Package 'MSTest' reference manual

Title:	Hypothesis Testing for Markov Switching Models
Description:	Implementation of hypothesis testing procedures described in Hansen (1992) <doi:10.1002/jae.3950070506>, Carrasco, Hu, & Ploberger (2014) <doi:10.3982/ECTA8609>, Dufour & Luger (2017) <doi:10.1080/07474938.2017.1307548>, and Rodriguez Rondon & Dufour (2022) <https://grodriguezrondon.com/files/RodriguezRondon_Dufour_MonteCarlo_LikelihoodRatioTest_MarkovSwitchingModels.pdf> that can be used to identify the number of regimes in Markov switching models.
Authors:	Gabriel Rodriguez Rondon [cre, aut], Jean-Marie Dufour [aut]
Maintainer:	Gabriel Rodriguez Rondon <[email protected]>
License:	GPL (>= 2)
Version:	0.1.2
Built:	2024-09-29 06:21:21 UTC
Source:	CRAN

Testing Markov Switching Models

Description

This package implements hypothesis testing procedures that can be used to identify the number of regimes in a Markov-Switching model.

Author(s)

Gabriel Rodriguez Rondon, [email protected] (Maintainer)

Jean-Marie Dufour, [email protected]

References

Carrasco, Marine, Liang Hu, and Werner Ploberger. 2014. “Optimal test for Markov switching parameters.” Econometrica 82 (2): 765–784.

Dempster, A. P., N. M. Laird, and D. B. Rubin. 1977. “Maximum Likelihood from Incomplete Data via the EM Algorithm.” Journal of the Royal Statistical Society. Series B 39 (1): 1–38..

Dufour, Jean-Marie, and Richard Luger. 2017. “Identification-robust moment-based tests for Markov switching in autoregressive models.” Econometric Reviews 36 (6-9): 713–727.

Kasahara, Hiroyuk, and Katsum Shimotsu. 2018. “Testing the number of regimes in Markov regime switching models.” arXiv preprint arXiv:1801.06862.

Krolzig, Hans-Martin. 1997. “The markov-switching vector autoregressive model.”. Springer.

Hamilton, James D. 1989. “A new approach to the economic analysis of nonstationary time series and the business cycle.” Econometrica 57 (2): 357–384.

Hamilton, James D. 1994. "Time series analysis". Princeton university press.

Hansen, Bruce E. 1992. “The likelihood ratio test under nonstandard conditions: testing the Markov switching model of GNP.” Journal of applied Econometrics 7 (S1): S61–S82.

Rodriguez Rondon, Gabriel and Jean-Marie Dufour. 2022. "Simulation-Based Inference for Markov Switching Models” JSM Proceedings, Business and Economic Statistics Section: American Statistical Association.

Rodriguez Rondon, Gabriel and Jean-Marie Dufour. 2022. “Monte Carlo Likelihood Ratio Tests for Markov Switching Models.” Unpublished manuscript.

Rodriguez Rondon, Gabriel and Jean-Marie Dufour. 2022. “MSTest: An R-package for Testing Markov-Switching Models.” Unpublished manuscript.

Qu, Zhongjun, and Fan Zhuo. 2021. “Likelihood Ratio-Based Tests for Markov Regime Switching.” The Review of Economic Studies 88 (2): 937–968.

Autoregressive Model

Description

This function estimates an autoregresive model with p lags. This can be used for the null hypothesis of a linear model against an alternative hypothesis of a Markov switching autoregressive model with k regimes.

Usage

ARmdl(Y, p, control = list())
ARmdl(Y, p, control = list())

Arguments

Y

A (T x 1) matrix of observations.

p

Integer determining the number of autoregressive lags.

control

List with model options including:

const: Boolean determining whether to estimate model with constant if TRUE or not if FALSE. Default is TRUE.
getSE: Boolean determining whether to compute standard errors of parameters if TRUE or not if FALSE. Default is TRUE.

