be
applicable. When an identified structural GMVAR, StMVAR, or G-StMVAR model is
provided in the argument gsmvar, the identification imposed by the model
is used. When a reduced form model is provided in the argument gsmvar,
lower triangular Cholesky identification is used to identify the shocks.
The confidence bounds reflect uncertainty about the initial state (but currently not about the parameter estimates) if initial values are not specified. If initial values are specified, there won't currently be confidence intervals. See the cited paper by Virolainen (2022) for details about the algorithm.
Note that if the argument scale is used, the scaled responses of
the mixing weights might be more than one in absolute value.
Returns a class 'girf' list with the GIRFs in the first
element ($girf_res) and the used arguments the rest. The first
element containing the GIRFs is a list with the th element
containing the point estimates for the GIRF in $point_est (the first
element) and confidence intervals in $conf_ints (the second
element). The first row is for the GIRF at impact , the second
for , the third for , and so on.
The element $all_girfs is a list containing results from all the individual GIRFs
obtained from the MC repetitions. Each element is for one shock and results are in
array of the form [horizon, variables, MC-repetitions].
plot(girf): plot method
print(girf): print method
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
@keywords internal
GFEVD, linear_IRF, fitGSMVAR, GSMVAR,
gsmvar_to_sgsmvar, reorder_W_columns,
swap_W_signs, simulate.gsmvar,
predict.gsmvar, profile_logliks,
quantile_residual_tests, LR_test,
Wald_test
# These are long-running examples that use parallel computing. # It takes approximately 30 seconds to run all the below examples. ## StMVAR(1, 2), d=2 model identified recursively by lower-triangular ## Cholesky decomposition (i.e., reduced form model is specified): params12t <- c(0.55, 0.11, 0.34, 0.05, -0.01, 0.72, 0.58, 0.01, 0.06, 0.17, 0.25, 0.34, 0.05, -0.01, 0.72, 0.50, -0.01, 0.20, 0.60, 3.00, 12.00) mod12t <- GSMVAR(gdpdef, p=1, M=2, params=params12t, model="StMVAR") # Estimating the GIRFs of both structural shocks with initial values # drawn from the stationary distribution of the process, # 12 periods ahead, confidence levels 0.95 and 0.8: girf0 <- GIRF(mod12t, N=12, R1=100, R2=100) girf0 plot(girf0) ## NOTE: Small R1 and R2 is used here to shorten the estimation time. ## Larger R1 and R2 should be considered in empirical applications! ## Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) mod22s # Alternatively, use: #fit22s <- fitGSMVAR(gdpdef, p=2, M=2, structural_pars=list(W=W_22), # ncalls=20, seeds=1:20) # To obtain an estimated version of the same model. # Estimating the GIRFs of both structural shocks with initial values # drawn from the stationary distribution of the process, # 12 periods ahead, confidence levels 0.95 and 0.8: girf1 <- GIRF(mod22s, N=12, R1=100, R2=100) girf1 plot(girf1) # Estimating the GIRF of the second shock only, 12 periods ahead # and shock size 1, initial values drawn from the stationary distribution # of the first regime, confidence level 0.9: girf2 <- GIRF(mod22s, which_shocks=2, shock_size=1, N=12, init_regimes=1, ci=0.9, R1=100, R2=100) # Estimating the GIRFs of both structural shocks, negative one standard # error shock, N=20 periods ahead, estimation based on 200 Monte Carlo # simulations, and fixed initial values given by the last p observations # of the data: girf3 <- GIRF(mod22s, shock_size=-1, N=20, R1=200, init_values=mod22s$data)# These are long-running examples that use parallel computing. # It takes approximately 30 seconds to run all the below examples. ## StMVAR(1, 2), d=2 model identified recursively by lower-triangular ## Cholesky decomposition (i.e., reduced form model is specified): params12t <- c(0.55, 0.11, 0.34, 0.05, -0.01, 0.72, 0.58, 0.01, 0.06, 0.17, 0.25, 0.34, 0.05, -0.01, 0.72, 0.50, -0.01, 0.20, 0.60, 3.00, 12.00) mod12t <- GSMVAR(gdpdef, p=1, M=2, params=params12t, model="StMVAR") # Estimating the GIRFs of both structural shocks with initial values # drawn from the stationary distribution of the process, # 12 periods ahead, confidence levels 0.95 and 0.8: girf0 <- GIRF(mod12t, N=12, R1=100, R2=100) girf0 plot(girf0) ## NOTE: Small R1 and R2 is used here to shorten the estimation time. ## Larger R1 and R2 should be considered in empirical applications! ## Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) mod22s # Alternatively, use: #fit22s <- fitGSMVAR(gdpdef, p=2, M=2, structural_pars=list(W=W_22), # ncalls=20, seeds=1:20) # To obtain an estimated version of the same model. # Estimating the GIRFs of both structural shocks with initial values # drawn from the stationary distribution of the process, # 12 periods ahead, confidence levels 0.95 and 0.8: girf1 <- GIRF(mod22s, N=12, R1=100, R2=100) girf1 plot(girf1) # Estimating the GIRF of the second shock only, 12 periods ahead # and shock size 1, initial values drawn from the stationary distribution # of the first regime, confidence level 0.9: girf2 <- GIRF(mod22s, which_shocks=2, shock_size=1, N=12, init_regimes=1, ci=0.9, R1=100, R2=100) # Estimating the GIRFs of both structural shocks, negative one standard # error shock, N=20 periods ahead, estimation based on 200 Monte Carlo # simulations, and fixed initial values given by the last p observations # of the data: girf3 <- GIRF(mod22s, shock_size=-1, N=20, R1=200, init_values=mod22s$data)
GSMVAR creates a class 'gsmvar' object that defines
a reduced form or structural GMVAR model DEPRECATED! USE THE FUNCTION GSMVAR INSTEAD!
GMVAR( data, p, M, d, params, conditional = TRUE, parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, structural_pars = NULL, calc_cond_moments, calc_std_errors = FALSE, stat_tol = 0.001, posdef_tol = 1e-08 )GMVAR( data, p, M, d, params, conditional = TRUE, parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, structural_pars = NULL, calc_cond_moments, calc_std_errors = FALSE, stat_tol = 0.001, posdef_tol = 1e-08 )
data |
a matrix or class |
p |
a positive integer specifying the autoregressive order of the model. |
M |
|
d |
number of times series in the system, i.e. |
params |
a real valued vector specifying the parameter values.
Above, In the GMVAR model, The notation is similar to the cited literature. |
conditional |
a logical argument specifying whether the conditional or exact log-likelihood function should be used. |
parametrization |
|
constraints |
a size |
same_means |
Restrict the mean parameters of some regimes to be the same? Provide a list of numeric vectors
such that each numeric vector contains the regimes that should share the common mean parameters. For instance, if
|
structural_pars |
If
See Virolainen (forthcoming) for the conditions required to identify the shocks and for the B-matrix as well (it is |
calc_cond_moments |
should conditional means and covariance matrices should be calculated?
Default is |
calc_std_errors |
should approximate standard errors be calculated? |
stat_tol |
numerical tolerance for stationarity of the AR parameters: if the "bold A" matrix of any regime
has eigenvalues larger that |
posdef_tol |
numerical tolerance for positive definiteness of the error term covariance matrices: if the error term covariance matrix of any regime has eigenvalues smaller than this, the model is classified as not satisfying positive definiteness assumption. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error. |
If data is provided, then also multivariate quantile residuals (Kalliovirta and Saikkonen 2010) are computed and included in the returned object.
If the function fails to calculate approximative standard errors and the parameter values are near the border of the parameter space, it might help to use smaller numerical tolerance for the stationarity and positive definiteness conditions.
The first plot displays the time series together with estimated mixing weights. The second plot displays a (Gaussian) kernel density estimates of the individual series together with the marginal stationary density implied by the model. The colored regimewise stationary densities are multiplied with the mixing weight parameter estimates.
Returns an object of class 'gsmvar' defining the specified reduced form or structural GMVAR,
StMVAR, or G-StMVAR model. Can be used to work with other functions provided in gmvarkit.
Note that the first autocovariance/correlation matrix in $uncond_moments is for the lag zero,
the second one for the lag one, etc.
Only the print method is available if data is not provided.
If data is provided, then in addition to the ones listed above, the predict method is also available.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Kalliovirta L. and Saikkonen P. 2010. Reliable Residuals for Multivariate Nonlinear Time Series Models. Unpublished Revision of HECER Discussion Paper No. 247.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
gmvar_to_gsmvar class 'gmvar' objects compatible with the functions using
s class 'gsmvar' objects
gmvar_to_gsmvar(gsmvar)gmvar_to_gsmvar(gsmvar)
gsmvar |
a class 'gmvar' or 'gsmvar' object. |
This exists so that models estimated with earlier versions of the package can be used conveniently.
If the provided object has the class 'gsmvar', the provided object
is returned without modifications. If the provided object has the class 'gmvar',
its element $model is given a new subelement called also model and this is
set to be "GMVAR". Also, the class of this object is changes to 'gsmvar' and then
it is returned.
DEPRECATED! USE THE FUNCTION fitGSMVAR INSTEAD!
gsmvar_to_sgsmvar constructs SGMVAR model based on a reduced
form GMVAR, StMVAR, or G-StMVAR model.
gmvar_to_sgmvar(gmvar, calc_std_errors = TRUE)gmvar_to_sgmvar(gmvar, calc_std_errors = TRUE)
gmvar |
object of class 'gmvar' |
calc_std_errors |
should approximate standard errors be calculated? |
The switch is made by simultaneously diagonalizing the two error term covariance matrices
with a well known matrix decomposition (Muirhead, 1982, Theorem A9.9) and then normalizing the
diagonal of the matrix W positive (which implies positive diagonal of the B-matrix). Models with
more that two regimes are not supported because the matrix decomposition does not generally
exists for more than two covariance matrices. If the model has only one regime (= regular SVAR model),
a symmetric and pos. def. square root matrix of the error term covariance matrix is used unless
cholesky = TRUE is set in the arguments, in which case Cholesky identification is employed.
In order to employ a structural model with Cholesky identification and multiple regimes (M > 1),
use the function GIRF directly with a reduced form model (see ?GIRF).
The columns of as well as the lambda parameters can be re-ordered (without changing the implied
reduced form model) afterwards with the function reorder_W_columns. Also all signs in any column
of can be swapped (without changing the implied reduced form model) afterwards with the function
swap_W_signs. These two functions work with models containing any number of regimes.
Returns an object of class 'gsmvar' defining a structural GMVAR, StMVAR, or G-StMVAR model based on a
two-regime reduced form GMVAR, StMVAR, or G-StMVAR model, with the main diagonal of the B-matrix normalized to be
positive.
Muirhead R.J. 1982. Aspects of Multivariate Statistical Theory, Wiley.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
GSMVAR creates a class 'gsmvar' object that defines
a reduced form or structural GMVAR, StMVAR, or G-StMVAR model
GSMVAR( data, p, M, d, params, conditional = TRUE, model = c("GMVAR", "StMVAR", "G-StMVAR"), parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, weight_constraints = NULL, structural_pars = NULL, calc_cond_moments, calc_std_errors = FALSE, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 ) ## S3 method for class 'gsmvar' logLik(object, ...) ## S3 method for class 'gsmvar' residuals(object, ...) ## S3 method for class 'gsmvar' summary(object, ..., digits = 2) ## S3 method for class 'gsmvar' plot(x, ..., type = c("both", "series", "density")) ## S3 method for class 'gsmvar' print(x, ..., digits = 2, summary_print = FALSE)GSMVAR( data, p, M, d, params, conditional = TRUE, model = c("GMVAR", "StMVAR", "G-StMVAR"), parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, weight_constraints = NULL, structural_pars = NULL, calc_cond_moments, calc_std_errors = FALSE, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 ) ## S3 method for class 'gsmvar' logLik(object, ...) ## S3 method for class 'gsmvar' residuals(object, ...) ## S3 method for class 'gsmvar' summary(object, ..., digits = 2) ## S3 method for class 'gsmvar' plot(x, ..., type = c("both", "series", "density")) ## S3 method for class 'gsmvar' print(x, ..., digits = 2, summary_print = FALSE)
data |
a matrix or class |
p |
a positive integer specifying the autoregressive order of the model. |
M |
|
d |
number of times series in the system, i.e. |
params |
a real valued vector specifying the parameter values.