Value

List of class ARmdl (S3 object) with model attributes including:

y: a (T-p x 1) matrix of observations.
X: a (T-p x p + const) matrix of lagged observations with a leading column of 1s if const=TRUE or not if const=FALSE.
x: a (T-p x p) matrix of lagged observations.
fitted: a (T-p x 1) matrix of fitted values.
resid: a (T-p x 1) matrix of residuals.
inter: estimated intercept of the process.
mu: estimated mean of the process.
coef: coefficient estimates. First value is the intercept (i.e., not mu) if const=TRUE. This is the same as phi if const=FALSE.
intercept: estimate of intercept.
phi: estimates of autoregressive coefficients.
stdev: estimated standard deviation of the process.
sigma: estimated variance of the process.
theta: vector containing: mu, sigma, and phi.
theta_mu_ind: vector indicating location of mean with 1 and 0 otherwise.
theta_sig_ind: vector indicating location of variance with 1 and 0 otherwise.
theta_var_ind: vector indicating location of variance with 1 and 0 otherwise. This is the same as theta_sig_ind in ARmdl.
theta_phi_ind: vector indicating location of autoregressive coefficients with 1 and 0 otherwise.
stationary: Boolean indicating if process is stationary if TRUE or non-stationary if FALSE.
n: number of observations after lag transformation (i.e., n = T-p).
p: number of autoregressive lags.
q: number of series. This is always 1 in ARmdl.
k: number of regimes. This is always 1 in ARmdl.
control: List with model options used.
logLike: log-likelihood.
AIC: Akaike information criterion.
BIC: Bayesian (Schwarz) information criterion.
Hess: Hessian matrix. Approximated using hessian and only returned if getSE=TRUE.
info_mat: Information matrix. Computed as the inverse of -Hess. If matrix is not PD then nearest PD matrix is obtained using nearest_spd. Only returned if getSE=TRUE.
nearPD_used: Boolean determining whether nearPD function was used on info_mat if TRUE or not if FALSE. Only returned if getSE=TRUE.
theta_se: standard errors of parameters in theta. Only returned if getSE=TRUE.

Examples

set.seed(1234)
# Define DGP of AR process
mdl_ar <- list(n     = 500, 
               mu    = 5,
               sigma = 2,
               phi   = c(0.5,0.2))

# Simulate process using simuAR() function
y_simu <- simuAR(mdl_ar)

# Set options for model estimation
control <- list(const  = TRUE, 
                getSE  = TRUE)

# Estimate model
y_ar_mdl <- ARmdl(y_simu$y, p = 2, control)
y_ar_mdl

set.seed(1234)
# Define DGP of AR process
mdl_ar <- list(n     = 500, 
               mu    = 5,
               sigma = 2,
               phi   = c(0.5,0.2))

# Simulate process using simuAR() function
y_simu <- simuAR(mdl_ar)

# Set options for model estimation
control <- list(const  = TRUE, 
                getSE  = TRUE)

# Estimate model
y_ar_mdl <- ARmdl(y_simu$y, p = 2, control)
y_ar_mdl

Bootstrap Likelihood Ratio Test

Description

This function performs the bootstrap likelihood ratio test discussed in Qu & Zhuo (2021) and Kasahara & Shimotsu (2018).

Usage

BootLRTest(Y, p, k0, k1, control = list())
BootLRTest(Y, p, k0, k1, control = list())

Arguments

`Y`	Series to be tested. Must be a (`T x q`) matrix.
`p`	Number of autoregressive lags. Must be greater than or equal to 0.
`k0`	Number of regimes under null hypothesis. Must be greater than or equal to 1.
`k1`	Number of regimes under alternative hypothesis. Must be greater than `k0`.
`control`	List with test procedure options including: B: Integer determining the number of bootstrap simulations. Default is set to `999`. burnin: Number of simulated observations to remove from beginning. Default is `100`. converge_check: String of NULL determining if convergence of model(s) should be verified. Allowed inputs are: "null", "alt", "both", or `NULL`. If `NULL` (default) no model convergence is verified. workers: Integer determining the number of workers to use for parallel computing version of test. Note that parallel pool must already be open. Default is `0`. mdl_h0_control: List with restricted model options. See `Nmdl`, `ARmdl`, `VARmdl`, `HMmdl`, `MSARmdl`, or `MSVARmdl` documentation for available and default values. mdl_h1_control: List with unrestricted model options. See `HMmdl`, `MSARmdl`, or `MSVARmdl` documentation for available and default values.

Value

List of class LMCLRTest (S3 object) with attributes including:

mdl_h0: List with restricted model attributes. See Nmdl, ARmdl, VARmdl, HMmdl, MSARmdl, or MSVARmdl documentation for return values.
mdl_h0: List with unrestricted model attributes. See HMmdl, MSARmdl, or MSVARmdl documentation for return values.
LRT_0: Value of test statistic from observed data.
LRN: A (N x 1) vector of test statistics from data simulated under the null hypothesis.
pval: P-value of Local Monte Carlo Likelihood Ratio Test.
LRN_cv: Vector with 90%, 95%, and 99% Monte Carlo critical values (from vector LRN).
control: List with test procedure options used.

References

Qu, Zhongjun, and Fan Zhuo. 2021. “Likelihood Ratio-Based Tests for Markov Regime Switching.” The Review of Economic Studies 88 (2): 937–968.

Kasahara, Hiroyuk, and Katsum Shimotsu. 2018. “Testing the number of regimes in Markov regime switching models.” arXiv preprint arXiv:1801.06862.

Rodriguez-Rondon, Gabriel and Jean-Marie Dufour. 2023. “Monte Carlo Likelihood Ratio Tests for Markov Switching Models.” Unpublished manuscript.

Carrasco, Hu, & Ploberger 2010 GNP data

Description

Carrasco, Hu, & Ploberger 2010 GNP data

Usage

chp10GNP
chp10GNP

Format

This data is the extension of the GNP series used in CHP (2014), Econometrica. This series ranges from 1951Q2 to 2010Q4.

DATE: Vector of dates
GNP_logdiff: log difference of US GNP series
GNP: US GNP series

Source

https://www.econometricsociety.org/content/supplement-optimal-test-markov-switching-parameters

References

Carrasco, Marine, Liang Hu, and Werner Ploberger. 2014. “Optimal test for Markov switching parameters.” Econometrica 82 (2): 765–784.

Carrasco, Hu, and Ploberger (2014) parameter stability test

Description

This function performs the CHP (2014) parameter stability test as outline in Carrasco, M., Hu, L. and Ploberger, W. (2014). Original source code can be found here.

Usage

CHPTest(Y, p, control = list())
CHPTest(Y, p, control = list())

Arguments

Y

A (T x 1) matrix of observations.

p

Integer determining the number of autoregressive lags.

control

List with test procedure options including:

N: Integer determining the number of Bootstrap iterations. Default is set to 3000 as in paper.
rho_b: Number determining bounds for distribution of rh0 (i.e. rho ~ [-rho_b,rho_b]).
msvar: Boolean indicator. If TRUE, there is a switch in variance. If FALSE only switch in mean is considered.
getSE: Boolean indicator. If TRUE, standard errors for restricted model are estimated. If FALSE no standard errors are estimated. Default is TRUE.

Value

List of class CHPTest (S3 object) with model attributes including:

mdl_h0: List with restricted model attributes. This will be of class ARmdl (S3 object). See ARmdl.
supTS: supTS test statistic value.
expTS: expTS test statistic value.
supTS_N: A (N x 1) vector with simulated supTS test statistics under null hypothesis.
expTS_N: A (N x 1) vector with simulated expTS test statistics under null hypothesis.
pval_supTS: P-value for supTS version of parameter stability test.
pval_expTS: P-value for expTS version of parameter stability test.
supTS_cv: Vector with 90%, 95%, and 99% bootstrap critical values for supTS version of parameter stability test.
expTS_cv: Vector with 90%, 95%, and 99% bootstrap critical values for expTS version of parameter stability test.
control: List with test procedure options used.

References

Carrasco, Marine, Liang Hu, and Werner Ploberger. 2014. “Optimal test for Markov switching parameters.” Econometrica 82 (2): 765–784.

Examples

# load data used in Hamilton 1989 
y84 <- as.matrix(hamilton84GNP$GNP_logdiff)


# Set test procedure options
control = list(N = 1000, 
               rho_b = 0.7, 
               msvar = FALSE)

# perform test with switch in mean only on Hamilton 1989 data

  mdl_84_msmu <- CHPTest(y84, p = 4, control = control)
  summary(mdl_84_msmu)

# load data used in Hamilton 1989 
y84 <- as.matrix(hamilton84GNP$GNP_logdiff)


# Set test procedure options
control = list(N = 1000, 
               rho_b = 0.7, 
               msvar = FALSE)

# perform test with switch in mean only on Hamilton 1989 data

  mdl_84_msmu <- CHPTest(y84, p = 4, control = control)
  summary(mdl_84_msmu)

Monte Carlo moment-based test for Markov switching model

Description

This function performs the Local Monte Carlo moment-based test for Markov switching autoregressive models proposed in Dufour & Luger (2017).