Above, In the GMVAR model, The notation is similar to the cited literature. |
conditional |
a logical argument specifying whether the conditional or exact log-likelihood function should be used. |
model |
is "GMVAR", "StMVAR", or "G-StMVAR" model considered? In the G-StMVAR model, the first |
parametrization |
|
constraints |
a size |
same_means |
Restrict the mean parameters of some regimes to be the same? Provide a list of numeric vectors
such that each numeric vector contains the regimes that should share the common mean parameters. For instance, if
|
weight_constraints |
a numeric vector of length |
structural_pars |
If
See Virolainen (forthcoming) for the conditions required to identify the shocks and for the B-matrix as well (it is |
calc_cond_moments |
should conditional means and covariance matrices should be calculated?
Default is |
calc_std_errors |
should approximate standard errors be calculated? |
stat_tol |
numerical tolerance for stationarity of the AR parameters: if the "bold A" matrix of any regime
has eigenvalues larger that |
posdef_tol |
numerical tolerance for positive definiteness of the error term covariance matrices: if the error term covariance matrix of any regime has eigenvalues smaller than this, the model is classified as not satisfying positive definiteness assumption. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error. |
df_tol |
the parameter vector is considered to be outside the parameter space if all degrees of
freedom parameters are not larger than |
object |
object of class |
... |
currently not used. |
digits |
number of digits to be printed. |
x |
object of class |
type |
which type figure should be produced? Or both? |
summary_print |
if set to |
If data is provided, then also multivariate quantile residuals (Kalliovirta and Saikkonen 2010) are computed and included in the returned object.
If the function fails to calculate approximative standard errors and the parameter values are near the border of the parameter space, it might help to use smaller numerical tolerance for the stationarity and positive definiteness conditions.
The first plot displays the time series together with estimated mixing weights. The second plot displays a (Gaussian) kernel density estimates of the individual series together with the marginal stationary density implied by the model. The colored regimewise stationary densities are multiplied with the mixing weight parameter estimates.
Returns an object of class 'gsmvar' defining the specified reduced form or structural GMVAR,
StMVAR, or G-StMVAR model. Can be used to work with other functions provided in gmvarkit.
Note that the first autocovariance/correlation matrix in $uncond_moments is for the lag zero,
the second one for the lag one, etc.
logLik(gsmvar): Log-likelihood method
residuals(gsmvar): residuals method to extract multivariate quantile residuals
summary(gsmvar): summary method
plot(gsmvar): plot method for class 'gsmvar'
print(gsmvar): print method
If data is not provided, only the print and simulate methods are available.
If data is provided, then in addition to the ones listed above, predict method is also available.
See ?simulate.gsmvar and ?predict.gsmvar for details about the usage.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Kalliovirta L. and Saikkonen P. 2010. Reliable Residuals for Multivariate Nonlinear Time Series Models. Unpublished Revision of HECER Discussion Paper No. 247.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
fitGSMVAR, add_data, swap_parametrization, GIRF,
gsmvar_to_sgsmvar, stmvar_to_gstmvar, reorder_W_columns,
swap_W_signs, update_numtols
# GMVAR(1, 2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) mod12 # GMVAR(1, 2), d=2 model without data mod12_2 <- GSMVAR(p=1, M=2, d=2, params=params12) mod12_2 # StMVAR(1, 2), d=2 model: mod12t <- GSMVAR(gdpdef, p=1, M=2, params=c(params12, 10, 20), model="StMVAR") mod12t # G-StMVAR(1, 1, 1), d=2 model: mod12gs <- GSMVAR(gdpdef, p=1, M=c(1, 1), params=c(params12, 20), model="G-StMVAR") mod12gs # GMVAR(2, 2), d=2 model with mean-parametrization: params22 <- c(0.869, 0.549, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.576, 1.168, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22, parametrization="mean") mod22 # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) mod22s# GMVAR(1, 2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) mod12 # GMVAR(1, 2), d=2 model without data mod12_2 <- GSMVAR(p=1, M=2, d=2, params=params12) mod12_2 # StMVAR(1, 2), d=2 model: mod12t <- GSMVAR(gdpdef, p=1, M=2, params=c(params12, 10, 20), model="StMVAR") mod12t # G-StMVAR(1, 1, 1), d=2 model: mod12gs <- GSMVAR(gdpdef, p=1, M=c(1, 1), params=c(params12, 20), model="G-StMVAR") mod12gs # GMVAR(2, 2), d=2 model with mean-parametrization: params22 <- c(0.869, 0.549, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.576, 1.168, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22, parametrization="mean") mod22 # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) mod22s
gsmvar_to_sgsmvar constructs SGMVAR, SStMVAR, or SG-StMVAR model based on a reduced
form GMVAR, StMVAR, or G-StMVAR model.
gsmvar_to_sgsmvar(gsmvar, calc_std_errors = TRUE, cholesky = FALSE)gsmvar_to_sgsmvar(gsmvar, calc_std_errors = TRUE, cholesky = FALSE)
gsmvar |
an object of class |
calc_std_errors |
should approximate standard errors be calculated? |
cholesky |
if |
The switch is made by simultaneously diagonalizing the two error term covariance matrices
with a well known matrix decomposition (Muirhead, 1982, Theorem A9.9) and then normalizing the
diagonal of the matrix W positive (which implies positive diagonal of the B-matrix). Models with
more that two regimes are not supported because the matrix decomposition does not generally
exists for more than two covariance matrices. If the model has only one regime (= regular SVAR model),
a symmetric and pos. def. square root matrix of the error term covariance matrix is used unless
cholesky = TRUE is set in the arguments, in which case Cholesky identification is employed.
In order to employ a structural model with Cholesky identification and multiple regimes (M > 1),
use the function GIRF directly with a reduced form model (see ?GIRF).
The columns of as well as the lambda parameters can be re-ordered (without changing the implied
reduced form model) afterwards with the function reorder_W_columns. Also all signs in any column
of can be swapped (without changing the implied reduced form model) afterwards with the function
swap_W_signs. These two functions work with models containing any number of regimes.
Returns an object of class 'gsmvar' defining a structural GMVAR, StMVAR, or G-StMVAR model based on a
two-regime reduced form GMVAR, StMVAR, or G-StMVAR model, with the main diagonal of the B-matrix normalized to be
positive.
Muirhead R.J. 1982. Aspects of Multivariate Statistical Theory, Wiley.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
fitGSMVAR, GSMVAR, GIRF, reorder_W_columns,
swap_W_signs, stmvar_to_gstmvar
# Reduced form GMVAR(1,2) model params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) # Form a structural model based on the reduced form model: mod12s <- gsmvar_to_sgsmvar(mod12) mod12s #' # Reduced form StMVAR(1,2) model mod12t <- GSMVAR(gdpdef, p=1, M=2, params=c(params12, 11, 12), model="StMVAR") # Form a structural model based on the reduced form model: mod12ts <- gsmvar_to_sgsmvar(mod12t) mod12ts# Reduced form GMVAR(1,2) model params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) # Form a structural model based on the reduced form model: mod12s <- gsmvar_to_sgsmvar(mod12) mod12s #' # Reduced form StMVAR(1,2) model mod12t <- GSMVAR(gdpdef, p=1, M=2, params=c(params12, 11, 12), model="StMVAR") # Form a structural model based on the reduced form model: mod12ts <- gsmvar_to_sgsmvar(mod12t) mod12ts
in_paramspace checks whether the given parameter vector lies in
the parameter space. Does NOT test the identification conditions!
in_paramspace( p, M, d, params, model = c("GMVAR", "StMVAR", "G-StMVAR"), constraints = NULL, same_means = NULL, weight_constraints = NULL, structural_pars = NULL, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 )in_paramspace( p, M, d, params, model = c("GMVAR", "StMVAR", "G-StMVAR"), constraints = NULL, same_means = NULL, weight_constraints = NULL, structural_pars = NULL, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 )
p |
a positive integer specifying the autoregressive order of the model. |
M |
|
d |
the number of time series in the system. |
params |
a real valued vector specifying the parameter values.
Above, In the GMVAR model, The notation is similar to the cited literature. |
model |
is "GMVAR", "StMVAR", or "G-StMVAR" model considered? In the G-StMVAR model, the first |
constraints |
a size |
same_means |
Restrict the mean parameters of some regimes to be the same? Provide a list of numeric vectors
such that each numeric vector contains the regimes that should share the common mean parameters. For instance, if
|
weight_constraints |
a numeric vector of length |
structural_pars |
If
See Virolainen (forthcoming) for the conditions required to identify the shocks and for the B-matrix as well (it is |
stat_tol |
numerical tolerance for stationarity of the AR parameters: if the "bold A" matrix of any regime
has eigenvalues larger that |
posdef_tol |
numerical tolerance for positive definiteness of the error term covariance matrices: if the error term covariance matrix of any regime has eigenvalues smaller than this, the model is classified as not satisfying positive definiteness assumption. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error. |
df_tol |
the parameter vector is considered to be outside the parameter space if all degrees of
freedom parameters are not larger than |
Returns TRUE if the given parameter vector lies in the parameter space
and FALSE otherwise.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
@keywords internal
# GMVAR(1,1), d=2 model: params11 <- c(1.07, 127.71, 0.99, 0.00, -0.01, 0.99, 4.05, 2.22, 8.87) in_paramspace(p=1, M=1, d=2, params=params11) # GMVAR(2,2), d=2 model: params22 <- c(1.39, -0.77, 1.31, 0.14, 0.09, 1.29, -0.39, -0.07, -0.11, -0.28, 0.92, -0.03, 4.84, 1.01, 5.93, 1.25, 0.08, -0.04, 1.27, -0.27, -0.07, 0.03, -0.31, 5.85, 3.57, 9.84, 0.74) in_paramspace(p=2, M=2, d=2, params=params22) # GMVAR(2,2), d=2 model with AR-parameters restricted to be # the same for both regimes: C_mat <- rbind(diag(2*2^2), diag(2*2^2)) params22c <- c(1.03, 2.36, 1.79, 3.00, 1.25, 0.06,0.04, 1.34, -0.29, -0.08, -0.05, -0.36, 0.93, -0.15, 5.20, 5.88, 3.56, 9.80, 0.37) in_paramspace(p=2, M=2, d=2, params=params22c, constraints=C_mat) # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(1.03, 2.36, 1.79, 3, 1.25, 0.06, 0.04, 1.34, -0.29, -0.08, -0.05, -0.36, 1.2, 0.05, 0.05, 1.3, -0.3, -0.1, -0.05, -0.4, 0.89, 0.72, -0.37, 2.16, 7.16, 1.3, 0.37) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) in_paramspace(p=2, M=2, d=2, params=params22s, structural_pars=list(W=W_22))# GMVAR(1,1), d=2 model: params11 <- c(1.07, 127.71, 0.99, 0.00, -0.01, 0.99, 4.05, 2.22, 8.87) in_paramspace(p=1, M=1, d=2, params=params11) # GMVAR(2,2), d=2 model: params22 <- c(1.39, -0.77, 1.31, 0.14, 0.09, 1.29, -0.39, -0.07, -0.11, -0.28, 0.92, -0.03, 4.84, 1.01, 5.93, 1.25, 0.08, -0.04, 1.27, -0.27, -0.07, 0.03, -0.31, 5.85, 3.57, 9.84, 0.74) in_paramspace(p=2, M=2, d=2, params=params22) # GMVAR(2,2), d=2 model with AR-parameters restricted to be # the same for both regimes: C_mat <- rbind(diag(2*2^2), diag(2*2^2)) params22c <- c(1.03, 2.36, 1.79, 3.00, 1.25, 0.06,0.04, 1.34, -0.29, -0.08, -0.05, -0.36, 0.93, -0.15, 5.20, 5.88, 3.56, 9.80, 0.37) in_paramspace(p=2, M=2, d=2, params=params22c, constraints=C_mat) # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(1.03, 2.36, 1.79, 3, 1.25, 0.06, 0.04, 1.34, -0.29, -0.08, -0.05, -0.36, 1.2, 0.05, 0.05, 1.3, -0.3, -0.1, -0.05, -0.4, 0.89, 0.72, -0.37, 2.16, 7.16, 1.3, 0.37) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) in_paramspace(p=2, M=2, d=2, params=params22s, structural_pars=list(W=W_22))
in_paramspace_int checks whether the parameter vector lies in the parameter
space.
in_paramspace_int( p, M, d, params, model = c("GMVAR", "StMVAR", "G-StMVAR"), all_boldA, alphas, all_Omega, W_constraints = NULL, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 )in_paramspace_int( p, M, d, params, model = c("GMVAR", "StMVAR", "G-StMVAR"), all_boldA, alphas, all_Omega, W_constraints = NULL, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 )
p |
a positive integer specifying the autoregressive order of the model. |
M |
|
d |
the number of time series in the system. |
params |
a real valued vector specifying the parameter values.