Usage

DLMCTest(Y, p, control = list())
DLMCTest(Y, p, control = list())

Arguments

Y

Series to be tested

p

Number of autoregressive lags.

control

List with test procedure options including:

N: Integer determining the number of Monte Carlo simulations. Default is set to 99 as in paper.
simdist_N: Integer determining the number of simulations for CDF distribution approximation. Default is set to 10000.
getSE: Boolean indicator. If TRUE, standard errors for restricted model are estimated. If FALSE no standard errors are estimated. Default is TRUE.

Value

List of class DLMCTest (S3 object) with attributes including:

mdl_h0: List with restricted model attributes. This will be of class ARmdl if p>0 or Nmdl otherwise (S3 objects). See ARmdl or Nmdl.
theta: Value of nuisance parameters. Specifically, these are the consistent estimates of nuisance parameters as discussed in Dufour & Luger (2017) LMC procedure.
S0: A (1 x 4)) matrix containing the four moment-based test statistics defined in (11) - (14) in Dufour & Luger (2017).
F0_min: Test statistic value for min version of Local Monte Carlo moment-based test.
F0_prod: Test statistic value for prod version of Local Monte Carlo moment-based test.
FN_min: A (N x 1) vector with simulated test statistics for min version of Local Monte Carlo moment-based test under null hypothesis.
FN_prod: A (N x 1) vector with simulated test statistics for prod version of Local Monte Carlo moment-based test under null hypothesis.
pval_min: P-value for min version of Local Monte Carlo moment-based test.
pval_prod: P-value for prod version of Local Monte Carlo moment-based test.
FN_min_cv: Vector with 90%, 95%, and 99% Monte Carlo critical values for min version of Local Monte Carlo moment-based test.
FN_prod_cv: Vector with 90%, 95%, and 99% Monte Carlo critical values for prod version of Local Monte Carlo moment-based test.
control: List with test procedure options used.

References

Dufour, J. M., & Luger, R. 2017. "Identification-robust moment-based tests for Markov switching in autoregressive models." Econometric Reviews, 36(6-9), 713-727.

Examples

set.seed(1234)
# load data used in Hamilton 1989 and extended data used in CHP 2014 
y84 <- as.matrix(hamilton84GNP$GNP_logdiff)
y10 <- as.matrix(chp10GNP$GNP_logdiff)

# Set test procedure options
lmc_control = list(N = 99,
                   simdist_N = 10000,
                   getSE = TRUE)

# perform test on Hamilton 1989 data
lmc_gnp84 <- DLMCTest(y84, p = 4, control = lmc_control)
summary(lmc_gnp84)

set.seed(1234)
# load data used in Hamilton 1989 and extended data used in CHP 2014 
y84 <- as.matrix(hamilton84GNP$GNP_logdiff)
y10 <- as.matrix(chp10GNP$GNP_logdiff)

# Set test procedure options
lmc_control = list(N = 99,
                   simdist_N = 10000,
                   getSE = TRUE)

# perform test on Hamilton 1989 data
lmc_gnp84 <- DLMCTest(y84, p = 4, control = lmc_control)
summary(lmc_gnp84)

Maximized Monte Carlo moment-based test for Markov switching model

Description

This function performs the maximized Monte Carlo moment-based test for Markov switching autoregressive models proposed in Dufour & Luger (2017).

Usage

DLMMCTest(Y, p, control = list())
DLMMCTest(Y, p, control = list())

Arguments

Y

Series to be tested

p

Number of autoregressive lags.

control

List with test procedure options including:

N: Integer determining the number of Monte Carlo simulations. Default is set to 99 as in paper.
simdist_N: Integer determining the number of simulations for CDF distribution approximation. Default is set to 10000.
getSE: Boolean indicator. If TRUE, standard errors for restricted model are estimated. If FALSE no standard errors are estimated. Default is TRUE.
eps: Fixed positive constant that does not depend on T used to determine lower and upper bounds on consistent set considered for nuisance parameter space.
CI_union: Boolean indicator determining if union between eps and confidence interval is used to determine lower and upper bound on consistent set considered for nuisance parameter space. If TRUE union is used and if FALSE only eps is used. Note that if standard errors obtained are not finite then only eps is used. Default is FALSE.
lambda: Numeric value for penalty on stationary constraint not being met. Default is 100.
stationary_ind: Boolean indicator determining if only stationary solutions should be considered if TRUE or any solution can be considered if FALSE. Default is TRUE.
phi_low: Vector with lower bound for autoregressive parameters when optimizing. Default is NULL.
phi_upp: Vector with upper bound for autoregressive parameters when optimizing. Default is NULL.
optim_type: String determining type of numerical optimization algorithm to use. Available options are: "pso", ""GenSA", "GA". Default is "GenSA".
silence: Boolean indicator determining if optimization updates should be silenced if TRUE or not if FALSE. Default is FALSE.
threshold_stop: Numeric value determining the maximum possible p-value attainable. Default is 1.
type_control: List containing other optimization options specific to the numerical optimization algorithm used. This includes maximum number of iterations which is 200 b y default. For other options see documentation of numerical algorithm chosen.

Value

List of class DLMCTest (S3 object) with attributes including:

mdl_h0: List with restricted model attributes. This will be of class ARmdl if p>0 or Nmdl otherwise (S3 objects). See ARmdl or Nmdl.
theta_max_min: Value of nuisance parameters when min version of p-value is maximized as discussed in Dufour & Luger (2017) MMC procedure.
theta_max_prod: Value of nuisance parameters when prod version of p-value is maximized as discussed in Dufour & Luger (2017) MMC procedure.
theta_low: Lower bound on nuisance parameter values used when searching for maximum p-value.
theta_upp: Upper bound on nuisance parameter values used when searching for maximum p-value.
S0_min: A (1 x 4)) matrix containing the four moment-based test statistics defined in (11) - (14) in Dufour & Luger (2017) when theta_min is used.
S0_prod: A (1 x 4)) matrix containing the four moment-based test statistics defined in (11) - (14) in Dufour & Luger (2017) when theta_prod is used.
F0_min: Test statistic value for min version of Maximized Monte Carlo moment-based test.
F0_prod: Test statistic value for prod version of Maximized Monte Carlo moment-based test.
FN_min: A (N x 1) vector with simulated test statistics for min version of Maximized Monte Carlo moment-based test under null hypothesis.
FN_prod: A (N x 1) vector with simulated test statistics for prod version of Maximized Monte Carlo moment-based test under null hypothesis.
pval_min: Maximum p-value for min version of Maximized Monte Carlo moment-based test.
pval_prod: Maximum p-value for prod version of Local Monte Carlo moment-based test.
FN_min_cv: Vector with 90%, 95%, and 99% Monte Carlo critical values for min version of Local Monte Carlo moment-based test.
FN_prod_cv: Vector with 90%, 95%, and 99% Monte Carlo critical values for prod version of Local Monte Carlo moment-based test.
control: List with test procedure options used.
optim_min_output: List with optimization output for min version of Maximized Monte Carlo moment-based test.
optim_prod_output: List with optimization output for prod version of Maximized Monte Carlo moment-based test.

References

Dufour, J. M., & Luger, R. 2017. "Identification-robust moment-based tests for Markov switching in autoregressive models." Econometric Reviews, 36(6-9), 713-727.

Examples

set.seed(1234)
# load data used in Hamilton 1989 and extended data used in CHP 2014 
y84 <- as.matrix(hamilton84GNP$GNP_logdiff)
y10 <- as.matrix(chp10GNP$GNP_logdiff)

# Set test procedure options
mmc_control <- list(N = 99,
                    simdist_N = 10000,
                    getSE = TRUE,
                    eps = 0.0000001, 
                    CI_union = TRUE,
                    lambda = 100,
                    stationary_ind = TRUE,
                    optim_type = "GenSA",
                    silence = FALSE,
                    threshold_stop = 1,
                    type_control = list(maxit = 200))


# perform test on Hamilton 1989 data

  mmc_gnp84 <- DLMMCTest(y84, p = 4, control = mmc_control)
  summary(mmc_gnp84)


set.seed(1234)
# load data used in Hamilton 1989 and extended data used in CHP 2014 
y84 <- as.matrix(hamilton84GNP$GNP_logdiff)
y10 <- as.matrix(chp10GNP$GNP_logdiff)