Above, In the GMVAR model, The notation is similar to the cited literature. |
model |
is "GMVAR", "StMVAR", or "G-StMVAR" model considered? In the G-StMVAR model, the first |
all_boldA |
3D array containing the |
alphas |
(Mx1) vector containing all mixing weight parameters, obtained from |
all_Omega |
3D array containing all covariance matrices |
W_constraints |
set |
stat_tol |
numerical tolerance for stationarity of the AR parameters: if the "bold A" matrix of any regime
has eigenvalues larger that |
posdef_tol |
numerical tolerance for positive definiteness of the error term covariance matrices: if the error term covariance matrix of any regime has eigenvalues smaller than this, the model is classified as not satisfying positive definiteness assumption. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error. |
df_tol |
the parameter vector is considered to be outside the parameter space if all degrees of
freedom parameters are not larger than |
The parameter vector in the argument params should be unconstrained and it is used for
structural models only.
Returns TRUE if the given parameter values are in the parameter space and FALSE otherwise.
This function does NOT consider the identifiability condition!
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
@keywords internal
iterate_more uses a variable metric algorithm to finalize maximum likelihood
estimation of a GMVAR, StMVAR, or G-StMVAR model (object of class 'gsmvar') which already has preliminary estimates.
iterate_more( gsmvar, maxit = 100, calc_std_errors = TRUE, custom_h = NULL, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 )iterate_more( gsmvar, maxit = 100, calc_std_errors = TRUE, custom_h = NULL, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 )
gsmvar |
an object of class |
maxit |
the maximum number of iterations in the variable metric algorithm. |
calc_std_errors |
calculate approximate standard errors for the estimates? |
custom_h |
A numeric vector with same the length as the parameter vector: i:th element of custom_h is the difference
used in central difference approximation for partial differentials of the log-likelihood function for the i:th parameter.
If |
stat_tol |
numerical tolerance for stationarity of the AR parameters: if the "bold A" matrix of any regime
has eigenvalues larger that |
posdef_tol |
numerical tolerance for positive definiteness of the error term covariance matrices: if the error term covariance matrix of any regime has eigenvalues smaller than this, the model is classified as not satisfying positive definiteness assumption. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error. |
df_tol |
the parameter vector is considered to be outside the parameter space if all degrees of
freedom parameters are not larger than |
The purpose of iterate_more is to provide a simple and convenient tool to finalize
the estimation when the maximum number of iterations is reached when estimating a GMVAR, StMVAR, or G-StMVAR model
with the main estimation function fitGSMVAR. iterate_more is essentially a wrapper
around the function optim from the package stats and GSMVAR from the package
gmvarkit.
Returns an object of class 'gsmvar' defining the estimated GMVAR, StMVAR, or G-StMVAR model.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Kalliovirta L. and Saikkonen P. 2010. Reliable Residuals for Multivariate Nonlinear Time Series Models. Unpublished Revision of HECER Discussion Paper No. 247.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
fitGSMVAR, GSMVAR, optim,
profile_logliks, update_numtols
## These are long running examples that use parallel computing! ## Running the below examples takes approximately 2 minutes # GMVAR(1,2) model, only 5 iterations of the variable metric # algorithm fit12 <- fitGSMVAR(gdpdef, p=1, M=2, ncalls=1, maxit=5, seeds=1) fit12 # Iterate more: fit12_2 <- iterate_more(fit12) fit12_2## These are long running examples that use parallel computing! ## Running the below examples takes approximately 2 minutes # GMVAR(1,2) model, only 5 iterations of the variable metric # algorithm fit12 <- fitGSMVAR(gdpdef, p=1, M=2, ncalls=1, maxit=5, seeds=1) fit12 # Iterate more: fit12_2 <- iterate_more(fit12) fit12_2
linear_IRF estimates linear impulse response function based on a single regime
of a structural GMVAR, StMVAR, or G-StMVAR model.
linear_IRF( gsmvar, N = 30, regime = 1, which_cumulative = numeric(0), scale = NULL, ci = NULL, bootstrap_reps = 100, ncores = 2, ncalls = 1, seeds = NULL, ... ) ## S3 method for class 'irf' plot(x, shocks_to_plot, ...) ## S3 method for class 'irf' print(x, ..., digits = 2, N_to_print, shocks_to_print)linear_IRF( gsmvar, N = 30, regime = 1, which_cumulative = numeric(0), scale = NULL, ci = NULL, bootstrap_reps = 100, ncores = 2, ncalls = 1, seeds = NULL, ... ) ## S3 method for class 'irf' plot(x, shocks_to_plot, ...) ## S3 method for class 'irf' print(x, ..., digits = 2, N_to_print, shocks_to_print)
gsmvar |
an object of class |
N |
a positive integer specifying the horizon how far ahead should the linear impulse responses be calculated. |
regime |
Based on which regime the linear IRF should be calculated?
An integer in |
which_cumulative |
a numeric vector with values in |
scale |
should the linear IRFs to some of the shocks be scaled so that they
correspond to a specific instantaneous response of some specific
variable? Provide a length three vector where the shock of interest
is given in the first element (an integer in |
ci |
a real number in |
bootstrap_reps |
the number of bootstrap repetitions for estimating confidence bounds. |
ncores |
the number of CPU cores to be used in parallel computing when bootstrapping confidence bounds. |
ncalls |
on how many estimation rounds should each bootstrap estimation be based on? Does not have to be very large since initial estimates used are based on the initially fitted model. Larger number of rounds gives more reliable results but is computationally more demanding. |
seeds |
a numeric vector of length |
... |
currently not used. |
x |
object of class |
shocks_to_plot |
IRFs of which shocks should be plotted? A numeric vector
with elements in |
digits |
the number of decimals to print |
N_to_print |
an integer specifying the horizon how far to print the estimates and confidence intervals. The default is that all the values are printed. |
shocks_to_print |
the responses to which should should be printed?
A numeric vector with elements in |
The model DOES NOT need to be structural in order for this function to be
applicable. When an identified structural GMVAR, StMVAR, or G-StMVAR model is
provided in the argument gsmvar, the identification imposed by the model
is used. When a reduced form model is provided in the argument gsmvar,
lower triangular Cholesky identification is used to identify the shocks.
If the autoregressive dynamics of the model are linear (i.e., either M == 1 or mean and AR parameters are constrained identical across the regimes), confidence bounds can be calculated based on a type of fixed-design wild residual bootstrap method. See Virolainen (forthcoming) for a related discussion. We employ the method described in Herwartz and Lütkepohl (2014); see also the relevant chapters in Kilian and Lütkepohl (2017).
Returns a class 'irf' list with the following elements:
$point_est:a 3D array [variables, shock, horizon] containing the point estimates of the IRFs.
Note that the first slice is for the impact responses and the slice i+1 for the period i. The response of the
variable 'i1' to the shock 'i2' is subsetted as $point_est[i1, i2, ].
$conf_ints:bootstrapped confidence intervals for the IRFs in a [variables, shock, horizon, bound]
4D array. The lower bound is obtained as $conf_ints[, , , 1], and similarly the upper bound as
$conf_ints[, , , 2]. The subsetted 3D array is then the bound in a form similar to $point_est.
$all_bootstrap_reps:IRFs from all of the bootstrap replications in a [variables, shock, horizon, rep].
4D array. The IRF from replication i1 is obtained as $all_bootstrap_reps[, , , i1], and the subsetted 3D array
is then the in a form similar to $point_est.
contains some of the arguments the linear_IRF was called with.
plot(irf): plot method
print(irf): print method
Herwartz H. and Lütkepohl H. 2014. Structural vector autoregressions with Markov switching: Combining conventional with statistical identification of shocks. Journal of Econometrics, 183, pp. 104-116.
Kilian L. and Lütkepohl H. 2017. Structural Vectors Autoregressive Analysis. Cambridge University Press, Cambridge.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
GIRF, GFEVD, fitGSMVAR, GSMVAR,
gsmvar_to_sgsmvar, reorder_W_columns, swap_W_signs
# These are long running examples that take a few minutes to run ## GMVAR, p=5, M=2, d=2 model with linear AR dynamics. # recursive identification, IRF based on the first regime: params52cm <- c(0.788, 0.559, 0.277, 0.038, -0.061, 0.463, 0.286, 0, 0.035, 0.161, -0.112, 0.031, -0.313, 0.183, 0.103, 0.014, 0.002, 0.143, -0.089, -0.013, 0.182, -0.04, 1.3, 0.008, 0.139, 0.277, -0.005, 0.032, 0.118) mod52cm <- GSMVAR(data=gdpdef, p=5, M=2, params=params52cm, constraints=rbind(diag(5*2^2), diag(5*2^2)), same_means=list(1:2), parametrization="mean") irf1 <- linear_IRF(mod52cm, regime=1, N=20, scale=cbind(c(1, 1, 1), c(2, 2, 1))) print(irf1, digits=3) plot(irf1) # Identification by heteroskedasticity, bootstrapped confidence intervals and # and scaled instantaneous effects of the shocks. Note that in actual # empirical application, a larger number of bootstrap reps should be used. mod52cms <- gsmvar_to_sgsmvar(mod52cm) irf2 <- linear_IRF(mod52cms, regime=1, N=20, ci=0.68, bootstrap_reps=10, ncalls=1, seeds=1:10, ncores=1) plot(irf2)# These are long running examples that take a few minutes to run ## GMVAR, p=5, M=2, d=2 model with linear AR dynamics. # recursive identification, IRF based on the first regime: params52cm <- c(0.788, 0.559, 0.277, 0.038, -0.061, 0.463, 0.286, 0, 0.035, 0.161, -0.112, 0.031, -0.313, 0.183, 0.103, 0.014, 0.002, 0.143, -0.089, -0.013, 0.182, -0.04, 1.3, 0.008, 0.139, 0.277, -0.005, 0.032, 0.118) mod52cm <- GSMVAR(data=gdpdef, p=5, M=2, params=params52cm, constraints=rbind(diag(5*2^2), diag(5*2^2)), same_means=list(1:2), parametrization="mean") irf1 <- linear_IRF(mod52cm, regime=1, N=20, scale=cbind(c(1, 1, 1), c(2, 2, 1))) print(irf1, digits=3) plot(irf1) # Identification by heteroskedasticity, bootstrapped confidence intervals and # and scaled instantaneous effects of the shocks. Note that in actual # empirical application, a larger number of bootstrap reps should be used. mod52cms <- gsmvar_to_sgsmvar(mod52cm) irf2 <- linear_IRF(mod52cms, regime=1, N=20, ci=0.68, bootstrap_reps=10, ncalls=1, seeds=1:10, ncores=1) plot(irf2)
loglikelihood computes log-likelihood of a GMVAR, StMVAR, or G-StMVAR model using parameter vector
instead of an object of class 'gsmvar'. Exists for convenience if one wants to for example
employ other estimation algorithms than the ones used in fitGSMVAR. Use minval to
control what happens when the parameter vector is outside the parameter space.
loglikelihood( data, p, M, params, model = c("GMVAR", "StMVAR", "G-StMVAR"), conditional = TRUE, parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, weight_constraints = NULL, structural_pars = NULL, minval = NA, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 )loglikelihood( data, p, M, params, model = c("GMVAR", "StMVAR", "G-StMVAR"), conditional = TRUE, parametrization = c("intercept", "mean"), constraints = NULL, same_means = NULL, weight_constraints = NULL, structural_pars = NULL, minval = NA, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 )
data |
a matrix or class |
p |
a positive integer specifying the autoregressive order of the model. |
M |
|
params |
a real valued vector specifying the parameter values.
Above, In the GMVAR model, The notation is similar to the cited literature. |
model |
is "GMVAR", "StMVAR", or "G-StMVAR" model considered? In the G-StMVAR model, the first |
conditional |
a logical argument specifying whether the conditional or exact log-likelihood function should be used. |
parametrization |
|
constraints |
a size |
same_means |
Restrict the mean parameters of some regimes to be the same? Provide a list of numeric vectors
such that each numeric vector contains the regimes that should share the common mean parameters. For instance, if
|
weight_constraints |
a numeric vector of length |
structural_pars |
If
See Virolainen (forthcoming) for the conditions required to identify the shocks and for the B-matrix as well (it is |
minval |
the value that will be returned if the parameter vector does not lie in the parameter space (excluding the identification condition). |
stat_tol |
numerical tolerance for stationarity of the AR parameters: if the "bold A" matrix of any regime
has eigenvalues larger that |
posdef_tol |
numerical tolerance for positive definiteness of the error term covariance matrices: if the error term covariance matrix of any regime has eigenvalues smaller than this, the model is classified as not satisfying positive definiteness assumption. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error. |
df_tol |
the parameter vector is considered to be outside the parameter space if all degrees of
freedom parameters are not larger than |
loglikelihood_int takes use of the function dmvn from the package mvnfast.