# Set test procedure options
mmc_control <- list(N = 99,
                    simdist_N = 10000,
                    getSE = TRUE,
                    eps = 0.0000001, 
                    CI_union = TRUE,
                    lambda = 100,
                    stationary_ind = TRUE,
                    optim_type = "GenSA",
                    silence = FALSE,
                    threshold_stop = 1,
                    type_control = list(maxit = 200))


# perform test on Hamilton 1989 data

  mmc_gnp84 <- DLMMCTest(y84, p = 4, control = mmc_control)
  summary(mmc_gnp84)

Hamilton 1984 & Hansen 1992 GNP data

Description

Hamilton 1984 & Hansen 1992 GNP data

Usage

hamilton84GNP
hamilton84GNP

Format

This data set is used in Hansen (1992) to test the US GNP model proposed by Hamilton (1989). This series ranges from 1951Q2 to 1984Q4.

DATE: Vector of dates
GNP_logdiff: US GNP log difference
GNP: US GNP series

Source

https://www.ssc.wisc.edu/~bhansen/progs/jae_92.html

References

Hansen, Bruce E. 1992. “The likelihood ratio test under nonstandard conditions: testing the Markov switching model of GNP.” Journal of applied Econometrics 7 (S1): S61–S82.

Hamilton, James D. 1989. “A new approach to the economic analysis of nonstationary time series and the business cycle.” Econometrica 57 (2): 357–384.

Hansen (1992) likelihood ratio test

Description

This function performs Hansen's likelihood ratio test as described in Hansen (1992). Original source code can be found here.

Usage

HLRTest(Y, p, control = list())
HLRTest(Y, p, control = list())

Arguments

Y

A (T x 1) matrix of observations.

p

Integer determining the number of autoregressive lags.

control

List with test procedure options including:

ix: List of Markov Switching parameters. 1 = just mean c(1,2) = mean and first param, (default: 1).
msvar: Boolean indicator. If TRUE, there is a switch in variance. If FALSE only switch in mean is considered. Default is FALSE.
qbound: Indicator that bounds q by 1-p (default: FALSE).
gridsize: Integer determining the number of grid points for markov switching parameters. Default is 20.
pgrid_from: Double determining the initial grid point for transition probabilities. Default is 0.1.
pgrid_by: Double determining the step size for grid points of transition probabilities. This, along with p_gridsize will determine the bounds of search space. Default is 0.075.
pgrid_to: Double determining the end grid point for transition probabilities. Default is 0.925.
mugrid_from: Double determining the minimum value of mean in second regime. Default is 0.1.
mugrid_by: Double determining the step size for grid points of mean in second regime. This, along with gridsize will determine the max value of mean in second regime. Default is 0.1.
siggrid_from: Double determining the minimum value of sigma in second regime (if msvar = TRUE). Default is 0.1.
siggrid_by: Double determining the step size for grid points of sigma in second regime. This, along with gridsize will determine the max value of sigma in second regime. Default is 0.1.
N: Integer determining the number of replications. Default is 1000.
nwband: Integer determining maximum bandwidth in Bartlett kernel. Critical values and p-values are returned for each bandwidth from 0:nwband as suggested in Hansen (1996). Default is 4.
theta_null_low: Vector determining lower bound on parameters under the null hypothesis. Length of vector should be number of model coefficients + 1 for variance. Default is to only bound variance at 0.01.
theta_null_upp: Vector determining upper bound on parameters under the null hypothesis. Length of vector should be number of model coefficients + 1 for variance. Default is to no bounds (i.e. Inf).
optim_method: String determining the type of optimization procedure used. Allowed options are "gp-optim" for general purpose optimization using optim from stats or "nl-optim" using slsqp from nloptr. Default is "gp-optim".

Value

List of class HLRTest (S3 object) with model attributes including:

mdl_h0: List with restricted model attributes. This will be of class ARmdl (S3 object). See ARmdl.
LR0: Likelihood ratio test statistic value.
LRN: A (N x 1) vector with simulated LRT statistics under null hypothesis.
pval: P-value.
LR_cv: A (nwband x 3) matrix with 90%, 95%, and 99% critical values in each column respectively.
coef: Vector of coefficients from restricted model and grid search that maximized standardized LRT.
control: List with test procedure options used.

References

Hansen, Bruce E. 1992. “The likelihood ratio test under nonstandard conditions: testing the Markov switching model of GNP.” Journal of applied Econometrics 7 (S1): S61–S82.