Returns log-likelihood if params is in the parameters space and minval if not.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Lütkepohl H. 2005. New Introduction to Multiple Time Series Analysis, Springer.
McElroy T. 2017. Computation of vector ARMA autocovariances. Statistics and Probability Letters, 124, 92-96.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
fitGSMVAR, GSMVAR, calc_gradient
# GMVAR(2, 2), d=2 model; params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) loglikelihood(data=gdpdef, p=2, M=2, params=params22) # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) loglikelihood(data=gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22))# GMVAR(2, 2), d=2 model; params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) loglikelihood(data=gdpdef, p=2, M=2, params=params22) # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) loglikelihood(data=gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22))
LR_test performs a likelihood ratio test for a GMVAR, StMVAR, or G-StMVAR model
LR_test(gsmvar1, gsmvar2)LR_test(gsmvar1, gsmvar2)
gsmvar1 |
an object of class |
gsmvar2 |
an object of class |
Performs a likelihood ratio test, testing the null hypothesis that the true parameter value lies
in the constrained parameter space. Under the null, the test statistic is asymptotically
-distributed with degrees of freedom, being the difference in the dimensions
of the unconstrained and constrained parameter spaces.
Note that this function does not verify that the two models are actually nested.
A list with class "hypotest" containing the test results and arguments used to calculate the test.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
@keywords internal
Wald_test, Rao_test, fitGSMVAR, GSMVAR, diagnostic_plot,
profile_logliks, quantile_residual_tests, cond_moment_plot
## These are long running examples that use parallel computing! ## The below examples take around 1 minute to run. # Structural GMVAR(2, 2), d=2 model with recursive identification W22 <- matrix(c(1, NA, 0, 1), nrow=2, byrow=FALSE) fit22s <- fitGSMVAR(gdpdef, p=2, M=2, structural_pars=list(W=W22), ncalls=1, seeds=2) # The same model but the AR coefficients restricted to be the same # in both regimes: C_mat <- rbind(diag(2*2^2), diag(2*2^2)) fit22sc <- fitGSMVAR(gdpdef, p=2, M=2, constraints=C_mat, structural_pars=list(W=W22), ncalls=1, seeds=1) # Test the AR constraints with likelihood ratio test: LR_test(fit22s, fit22sc)## These are long running examples that use parallel computing! ## The below examples take around 1 minute to run. # Structural GMVAR(2, 2), d=2 model with recursive identification W22 <- matrix(c(1, NA, 0, 1), nrow=2, byrow=FALSE) fit22s <- fitGSMVAR(gdpdef, p=2, M=2, structural_pars=list(W=W22), ncalls=1, seeds=2) # The same model but the AR coefficients restricted to be the same # in both regimes: C_mat <- rbind(diag(2*2^2), diag(2*2^2)) fit22sc <- fitGSMVAR(gdpdef, p=2, M=2, constraints=C_mat, structural_pars=list(W=W22), ncalls=1, seeds=1) # Test the AR constraints with likelihood ratio test: LR_test(fit22s, fit22sc)
Pearson_residuals calculates multivariate Pearson residuals for a GMVAR, StMVAR, or G-StMVAR model.
Pearson_residuals(gsmvar, standardize = TRUE)Pearson_residuals(gsmvar, standardize = TRUE)
gsmvar |
an object of class |
standardize |
Should the residuals be standardized? Use |
Returns matrix containing the residuals,
:th column corresponds to the time series in the :th column of the data.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Kalliovirta L. and Saikkonen P. 2010. Reliable Residuals for Multivariate Nonlinear Time Series Models. Unpublished Revision of HECER Discussion Paper No. 247.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
fitGSMVAR, GSMVAR, quantile_residuals,
diagnostic_plot
# GMVAR(1,2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) Pearson_residuals(mod12, standardize=FALSE) # Raw residuals Pearson_residuals(mod12, standardize=TRUE) # Standardized to identity cov.matrix. # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) Pearson_residuals(mod22s, standardize=FALSE) # Raw residuals Pearson_residuals(mod22s, standardize=TRUE) # Standardized to identity cov.matrix.# GMVAR(1,2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) Pearson_residuals(mod12, standardize=FALSE) # Raw residuals Pearson_residuals(mod12, standardize=TRUE) # Standardized to identity cov.matrix. # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) Pearson_residuals(mod22s, standardize=FALSE) # Raw residuals Pearson_residuals(mod22s, standardize=TRUE) # Standardized to identity cov.matrix.
plot.gmvarpred is plot method for gsmvarpred objects.
EXISTS FOR BACKWARD COMPATIBILITY. THE CLASS 'gmvarpred' IS DEPRECATED FROM
THE VERSION 2.0.0 ONWARD: WE USE THE CLASS 'gsmvarpred' NOW.
## S3 method for class 'gmvarpred' plot(x, ..., nt, mix_weights = TRUE, add_grid = TRUE) ## S3 method for class 'gmvarpred' print(x, ..., digits = 2)## S3 method for class 'gmvarpred' plot(x, ..., nt, mix_weights = TRUE, add_grid = TRUE) ## S3 method for class 'gmvarpred' print(x, ..., digits = 2)
x |
object of class |
... |
arguments passed to |
nt |
a positive integer specifying the number of observations to be plotted
along with the prediction (ignored if |
mix_weights |
|
add_grid |
should grid be added to the plots? |
digits |
how many digits to print? |
These methods exist so that objects created with earlier versions of the package can be used normally.
plot.gsmvarpred is plot method for gsmvarpred objects.
## S3 method for class 'gsmvarpred' plot(x, ..., nt, mix_weights = TRUE, add_grid = TRUE)## S3 method for class 'gsmvarpred' plot(x, ..., nt, mix_weights = TRUE, add_grid = TRUE)
x |
object of class |
... |
arguments passed to |
nt |
a positive integer specifying the number of observations to be plotted
along with the prediction (ignored if |
mix_weights |
|
add_grid |
should grid be added to the plots? |
This method is used plot forecasts of GSMVAR processes
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
@keywords internal
quantile_residual_tests performs quantile residual tests described
by Kalliovirta and Saikkonen 2010, testing autocorrelation, conditional heteroskedasticity,
and normality.
## S3 method for class 'qrtest' plot(x, ...) ## S3 method for class 'qrtest' print(x, ..., digits = 3) quantile_residual_tests( gsmvar, lags_ac = c(1, 3, 6, 12), lags_ch = lags_ac, nsim = 1, ncores = 1, print_res = TRUE, stat_tol, posdef_tol, df_tol )## S3 method for class 'qrtest' plot(x, ...) ## S3 method for class 'qrtest' print(x, ..., digits = 3) quantile_residual_tests( gsmvar, lags_ac = c(1, 3, 6, 12), lags_ch = lags_ac, nsim = 1, ncores = 1, print_res = TRUE, stat_tol, posdef_tol, df_tol )
x |
object of class |
... |
currently not used. |
digits |
the number of decimals to print |
gsmvar |
an object of class |
lags_ac |
a positive integer vector specifying the lags used to test autocorrelation. |
lags_ch |
a positive integer vector specifying the lags used to test conditional heteroskedasticity. |
nsim |
to how many simulations should the covariance matrix Omega used in the qr-tests be based on? If smaller than sample size, then the covariance matrix will be evaluated from the sample. Larger number of simulations might improve the tests size properties but it increases the computation time. |
ncores |
the number of CPU cores to be used in numerical differentiation. Multiple cores are not supported on Windows, though. |
print_res |
should the test results be printed while computing the tests? |
stat_tol |
numerical tolerance for stationarity of the AR parameters: if the "bold A" matrix of any regime
has eigenvalues larger that |
posdef_tol |
numerical tolerance for positive definiteness of the error term covariance matrices: if the error term covariance matrix of any regime has eigenvalues smaller than this, the model is classified as not satisfying positive definiteness assumption. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error. |
df_tol |
the parameter vector is considered to be outside the parameter space if all degrees of
freedom parameters are not larger than |
If the function fails to calculate the tests because of numerical problems and the parameter values
are near the border of the parameter space, it might help to use smaller numerical tolerance for the
stationarity and positeve definiteness conditions. The numerical tolerance of an existing model
can be changed with the function update_numtols or you can set it directly with the arguments
stat_tol and posdef_tol.
Returns an object of class 'qrtest' which has its own print method. The returned object
is a list containing the quantile residual test results for normality, autocorrelation, and conditional
heteroskedasticity. The autocorrelation and conditional heteroskedasticity results also contain the
associated (vectorized) individual statistics divided by their standard errors
(see Kalliovirta and Saikkonen 2010, s.17-20) under the label $ind_stats.
plot(qrtest): Plot p-values of the autocorrelation and conditional
heteroskedasticity tests.
print(qrtest): Print method for class 'qrtest'
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Kalliovirta L. and Saikkonen P. 2010. Reliable Residuals for Multivariate Nonlinear Time Series Models. Unpublished Revision of HECER Discussion Paper No. 247.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
fitGSMVAR, GSMVAR, quantile_residuals, GIRF,
diagnostic_plot, predict.gsmvar, profile_logliks,
LR_test, Wald_test, cond_moment_plot, update_numtols
# GMVAR(3,2) model fit32 <- fitGSMVAR(gdpdef, p=3, M=2, ncalls=1, seeds=2) qrtests32 <- quantile_residual_tests(fit32) qrtests32 plot(qrtests32) # Structural GMVAR(1,2) model identified with sign # constraints and build with hand-specified parameter values. # Tests based on simulation procedure with nsim=1000: params12s <- c(0.55, 0.112, 0.619, 0.173, 0.344, 0.055, -0.009, 0.718, 0.255, 0.017, -0.136, 0.858, 0.541, 0.057, -0.162, 0.162, 3.623, 4.726, 0.674) W_12 <- matrix(c(1, 1, -1, 1), nrow=2) mod12s <- GSMVAR(gdpdef, p=1, M=2, params=params12s, structural_pars=list(W=W_12)) qrtests12s <- quantile_residual_tests(mod12s, nsim=1000) qrtests12s# GMVAR(3,2) model fit32 <- fitGSMVAR(gdpdef, p=3, M=2, ncalls=1, seeds=2) qrtests32 <- quantile_residual_tests(fit32) qrtests32 plot(qrtests32) # Structural GMVAR(1,2) model identified with sign # constraints and build with hand-specified parameter values. # Tests based on simulation procedure with nsim=1000: params12s <- c(0.55, 0.112, 0.619, 0.173, 0.344, 0.055, -0.009, 0.718, 0.255, 0.017, -0.136, 0.858, 0.541, 0.057, -0.162, 0.162, 3.623, 4.726, 0.674) W_12 <- matrix(c(1, 1, -1, 1), nrow=2) mod12s <- GSMVAR(gdpdef, p=1, M=2, params=params12s, structural_pars=list(W=W_12)) qrtests12s <- quantile_residual_tests(mod12s, nsim=1000) qrtests12s
predict.gsmvar is a predict method for class 'gsmvar' objects. The forecasts of
the GMVAR model are computed by performing independent simulations and using the
sample medians or means as point forecasts and empirical quantiles as prediction intervals.
For one-step-ahead predictions using the exact conditional mean is also supported.
## S3 method for class 'gmvar' predict( object, ..., n_ahead, n_simu = 2000, pi = c(0.95, 0.8), pi_type = c("two-sided", "upper", "lower", "none"), pred_type = c("median", "mean", "cond_mean"), plot_res = TRUE, mix_weights = TRUE, nt )## S3 method for class 'gmvar' predict( object, ..., n_ahead, n_simu = 2000, pi = c(0.95, 0.8), pi_type = c("two-sided", "upper", "lower", "none"), pred_type = c("median", "mean", "cond_mean"), plot_res = TRUE, mix_weights = TRUE, nt )
object |
an object of class 'gmvar' |
... |
additional arguments passed to |
n_ahead |
how many steps ahead should be predicted? |
n_simu |
to how many independent simulations should the forecast be based on? |
pi |
a numeric vector specifying the confidence levels of the prediction intervals. |
pi_type |
should the prediction intervals be "two-sided", "upper", or "lower"? |
pred_type |
should the prediction be based on sample "median" or "mean"? Or should it
be one-step-ahead forecast based on the exact conditional mean ( |
plot_res |
should the results be plotted? |
mix_weights |
|
nt |
a positive integer specifying the number of observations to be plotted
along with the prediction (ignored if |
Returns a class 'gsmvarpred' object containing, among the specifications,...
Point forecasts
Prediction intervals, as [, , d].
Point forecasts for the mixing weights
Individual prediction intervals for mixing weights, as [, , m], m=1,..,M.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
@keywords internal
predict.gsmvar is a predict method for class 'gsmvar' objects. The forecasts of
the GMVAR, StMVAR, and G-StMVAR models are computed by performing independent simulations and using the
sample medians or means as point forecasts and empirical quantiles as prediction intervals.
For one-step-ahead predictions using the exact conditional mean is also supported.
## S3 method for class 'gsmvar' predict( object, ..., n_ahead, nsim = 2000, pi = c(0.95, 0.8), pi_type = c("two-sided", "upper", "lower", "none"), pred_type = c("median", "mean", "cond_mean"), plot_res = TRUE, mix_weights = TRUE, nt )## S3 method for class 'gsmvar' predict( object, ..., n_ahead, nsim = 2000, pi = c(0.95, 0.8), pi_type = c("two-sided", "upper", "lower", "none"), pred_type = c("median", "mean", "cond_mean"), plot_res = TRUE, mix_weights = TRUE, nt )
object |
an object of class |
... |
additional arguments passed to |
n_ahead |
how many steps ahead should be predicted? |
nsim |
to how many independent simulations should the forecast be based on? |
pi |
a numeric vector specifying the confidence levels of the prediction intervals. |
pi_type |
should the prediction intervals be "two-sided", "upper", or "lower"? |
pred_type |
should the prediction be based on sample "median" or "mean"? Or should it
be one-step-ahead forecast based on the exact conditional mean ( |
plot_res |
should the results be plotted? |
mix_weights |
|
nt |
a positive integer specifying the number of observations to be plotted
along with the prediction (ignored if |
Returns a class 'gsmvarpred' object containing, among the specifications,...
Point forecasts
Prediction intervals, as [, , d].
Point forecasts for the mixing weights
Individual prediction intervals for mixing weights, as [, , m], m=1,..,M.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
@keywords internal
# GMVAR(2, 2), d=2 model params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, d=2, params=params22) p1 <- predict(mod22, n_ahead=10, pred_type="median", nsim=500) p1 p2 <- predict(mod22, n_ahead=10, nt=20, lty=1, nsim=500) p2 p3 <- predict(mod22, n_ahead=10, pi=c(0.99, 0.90, 0.80, 0.70), nt=30, lty=0, nsim=500) p3 # StMVAR(2, 2), d=2 model params22t <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58, 3, 4) mod22t <- GSMVAR(gdpdef, p=2, M=2, d=2, params=params22t, model="StMVAR") p1 <- predict(mod22t, n_ahead=12, pred_type="median", nsim=500, pi=0.9) p1# GMVAR(2, 2), d=2 model params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, d=2, params=params22) p1 <- predict(mod22, n_ahead=10, pred_type="median", nsim=500) p1 p2 <- predict(mod22, n_ahead=10, nt=20, lty=1, nsim=500) p2 p3 <- predict(mod22, n_ahead=10, pi=c(0.99, 0.90, 0.80, 0.70), nt=30, lty=0, nsim=500) p3 # StMVAR(2, 2), d=2 model params22t <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58, 3, 4) mod22t <- GSMVAR(gdpdef, p=2, M=2, d=2, params=params22t, model="StMVAR") p1 <- predict(mod22t, n_ahead=12, pred_type="median", nsim=500, pi=0.9) p1
print_std_errors prints the approximate standard errors of a GMVAR, StMVAR, or G-StMVAR model in the
same form as the parameters of objects of class 'gsmvar' are printed.
print_std_errors(gsmvar, digits = 3)print_std_errors(gsmvar, digits = 3)
gsmvar |
an object of class |
digits |
how many digits should be printed? |
The main purpose of print_std_errors is to provide a convenient tool to match the standard
errors to certain parameter estimates. Note that if the model is intercept parametrized, there won't
be standard errors for the unconditional means, and vice versa. Also, there is no standard error for the
last mixing weight alpha_M because it is not parametrized.
Note that if linear constraints are imposed and they involve summations or multiplications, then the AR parameter standard errors are printed separately as they don't correspond one-to-one to the model parameter standard errors.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Kalliovirta L. and Saikkonen P. 2010. Reliable Residuals for Multivariate Nonlinear Time Series Models. Unpublished Revision of HECER Discussion Paper No. 247.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
profile_logliks, fitGSMVAR, GSMVAR, print.gsmvar,
swap_parametrization
# GMVAR(1,2) model fit12 <- fitGSMVAR(gdpdef, p=1, M=2, ncalls=1, seeds=1) fit12 print_std_errors(fit12)# GMVAR(1,2) model fit12 <- fitGSMVAR(gdpdef, p=1, M=2, ncalls=1, seeds=1) fit12 print_std_errors(fit12)
Deprecated S3 methods for the deprecated class 'gmvar'. From the gmvarkit version 2.0.0 onwards, class 'gsmvar' is used instead.
## S3 method for class 'gmvar' print(x, ..., digits = 2) ## S3 method for class 'gmvar' summary(object, ..., digits) ## S3 method for class 'gmvar' plot(x, ...) ## S3 method for class 'gmvar' logLik(object, ...) ## S3 method for class 'gmvar' residuals(object, ...)## S3 method for class 'gmvar' print(x, ..., digits = 2) ## S3 method for class 'gmvar' summary(object, ..., digits) ## S3 method for class 'gmvar' plot(x, ...) ## S3 method for class 'gmvar' logLik(object, ...) ## S3 method for class 'gmvar' residuals(object, ...)
x |
a class 'gmvar' object. THIS CLASS IS DEPRECATED FROM THE VERSION 2.0.0 ONWARDS. |
... |
See the usage from the documentation of the appropriate class 'gsmvar' S3 method. |
digits |
number of digits to be printed. |
object |
object of class |
These methods exist so that models estimated with earlier versions of the package can be used normally.
print.gmvarsum is a print method for object 'gmvarsum'.
EXISTS FOR BACKWARD COMPATIBILITY. CLASS 'gmvarsum' IS DEPRECATED FROM THE VERSION
2.0.0. ONWARDS. NOW, WE USE THE CLASS 'gsmvarsum'.
## S3 method for class 'gmvarsum' print(x, ..., digits)## S3 method for class 'gmvarsum' print(x, ..., digits)
x |
object of class 'gsmvarsum' generated by |
... |
currently not used. |
digits |
the number of digits to be printed. |
print.gsmvarpred is a print method for object generated
by predict.gsmvar.
## S3 method for class 'gsmvarpred' print(x, ..., digits = 2)## S3 method for class 'gsmvarpred' print(x, ..., digits = 2)
x |
object of class |
... |
currently not used. |
digits |
the number of decimals to print |
# GMVAR(2, 2), d=2 model; params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22) pred22 <- predict(mod22, n_ahead=3, plot_res=FALSE) print(pred22) print(pred22, digits=3)# GMVAR(2, 2), d=2 model; params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22) pred22 <- predict(mod22, n_ahead=3, plot_res=FALSE) print(pred22) print(pred22, digits=3)
print.gsmvarsum is a print method for object 'gsmvarsum' generated
by summary.gsmvar.
## S3 method for class 'gsmvarsum' print(x, ..., digits)## S3 method for class 'gsmvarsum' print(x, ..., digits)
x |
object of class 'gsmvarsum' generated by |
... |
currently not used. |
digits |
the number of digits to be printed. |
# GMVAR(2, 2), d=2 model; params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22) sumry22 <- summary(mod22) print(sumry22)# GMVAR(2, 2), d=2 model; params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22) sumry22 <- summary(mod22) print(sumry22)
print.hypotest is the print method for the class hypotest
objects.
## S3 method for class 'hypotest' print(x, ..., digits = 4)## S3 method for class 'hypotest' print(x, ..., digits = 4)
x |
object of class |
... |
currently not in use. |
digits |
how many significant digits to print? |
profile_logliks plots profile log-likelihoods around the estimates.
profile_logliks( gsmvar, which_pars, scale = 0.02, nrows, ncols, precision = 200, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 )profile_logliks( gsmvar, which_pars, scale = 0.02, nrows, ncols, precision = 200, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 )
gsmvar |
an object of class |
which_pars |
the profile log-likelihood function of which parameters should be plotted? An integer vector specifying the positions of the parameters in the parameter vector. The parameter vector has the form...
Above, The default is that profile log-likelihood functions for all parameters are plotted. |
scale |
a numeric scalar specifying the interval plotted for each estimate:
the estimate plus-minus |
nrows |
how many rows should be in the plot-matrix? The default is |
ncols |
how many columns should be in the plot-matrix? The default is |
precision |
at how many points should each profile log-likelihood be evaluated at? |
stat_tol |
numerical tolerance for stationarity of the AR parameters: if the "bold A" matrix of any regime
has eigenvalues larger that |
posdef_tol |
numerical tolerance for positive definiteness of the error term covariance matrices: if the error term covariance matrix of any regime has eigenvalues smaller than this, the model is classified as not satisfying positive definiteness assumption. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error. |
df_tol |
the parameter vector is considered to be outside the parameter space if all degrees of
freedom parameters are not larger than |
When the number of parameters is large, it might be better to plot a smaller number of profile
log-likelihood functions at a time using the argument which_pars.
The red vertical line points the estimate.
Only plots to a graphical device and doesn't return anything.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Lütkepohl H. 2005. New Introduction to Multiple Time Series Analysis, Springer.
McElroy T. 2017. Computation of vector ARMA autocovariances. Statistics and Probability Letters, 124, 92-96.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
get_soc, diagnostic_plot, fitGSMVAR, GSMVAR,
GIRF, LR_test, Wald_test, cond_moment_plot
# Running all the below examples takes approximately 2 minutes. # GMVAR(1,2) model fit12 <- fitGSMVAR(gdpdef, p=1, M=2, ncalls=1, seeds=1) fit12 profile_logliks(fit12) # Structural GMVAR(1,2) model identified with sign # constraints: model build based on inaccurate hand-given estimates. W_122 <- matrix(c(1, 1, -1, 1), nrow=2) params12s <- c(0.55, 0.11, 0.62, 0.17, 0.34, 0.05, -0.01, 0.72, 0.25, 0.02, -0.14, 0.86, 0.54, 0.06, -0.16, 0.16, 3.62, 4.73, 0.67) mod12s <- GSMVAR(gdpdef, p=1, M=2, params=params12s, structural_pars=list(W=W_122)) profile_logliks(mod12s) #' # G-StMVAR(2, 1, 1), d=2 model: params22gs <- c(0.697, 0.154, 0.049, 0.374, 0.476, 0.318, -0.645, -0.302, -0.222, 0.193, 0.042, -0.013, 0.048, 0.554, 0.033, 0.184, 0.005, -0.186, 0.683, 0.256, 0.031, 0.026, 0.204, 0.583, -0.002, 0.048, 0.182, 4.334) mod22gs <- GSMVAR(gdpdef, p=2, M=c(1, 1), params=params22gs, model="G-StMVAR") profile_logliks(mod22gs, which_pars=c(1, 3, 28))# Running all the below examples takes approximately 2 minutes. # GMVAR(1,2) model fit12 <- fitGSMVAR(gdpdef, p=1, M=2, ncalls=1, seeds=1) fit12 profile_logliks(fit12) # Structural GMVAR(1,2) model identified with sign # constraints: model build based on inaccurate hand-given estimates. W_122 <- matrix(c(1, 1, -1, 1), nrow=2) params12s <- c(0.55, 0.11, 0.62, 0.17, 0.34, 0.05, -0.01, 0.72, 0.25, 0.02, -0.14, 0.86, 0.54, 0.06, -0.16, 0.16, 3.62, 4.73, 0.67) mod12s <- GSMVAR(gdpdef, p=1, M=2, params=params12s, structural_pars=list(W=W_122)) profile_logliks(mod12s) #' # G-StMVAR(2, 1, 1), d=2 model: params22gs <- c(0.697, 0.154, 0.049, 0.374, 0.476, 0.318, -0.645, -0.302, -0.222, 0.193, 0.042, -0.013, 0.048, 0.554, 0.033, 0.184, 0.005, -0.186, 0.683, 0.256, 0.031, 0.026, 0.204, 0.583, -0.002, 0.048, 0.182, 4.334) mod22gs <- GSMVAR(gdpdef, p=2, M=c(1, 1), params=params22gs, model="G-StMVAR") profile_logliks(mod22gs, which_pars=c(1, 3, 28))
quantile_residuals calculates multivariate quantile residuals
(proposed by Kalliovirta and Saikkonen 2010) for a GMVAR, StMVAR, or G-StMVAR model.
quantile_residuals(gsmvar)quantile_residuals(gsmvar)
gsmvar |
an object of class |
Returns matrix containing the multivariate quantile residuals,
:th column corresponds to the time series in the :th column of the data. The multivariate
quantile residuals are calculated so that the first column quantile residuals are the "unconditioned ones"
and the rest condition on all the previous ones in numerical order. Read the cited article by
Kalliovirta and Saikkonen 2010 for details.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Kalliovirta L. and Saikkonen P. 2010. Reliable Residuals for Multivariate Nonlinear Time Series Models. Unpublished Revision of HECER Discussion Paper No. 247.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
fitGSMVAR, GSMVAR, quantile_residual_tests,
diagnostic_plot, predict.gsmvar, profile_logliks
# GMVAR(1,2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) quantile_residuals(mod12) # GMVAR(2,2), d=2 model with mean-parametrization: params22 <- c(0.869, 0.549, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.576, 1.168, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22, parametrization="mean") quantile_residuals(mod22) # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) quantile_residuals(mod22s)# GMVAR(1,2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) quantile_residuals(mod12) # GMVAR(2,2), d=2 model with mean-parametrization: params22 <- c(0.869, 0.549, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.576, 1.168, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22, parametrization="mean") quantile_residuals(mod22) # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) quantile_residuals(mod22s)
random_ind2 generates random mean-parametrized parameter vector
that is always stationary.
random_ind2( p, M, d, model = c("GMVAR", "StMVAR", "G-StMVAR"), same_means = NULL, weight_constraints = NULL, structural_pars = NULL, mu_scale, mu_scale2, omega_scale, ar_scale = 1, W_scale, lambda_scale )random_ind2( p, M, d, model = c("GMVAR", "StMVAR", "G-StMVAR"), same_means = NULL, weight_constraints = NULL, structural_pars = NULL, mu_scale, mu_scale2, omega_scale, ar_scale = 1, W_scale, lambda_scale )
p |
a positive integer specifying the autoregressive order of the model. |
M |
|
d |
the number of time series in the system. |
model |
is "GMVAR", "StMVAR", or "G-StMVAR" model considered? In the G-StMVAR model, the first |
same_means |
Restrict the mean parameters of some regimes to be the same? Provide a list of numeric vectors
such that each numeric vector contains the regimes that should share the common mean parameters. For instance, if
|
weight_constraints |
a numeric vector of length |
structural_pars |
If
See Virolainen (forthcoming) for the conditions required to identify the shocks and for the B-matrix as well (it is |
mu_scale |
a size |
mu_scale2 |
a size |
omega_scale |
a size |
ar_scale |
a positive real number between zero and one, adjusting how large AR parameter values are typically
proposed in construction of the initial population: larger value implies larger coefficients (in absolute value).
After construction of the initial population, a new scale is drawn from |
W_scale |
a size |
lambda_scale |
a length If the lambda parameters are constrained with the This argument is ignored if As with omega_scale and W_scale, this argument should be adjusted carefully if specified by hand. NOTE that if lambdas are constrained in some other way than restricting some of them to be identical, this parameter should be adjusted accordingly in order to the estimation succeed! |
The coefficient matrices are generated using the algorithm proposed by Ansley
and Kohn (1986) which forces stationarity. It's not clear in detail how ar_scale
exactly affects the coefficient matrices but larger ar_scale seems to result in larger
AR coefficients. Read the cited article by Ansley and Kohn (1986) and the source code
for more information.
The covariance matrices are generated from (scaled) Wishart distribution.
Models with AR parameters constrained are not supported!
Returns random mean-parametrized parameter vector that has the same form as the argument params
in the other functions, for instance, in the function loglikelihood.
Ansley C.F., Kohn R. 1986. A note on reparameterizing a vector autoregressive moving average model to enforce stationarity. Journal of statistical computation and simulation, 24:2, 99-106.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
@keywords internal
Rao_test performs Rao's score test for a GSMVAR model
Rao_test(gsmvar)Rao_test(gsmvar)
gsmvar |
an object of class |
Tests the constraints imposed in the model given in the argument GSMVAR.
This implementation uses the outer product of gradients approximation in the test statistic.
A list with class "hypotest" containing the test results and arguments used to calculate the test.
Buse A. (1982). The Likelihood Ratio, Wald, and Lagrange Multiplier Tests: An Expository Note. The American Statistician, 36(3a), 153-157.
LR_test, Wald_test, fitGSMVAR, GSMVAR, diagnostic_plot,
profile_logliks
## These are long running examples that use parallel computing! ## The below examples take around 30 seconds to run. # Structural GMVAR(2, 2), d=2 model with recursive identification # with the AR matrices restricted to be the identical across the regimes: W22 <- matrix(c(1, NA, 0, 1), nrow=2, byrow=FALSE) C_mat <- rbind(diag(2*2^2), diag(2*2^2)) fit22sc <- fitGSMVAR(gdpdef, p=2, M=2, constraints=C_mat, structural_pars=list(W=W22), ncalls=1, seeds=1) # Test the null: Rao_test(fit22sc)## These are long running examples that use parallel computing! ## The below examples take around 30 seconds to run. # Structural GMVAR(2, 2), d=2 model with recursive identification # with the AR matrices restricted to be the identical across the regimes: W22 <- matrix(c(1, NA, 0, 1), nrow=2, byrow=FALSE) C_mat <- rbind(diag(2*2^2), diag(2*2^2)) fit22sc <- fitGSMVAR(gdpdef, p=2, M=2, constraints=C_mat, structural_pars=list(W=W22), ncalls=1, seeds=1) # Test the null: Rao_test(fit22sc)
redecompose_Omegas exchanges the order of the covariance matrices in
the decomposition of Muirhead (1982, Theorem A9.9) and returns the new decomposition.
redecompose_Omegas(M, d, W, lambdas, perm = 1:sum(M))redecompose_Omegas(M, d, W, lambdas, perm = 1:sum(M))
M |
|
d |
the number of time series in the system. |
W |
a length |
lambdas |
a length |
perm |
a vector of length |
We consider the following decomposition of positive definite covariannce matrices:
, , where
contains the strictly postive eigenvalues of and the column of the invertible are the
corresponding eigenvectors. Note that this decomposition does not necessarily exists for .
See Muirhead (1982), Theorem A9.9 for more details on the decomposition and the source code for more details on the reparametrization.
Returns a vector of the form c(vec(new_W), new_lambdas)
where the lambdas parameters are in the regimewise order (first regime 2, then 3, etc) and the
"new W" and "new lambdas" are constitute the new decomposition with the order of the covariance
matrices given by the argument perm. Notice that if the first element of perm
is one, the W matrix will be the same and the lambdas are just re-ordered.
Note that unparametrized zero elements ARE present in the returned W!
No argument checks! Does not work with dimension or with only
one mixture component .
Muirhead R.J. 1982. Aspects of Multivariate Statistical Theory, Wiley.
d <- 2 M <- 2 Omega1 <- matrix(c(2, 0.5, 0.5, 2), nrow=d) Omega2 <- matrix(c(1, -0.2, -0.2, 1), nrow=d) # Decomposition with Omega1 as the first covariance matrix: decomp1 <- diag_Omegas(Omega1, Omega2) W <- matrix(decomp1[1:d^2], nrow=d, ncol=d) lambdas <- decomp1[(d^2 + 1):length(decomp1)] tcrossprod(W) # = Omega1 W%*%tcrossprod(diag(lambdas), W) # = Omega2 # Reorder the covariance matrices in the decomposition so that now # the first covariance matrix is Omega2: decomp2 <- redecompose_Omegas(M=M, d=d, W=as.vector(W), lambdas=lambdas, perm=2:1) new_W <- matrix(decomp2[1:d^2], nrow=d, ncol=d) new_lambdas <- decomp2[(d^2 + 1):length(decomp2)] tcrossprod(new_W) # = Omega2 new_W%*%tcrossprod(diag(new_lambdas), new_W) # = Omega1d <- 2 M <- 2 Omega1 <- matrix(c(2, 0.5, 0.5, 2), nrow=d) Omega2 <- matrix(c(1, -0.2, -0.2, 1), nrow=d) # Decomposition with Omega1 as the first covariance matrix: decomp1 <- diag_Omegas(Omega1, Omega2) W <- matrix(decomp1[1:d^2], nrow=d, ncol=d) lambdas <- decomp1[(d^2 + 1):length(decomp1)] tcrossprod(W) # = Omega1 W%*%tcrossprod(diag(lambdas), W) # = Omega2 # Reorder the covariance matrices in the decomposition so that now # the first covariance matrix is Omega2: decomp2 <- redecompose_Omegas(M=M, d=d, W=as.vector(W), lambdas=lambdas, perm=2:1) new_W <- matrix(decomp2[1:d^2], nrow=d, ncol=d) new_lambdas <- decomp2[(d^2 + 1):length(decomp2)] tcrossprod(new_W) # = Omega2 new_W%*%tcrossprod(diag(new_lambdas), new_W) # = Omega1
reorder_W_columns reorder columns of the W-matrix and lambda parameters
of a structural GMVAR, StMVAR, or G-StMVAR model.
reorder_W_columns(gsmvar, perm)reorder_W_columns(gsmvar, perm)
gsmvar |
an object of class |
perm |
an integer vector of length |
The order of the columns of can be changed without changing the implied reduced
form model as long as the order of lambda parameters is also changed accordingly. Note that the
constraints imposed on (or the B-matrix) will also be modified accordingly.
This function does not support models with constraints imposed on the lambda parameters!
Also all signs in any column of can be swapped (without changing the implied reduced form model)
with the function swap_W_signs but this obviously also swaps the sign constraints in the
corresponding columns of .
Returns an object of class 'gsmvar' defining a structural GMVAR, StMVAR, or G-StMVAR model with the modified
structural parameters and constraints.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
@keywords internal
fitGSMVAR, GSMVAR, GIRF, gsmvar_to_sgsmvar,
stmvar_to_gstmvar, swap_W_signs
# Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(p=2, M=2, d=2, params=params22s, structural_pars=list(W=W_22)) mod22s # The same reduced form model, reordered W and lambda in the structual model: mod22s_2 <- reorder_W_columns(mod22s, perm=2:1) mod22s_2 # Structural StMVAR(2, 2), d=2 model identified with sign-constraints: mod22ts <- GSMVAR(p=2, M=2, d=2, params=c(params22s, 10, 20), model="StMVAR", structural_pars=list(W=W_22)) mod22ts # The same reduced form model, reordered W and lambda in the structual model: mod22ts_2 <- reorder_W_columns(mod22ts, perm=2:1) mod22ts_2# Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(p=2, M=2, d=2, params=params22s, structural_pars=list(W=W_22)) mod22s # The same reduced form model, reordered W and lambda in the structual model: mod22s_2 <- reorder_W_columns(mod22s, perm=2:1) mod22s_2 # Structural StMVAR(2, 2), d=2 model identified with sign-constraints: mod22ts <- GSMVAR(p=2, M=2, d=2, params=c(params22s, 10, 20), model="StMVAR", structural_pars=list(W=W_22)) mod22ts # The same reduced form model, reordered W and lambda in the structual model: mod22ts_2 <- reorder_W_columns(mod22ts, perm=2:1) mod22ts_2
simulate.gsmvar is a simulate method for class 'gsmvar' objects.
It allows to simulate observations from a GMVAR, StMVAR, or G-StMVAR process.
## S3 method for class 'gsmvar' simulate( object, nsim = 1, seed = NULL, ..., init_values = NULL, init_regimes = 1:sum(gsmvar$model$M), ntimes = 1, drop = TRUE, girf_pars = NULL )## S3 method for class 'gsmvar' simulate( object, nsim = 1, seed = NULL, ..., init_values = NULL, init_regimes = 1:sum(gsmvar$model$M), ntimes = 1, drop = TRUE, girf_pars = NULL )
object |
an object of class |
nsim |
number of observations to be simulated. |
seed |
set seed for the random number generator? |
... |
currently not in use. |
init_values |
a size |
init_regimes |
a numeric vector of length at most |
ntimes |
how many sets of simulations should be performed? |
drop |
if |
girf_pars |
This argument is used internally in the estimation of generalized impulse response functions (see |
The argument ntimes is intended for forecasting: a GMVAR, StMVAR, or G-StMVAR process can be forecasted by simulating
its possible future values. One can easily perform a large number simulations and calculate the sample quantiles from the simulated
values to obtain prediction intervals (see the forecasting example).
If drop==TRUE and ntimes==1 (default): $sample, $component, and $mixing_weights are matrices.
Otherwise, returns a list with...
$samplea size (nsimntimes) array containing the samples: the dimension [t, , ] is
the time index, the dimension [, d, ] indicates the marginal time series, and the dimension [, , i] indicates
the i:th set of simulations.
$componenta size (nsimntimes) matrix containing the information from which mixture component
each value was generated from.
$mixing_weightsa size (nsimntimes) array containing the mixing weights corresponding to
the sample: the dimension [t, , ] is the time index, the dimension [, m, ] indicates the regime, and the dimension
[, , i] indicates the i:th set of simulations.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Lütkepohl H. 2005. New Introduction to Multiple Time Series Analysis, Springer.
McElroy T. 2017. Computation of vector ARMA autocovariances. Statistics and Probability Letters, 124, 92-96.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
fitGSMVAR, GSMVAR, diagnostic_plot, predict.gsmvar,
profile_logliks, quantile_residual_tests, GIRF, GFEVD
# GMVAR(1,2), d=2 process, initial values from the stationary # distribution params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(p=1, M=2, d=2, params=params12) set.seed(1) sim12 <- simulate(mod12, nsim=500) plot.ts(sim12$sample) ts.plot(sim12$mixing_weights, col=c("blue", "red"), lty=2) plot(sim12$component, type="l") # StMVAR(2, 2), d=2 model params22t <- c(0.554, 0.033, 0.184, 0.005, -0.186, 0.683, 0.256, 0.031, 0.026, 0.204, 0.583, -0.002, 0.048, 0.697, 0.154, 0.049, 0.374, 0.476, 0.318, -0.645, -0.302, -0.222, 0.193, 0.042, -0.013, 0.048, 0.818, 4.334, 20) mod22t <- GSMVAR(gdpdef, p=2, M=2, params=params22t, model="StMVAR") sim22t <- simulate(mod22t, nsim=100) plot.ts(sim22t$mixing_weights) ## FORECASTING EXAMPLE ## # Forecast 5-steps-ahead, 500 sets of simulations with initial # values from the data: # GMVAR(2,2), d=2 model params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22) sim22 <- simulate(mod22, nsim=5, ntimes=500) # Point forecast + 95% prediction intervals: apply(sim22$sample, MARGIN=1:2, FUN=quantile, probs=c(0.025, 0.5, 0.972)) # Similar forecast for the mixing weights: apply(sim22$mixing_weights, MARGIN=1:2, FUN=quantile, probs=c(0.025, 0.5, 0.972))# GMVAR(1,2), d=2 process, initial values from the stationary # distribution params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(p=1, M=2, d=2, params=params12) set.seed(1) sim12 <- simulate(mod12, nsim=500) plot.ts(sim12$sample) ts.plot(sim12$mixing_weights, col=c("blue", "red"), lty=2) plot(sim12$component, type="l") # StMVAR(2, 2), d=2 model params22t <- c(0.554, 0.033, 0.184, 0.005, -0.186, 0.683, 0.256, 0.031, 0.026, 0.204, 0.583, -0.002, 0.048, 0.697, 0.154, 0.049, 0.374, 0.476, 0.318, -0.645, -0.302, -0.222, 0.193, 0.042, -0.013, 0.048, 0.818, 4.334, 20) mod22t <- GSMVAR(gdpdef, p=2, M=2, params=params22t, model="StMVAR") sim22t <- simulate(mod22t, nsim=100) plot.ts(sim22t$mixing_weights) ## FORECASTING EXAMPLE ## # Forecast 5-steps-ahead, 500 sets of simulations with initial # values from the data: # GMVAR(2,2), d=2 model params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22) sim22 <- simulate(mod22, nsim=5, ntimes=500) # Point forecast + 95% prediction intervals: apply(sim22$sample, MARGIN=1:2, FUN=quantile, probs=c(0.025, 0.5, 0.972)) # Similar forecast for the mixing weights: apply(sim22$mixing_weights, MARGIN=1:2, FUN=quantile, probs=c(0.025, 0.5, 0.972))
DEPRECATED! USE THE FUNCTION simulate.gsmvar INSTEAD!
simulateGMVAR simulates observations from a GMVAR
simulateGMVAR( gmvar, nsimu, init_values = NULL, ntimes = 1, drop = TRUE, seed = NULL, girf_pars = NULL )simulateGMVAR( gmvar, nsimu, init_values = NULL, ntimes = 1, drop = TRUE, seed = NULL, girf_pars = NULL )
gmvar |
object of class 'gmvar' |
nsimu |
number of observations to be simulated. |
init_values |
a size |
ntimes |
how many sets of simulations should be performed? |
drop |
if |
seed |
set seed for the random number generator? |
girf_pars |
This argument is used internally in the estimation of generalized impulse response functions (see |
The argument ntimes is intended for forecasting: a GMVAR, StMVAR, or G-StMVAR process can be forecasted by simulating
its possible future values. One can easily perform a large number simulations and calculate the sample quantiles from the simulated
values to obtain prediction intervals (see the forecasting example).
If drop==TRUE and ntimes==1 (default): $sample, $component, and $mixing_weights are matrices.
Otherwise, returns a list with...
$samplea size (nsimntimes) array containing the samples: the dimension [t, , ] is
the time index, the dimension [, d, ] indicates the marginal time series, and the dimension [, , i] indicates
the i:th set of simulations.
$componenta size (nsimntimes) matrix containing the information from which mixture component
each value was generated from.
$mixing_weightsa size (nsimntimes) array containing the mixing weights corresponding to
the sample: the dimension [t, , ] is the time index, the dimension [, m, ] indicates the regime, and the dimension
[, , i] indicates the i:th set of simulations.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Lütkepohl H. 2005. New Introduction to Multiple Time Series Analysis, Springer.
McElroy T. 2017. Computation of vector ARMA autocovariances. Statistics and Probability Letters, 124, 92-96.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
stmvar_to_gstmvar estimates a G-StMVAR model based on a StMVAR model that has
large degrees of freedom parameters.
stmvar_to_gstmvar( gsmvar, estimate, calc_std_errors = estimate, maxdf = 100, maxit = 100 )stmvar_to_gstmvar( gsmvar, estimate, calc_std_errors = estimate, maxdf = 100, maxit = 100 )
gsmvar |
an object of class |
estimate |
set |
calc_std_errors |
set |
maxdf |
regimes with degrees of freedom parameter value larger than this will be turned into GMVAR type. |
maxit |
the maximum number of iterations for the variable metric algorithm. Ignored if |
If a StMVAR model contains large estimates for the degrees of freedom parameters,
one should consider switching to the corresponding G-StMAR model that lets the corresponding
regimes to be GMVAR type. stmvar_to_gstmvar does this switch conveniently. Also G-StMVAR models
are supported if some of the StMVAR type regimes have large degrees of freedom paraters.
Note that if the model imposes constraints on the autoregressive parameters, or if a structural model imposes
constraints on the lambda parameters, and the ordering the regimes changes, the constraints are removed from
the model. This is because of the form of the constraints that does not generally allow to switch the ordering
of the regimes. If you wish to keep the constraints, you may construct the resulting G-StMVAR model parameter
vector by hand, redefine your constraints accordingly, build the model with the function GSMVAR, and then
estimate it with the function iterate_more. Alternatively, you can always directly estimate the constrained
G-StMVAR model with the function fitGSMVAR.
Returns an object of class 'gsmvar' defining a G-StMVAR model based on the provided StMVAR (or G-StMVAR)
model with the regimes that had large degrees of freedom parameters changed to GMVAR type.
Muirhead R.J. 1982. Aspects of Multivariate Statistical Theory, Wiley.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
fitGSMVAR, GSMVAR, GIRF, reorder_W_columns,
swap_W_signs, gsmvar_to_sgsmvar
# StMVAR(1, 2), d=2 model: params12t <- c(0.5453, 0.1157, 0.331, 0.0537, -0.0422, 0.7089, 0.4181, 0.0018, 0.0413, 1.6004, 0.4843, 0.1256, -0.0311, -0.6139, 0.7221, 1.2123, -0.0357, 0.1381, 0.8337, 7.5564, 90000) mod12t <- GSMVAR(gdpdef, p=1, M=2, params=params12t, model="StMVAR") mod12t # Switch to the G-StMVAR model: mod12gs <- stmvar_to_gstmvar(mod12t) mod12gs# StMVAR(1, 2), d=2 model: params12t <- c(0.5453, 0.1157, 0.331, 0.0537, -0.0422, 0.7089, 0.4181, 0.0018, 0.0413, 1.6004, 0.4843, 0.1256, -0.0311, -0.6139, 0.7221, 1.2123, -0.0357, 0.1381, 0.8337, 7.5564, 90000) mod12t <- GSMVAR(gdpdef, p=1, M=2, params=params12t, model="StMVAR") mod12t # Switch to the G-StMVAR model: mod12gs <- stmvar_to_gstmvar(mod12t) mod12gs
swap_parametrization swaps the parametrization of a GMVAR, StMVAR or G-StMVAR, model
to "mean" if the current parametrization is "intercept", and vice versa.
swap_parametrization(gsmvar)swap_parametrization(gsmvar)
gsmvar |
an object of class |
swap_parametrization is a convenient tool if you have estimated the model in
"intercept"-parametrization, but wish to work with "mean"-parametrization in the future, or vice versa.
In gmvarkit, the approximate standard errors are only available for parametrized parameters.
Returns an object of class 'gsmvar' defining the specified reduced form or structural GMVAR,
StMVAR, or G-StMVAR model. Can be used to work with other functions provided in gmvarkit.
Note that the first autocovariance/correlation matrix in $uncond_moments is for the lag zero,
the second one for the lag one, etc.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Kalliovirta L. and Saikkonen P. 2010. Reliable Residuals for Multivariate Nonlinear Time Series Models. Unpublished Revision of HECER Discussion Paper No. 247.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
fitGSMVAR, GSMVAR, iterate_more, update_numtols
# GMVAR(2, 2), d=2 model with mean-parametrization: params22 <- c(0.869, 0.549, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.576, 1.168, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22, parametrization="mean") mod22 # mean parametrization mod22_2 <- swap_parametrization(mod22) mod22_2 # intercept parametrization # G-StMVAR(2, 1, 1), d=2 model with mean-parametrization: mod22gs <- GSMVAR(gdpdef, p=2, M=c(1, 1), params=c(params22, 10), model="G-StMVAR", parametrization="mean") mod22gs # mean parametrization mod22gs_2 <- swap_parametrization(mod22gs) mod22gs_2 # intercept parametrization # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(p=2, M=2, d=2, params=params22s, structural_pars=list(W=W_22)) mod22s # intercept parametrization mod22s_2 <- swap_parametrization(mod22s) mod22s_2 # mean parametrization# GMVAR(2, 2), d=2 model with mean-parametrization: params22 <- c(0.869, 0.549, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.215, 0.002, 0.03, 0.576, 1.168, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58) mod22 <- GSMVAR(gdpdef, p=2, M=2, params=params22, parametrization="mean") mod22 # mean parametrization mod22_2 <- swap_parametrization(mod22) mod22_2 # intercept parametrization # G-StMVAR(2, 1, 1), d=2 model with mean-parametrization: mod22gs <- GSMVAR(gdpdef, p=2, M=c(1, 1), params=c(params22, 10), model="G-StMVAR", parametrization="mean") mod22gs # mean parametrization mod22gs_2 <- swap_parametrization(mod22gs) mod22gs_2 # intercept parametrization # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(p=2, M=2, d=2, params=params22s, structural_pars=list(W=W_22)) mod22s # intercept parametrization mod22s_2 <- swap_parametrization(mod22s) mod22s_2 # mean parametrization
swap_W_signs swaps all signs in pointed columns a the matrix
of a structural GMVAR, StMVAR, or G-StMVAR model. Consequently, signs in the columns of the B-matrix are also swapped
accordingly.
swap_W_signs(gsmvar, which_to_swap)swap_W_signs(gsmvar, which_to_swap)
gsmvar |
an object of class |
which_to_swap |
a numeric vector of length at most |
All signs in any column of can be swapped without changing the implied reduced form model.
Consequently, also the signs in the columns of the B-matrix are swapped. Note that the sign constraints
imposed on (or the B-matrix) are also swapped in the corresponding columns accordingly.
Also the order of the columns of can be changed (without changing the implied reduced
form model) as long as the order of lambda parameters is also changed accordingly. This can be
done with the function reorder_W_columns.
Returns an object of class 'gsmvar' defining a structural GMVAR, StMVAR, or G-StMVAR model with the modified
structural parameters and constraints.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
@keywords internal
fitGSMVAR, GSMVAR, GIRF, reorder_W_columns,
gsmvar_to_sgsmvar, stmvar_to_gstmvar
# Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(p=2, M=2, d=2, params=params22s, structural_pars=list(W=W_22)) mod22s # The same reduced form model, with signs in the second column of W swapped: swap_W_signs(mod22s, which_to_swap=2) # The same reduced form model, with signs in both column of W swapped: swap_W_signs(mod22s, which_to_swap=1:2) #' # Structural G-StMVAR(2, 1, 1), d=2 model identified with sign-constraints: mod22gss <- GSMVAR(p=2, M=c(1, 1), d=2, params=c(params22s, 10), model="G-StMVAR", structural_pars=list(W=W_22)) mod22gss # The same reduced form model, with signs in the first column of W swapped: swap_W_signs(mod22gss, which_to_swap=1)# Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(p=2, M=2, d=2, params=params22s, structural_pars=list(W=W_22)) mod22s # The same reduced form model, with signs in the second column of W swapped: swap_W_signs(mod22s, which_to_swap=2) # The same reduced form model, with signs in both column of W swapped: swap_W_signs(mod22s, which_to_swap=1:2) #' # Structural G-StMVAR(2, 1, 1), d=2 model identified with sign-constraints: mod22gss <- GSMVAR(p=2, M=c(1, 1), d=2, params=c(params22s, 10), model="G-StMVAR", structural_pars=list(W=W_22)) mod22gss # The same reduced form model, with signs in the first column of W swapped: swap_W_signs(mod22gss, which_to_swap=1)
uncond_moments calculates the unconditional mean, variance, the first p autocovariances,
and the first p autocorrelations of the given GMVAR, StMVAR, or G-StMVAR process.
uncond_moments(gsmvar)uncond_moments(gsmvar)
gsmvar |
an object of class |
The unconditional moments are based on the stationary distribution of the process.
Returns a list with three components:
$uncond_meana length d vector containing the unconditional mean of the process.
$autocovsan array containing the lag 0,1,...,p autocovariances of
the process. The subset [, , j] contains the lag j-1 autocovariance matrix (lag zero for the variance).
$autocorsthe autocovariance matrices scaled to autocorrelation matrices.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Lütkepohl H. 2005. New Introduction to Multiple Time Series Analysis, Springer.
McElroy T. 2017. Computation of vector ARMA autocovariances. Statistics and Probability Letters, 124, 92-96.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
Other moment functions:
cond_moments(),
get_regime_autocovs(),
get_regime_means()
# GMVAR(1,2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) uncond_moments(mod12) # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) mod22s uncond_moments(mod22s)# GMVAR(1,2), d=2 model: params12 <- c(0.55, 0.112, 0.344, 0.055, -0.009, 0.718, 0.319, 0.005, 0.03, 0.619, 0.173, 0.255, 0.017, -0.136, 0.858, 1.185, -0.012, 0.136, 0.674) mod12 <- GSMVAR(gdpdef, p=1, M=2, params=params12) uncond_moments(mod12) # Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(gdpdef, p=2, M=2, params=params22s, structural_pars=list(W=W_22)) mod22s uncond_moments(mod22s)
update_numtols updates the stationarity and positive definiteness
numerical tolerances of an existing class 'gsmvar' model.
update_numtols(gsmvar, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08)update_numtols(gsmvar, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08)
gsmvar |
an object of class |
stat_tol |
numerical tolerance for stationarity of the AR parameters: if the "bold A" matrix of any regime
has eigenvalues larger that |
posdef_tol |
numerical tolerance for positive definiteness of the error term covariance matrices: if the error term covariance matrix of any regime has eigenvalues smaller than this, the model is classified as not satisfying positive definiteness assumption. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error. |
df_tol |
the parameter vector is considered to be outside the parameter space if all degrees of
freedom parameters are not larger than |
All signs in any column of can be swapped without changing the implied reduced form model.
Consequently, also the signs in the columns of the B-matrix are swapped. Note that the sign constraints
imposed on (or the B-matrix) are also swapped in the corresponding columns accordingly.
Also the order of the columns of can be changed (without changing the implied reduced
form model) as long as the order of lambda parameters is also changed accordingly. This can be
done with the function reorder_W_columns.
Returns an object of class 'gsmvar' defining a structural GSMVAR model with the modified
structural parameters and constraints.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
@keywords internal
fitGSMVAR, GSMVAR, GIRF, reorder_W_columns,
gsmvar_to_sgsmvar, stmvar_to_gstmvar
# Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(p=2, M=2, d=2, params=params22s, structural_pars=list(W=W_22)) mod22s # Update numerical tolerances: mod22s <- update_numtols(mod22s, stat_tol=1e-4, posdef_tol=1e-9, df_tol=1e-10) mod22s # The same model# Structural GMVAR(2, 2), d=2 model identified with sign-constraints: params22s <- c(0.36, 0.121, 0.484, 0.072, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005, 0.083, 0.299, 0.218, 0.02, -0.119, 0.722, 0.093, 0.032, 0.044, 0.191, 0.057, 0.172, -0.46, 0.016, 3.518, 5.154, 0.58) W_22 <- matrix(c(1, 1, -1, 1), nrow=2, byrow=FALSE) mod22s <- GSMVAR(p=2, M=2, d=2, params=params22s, structural_pars=list(W=W_22)) mod22s # Update numerical tolerances: mod22s <- update_numtols(mod22s, stat_tol=1e-4, posdef_tol=1e-9, df_tol=1e-10) mod22s # The same model
A quarterly U.S. data covering the period from 1954Q3 to 2021Q4 (270 observations) and consisting four variables: the log-difference of real GDP, the log-difference of GDP implicit price deflator, the log-difference of producer price index (all commodities), and an interest rate variable. The interest rate variable is the effective federal funds rate from 1954Q3 to 2008Q2 and after that the Wu and Xia (2016) shadow rate, which is not constrained by the zero lower bound and also quantifies unconventional monetary policy measures. The log-differences of the GDP, GDP deflator, and producer price index are multiplied by hundred. This data is used in Virolainen (forthcoming).
usamonusamon
A numeric matrix of class 'ts' with 270 rows and 4 columns with one time series in each column:
The log-difference of real GDP, https://fred.stlouisfed.org/series/GDPC1.
The log-difference of GDP implicit price deflator, https://fred.stlouisfed.org/series/GDPDEF.
The log-difference of producer price index (all commodities), https://fred.stlouisfed.org/series/PPIACO.
The Federal funds rate from 1954Q3 to 2008Q2 and after that the Wu and Xia (2016) shadow rate, https://fred.stlouisfed.org/series/FEDFUNDS, https://www.atlantafed.org/cqer/research/wu-xia-shadow-federal-funds-rate.
The Federal Reserve Bank of St. Louis database and the Federal Reserve Bank of Atlanta's website
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Wu J. and Xia F. 2016. Measuring the macroeconomic impact of monetary policy at the zero lower bound. Journal of Money, Credit and Banking, 48(2-3): 253-291.
The cyclical component of the log of real GDP was obtained by applying a one-sided Hodrick-Prescott (HP) filter with the standard smoothing parameter lambda=1600. The one-sided filter was obtained from the two-sided HP filter by applying the filter up to horizon t, taking the last observation, and repeating this procedure for the full sample t=1,...,T. In order to allow the series to start from any phase of the cycle, we applied the one-sided filter to the full available sample from 1947Q1 to 2021Q1 before extracting our sample period from it. We computed the two-sided HP filters with the R package lpirfs (Adämmer, 2021)
usamoneusamone
A numeric matrix of class 'ts' with 270 rows and 4 columns with one time series in each column:
The cyclical component of the log of real GDP, https://fred.stlouisfed.org/series/GDPC1.
The log-difference of GDP implicit price deflator, https://fred.stlouisfed.org/series/GDPDEF.
The log-difference of producer price index (all commodities), https://fred.stlouisfed.org/series/PPIACO.
The Federal funds rate from 1954Q3 to 2008Q2 and after that the Wu and Xia (2016) shadow rate, https://fred.stlouisfed.org/series/FEDFUNDS, https://www.atlantafed.org/cqer/research/wu-xia-shadow-federal-funds-rate.
The Federal Reserve Bank of St. Louis database and the Federal Reserve Bank of Atlanta's website
Adämmer P. 2021. lprfs: Local Projections Impulse Response Functions. R package version: 0.2.0, https://CRAN.R-project.org/package=lpirfs.
Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.
Wu J. and Xia F. 2016. Measuring the macroeconomic impact of monetary policy at the zero lower bound. Journal of Money, Credit and Banking, 48(2-3): 253-291.
Wald_test performs a Wald test for a GMVAR, StMVAR, or G-StMVAR model
Wald_test(gsmvar, A, c, custom_h = NULL)Wald_test(gsmvar, A, c, custom_h = NULL)
gsmvar |
an object of class |
A |
a size |
c |
a length |
custom_h |
a numeric vector with the same length as |
Denoting the true parameter value by , we test the null hypothesis .
Under the null, the test statistic is asymptotically -distributed with
(=nrow(A)) degrees of freedom. The parameter is assumed to have the same form as in
the model supplied in the argument gsmvar and it is presented in the documentation of the argument
params in the function GSMVAR (see ?GSMVAR).
Finally, note that this function does not check whether the specified constraints are feasible (e.g. whether the implied constrained model would be stationary or have positive definite error term covariance matrices).
A list with class "hypotest" containing the test results and arguments used to calculate the test.
Buse A. (1982). The Likelihood Ratio, Wald, and Lagrange Multiplier Tests: An Expository Note. The American Statistician, 36(3a), 153-157.
LR_test, Rao_test, fitGSMVAR, GSMVAR, diagnostic_plot,
profile_logliks, quantile_residual_tests, cond_moment_plot
# Structural GMVAR(2, 2), d=2 model with recursive identification W22 <- matrix(c(1, NA, 0, 1), nrow=2, byrow=FALSE) fit22s <- fitGSMVAR(gdpdef, p=2, M=2, structural_pars=list(W=W22), ncalls=1, seeds=2) fit22s # Test whether the lambda parameters (of the second regime) are identical # (due to the zero constraint, the model is identified under the null): # fit22s has parameter vector of length 26 with the lambda parameters # in elements 24 and 25. A <- matrix(c(rep(0, times=23), 1, -1, 0), nrow=1, ncol=26) c <- 0 Wald_test(fit22s, A=A, c=c) # Test whether the off-diagonal elements of the first regime's first # AR coefficient matrix (A_11) are both zero: # fit22s has parameter vector of length 26 and the off-diagonal elements # of the 1st regime's 1st AR coefficient matrix are in the elements 6 and 7. A <- rbind(c(rep(0, times=5), 1, rep(0, times=20)), c(rep(0, times=6), 1, rep(0, times=19))) c <- c(0, 0) Wald_test(fit22s, A=A, c=c)# Structural GMVAR(2, 2), d=2 model with recursive identification W22 <- matrix(c(1, NA, 0, 1), nrow=2, byrow=FALSE) fit22s <- fitGSMVAR(gdpdef, p=2, M=2, structural_pars=list(W=W22), ncalls=1, seeds=2) fit22s # Test whether the lambda parameters (of the second regime) are identical # (due to the zero constraint, the model is identified under the null): # fit22s has parameter vector of length 26 with the lambda parameters # in elements 24 and 25. A <- matrix(c(rep(0, times=23), 1, -1, 0), nrow=1, ncol=26) c <- 0 Wald_test(fit22s, A=A, c=c) # Test whether the off-diagonal elements of the first regime's first # AR coefficient matrix (A_11) are both zero: # fit22s has parameter vector of length 26 and the off-diagonal elements # of the 1st regime's 1st AR coefficient matrix are in the elements 6 and 7. A <- rbind(c(rep(0, times=5), 1, rep(0, times=20)), c(rep(0, times=6), 1, rep(0, times=19))) c <- c(0, 0) Wald_test(fit22s, A=A, c=c)