Hansen, Bruce E. 1996. “Erratum: The likelihood ratio test under nonstandard conditions: testing the Markov switching model of GNP.” Journal of applied Econometrics 7 (S1): S61–S82.

Examples

# --------------------------- Use simulated process ----------------------------
set.seed(1234)
# Define DGP of MS AR process
mdl_ms2 <- list(n     = 200, 
                mu    = c(5,1),
                sigma = c(1,1),
                phi   = c(0.5),
                k     = 2,
                P     = rbind(c(0.90, 0.10),
                              c(0.10, 0.90)))

# Simulate process using simuMSAR() function
y_ms_simu <- simuMSAR(mdl_ms2)

hlrt_control  <- list(ix          = 1, 
                      gridsize    = 5,
                      p_gridsize  = 9,
                      p_stepsize  = 0.1,
                      mugrid_from = 0,
                      mugrid_by   = 1)


  hlrt <- HLRTest(y_ms_simu$y, p = 1, control = hlrt_control)
  summary(hlrt)


# --------------------------- Use simulated process ----------------------------
set.seed(1234)
# Define DGP of MS AR process
mdl_ms2 <- list(n     = 200, 
                mu    = c(5,1),
                sigma = c(1,1),
                phi   = c(0.5),
                k     = 2,
                P     = rbind(c(0.90, 0.10),
                              c(0.10, 0.90)))

# Simulate process using simuMSAR() function
y_ms_simu <- simuMSAR(mdl_ms2)

hlrt_control  <- list(ix          = 1, 
                      gridsize    = 5,
                      p_gridsize  = 9,
                      p_stepsize  = 0.1,
                      mugrid_from = 0,
                      mugrid_by   = 1)


  hlrt <- HLRTest(y_ms_simu$y, p = 1, control = hlrt_control)
  summary(hlrt)

Hidden Markov model

Description

This function estimates a Hidden Markov model with k regimes.

Usage

HMmdl(Y, k, control = list())
HMmdl(Y, k, control = list())

Arguments

Y

a (T x q) matrix of observations.

k

integer determining the number of regimes to use in estimation. Must be greater than or equal to 2.

control

List with model options including:

getSE: Boolean. If TRUE standard errors are computed and returned. If FALSE standard errors are not computed. Default is TRUE.
msmu: Boolean. If TRUE model is estimated with switch in mean. If FALSE model is estimated with constant mean. Default is TRUE.
msvar: Boolean. If TRUE model is estimated with switch in variance. If FALSE model is estimated with constant variance. Default is TRUE.
init_theta: vector of initial values. vector must contain (1 x q) vector mu, vech(sigma), and vec(P) where sigma is a (q x q) covariance matrix.This is optional. Default is NULL, in which case initVals_MSARmdl is used to generate initial values.
method: string determining which method to use. Options are 'EM' for EM algorithm or 'MLE' for Maximum Likelihood Estimation.
maxit: integer determining the maximum number of EM iterations.
thtol: double determining the convergence criterion for the absolute difference in parameter estimates theta between iterations. Default is 1e-6.
maxit_converge: integer determining the maximum number of initial values attempted until solution is finite. For example, if parameters in theta or logLike are NaN another set of initial values (up to maxit_converge) is attempted until finite values are returned. This does not occur frequently for most types of data but may be useful in some cases. Once finite values are obtained, this counts as one iteration towards use_diff_init. Default is 500.
use_diff_init: integer determining how many different initial values to try (that do not return NaN; see maxit_converge). Default is 1.
mle_variance_constraint: double used to determine the lower bound on the smallest eigenvalue for the covariance matrix of each regime. Default is 1e-3.
mle_theta_low: Vector with lower bounds on parameters (Used only if method = "MLE"). Default is NULL.
mle_theta_upp: Vector with upper bounds on parameters (Used only if method = "MLE"). Default is NULL.

Value

List of class HMmdl (S3 object) with model attributes including:

y: a (T x q) matrix of observations.
fitted: a (T x q) matrix of fitted values.
resid: a (T x q) matrix of residuals.
mu: a (k x q) matrix of estimated means of each process.
stdev: List with k (q x 1) vector of estimated standard deviation of each process.
sigma: List with k (q x q) estimated covariance matrix.
theta: vector containing: mu and vech(sigma).
theta_mu_ind: vector indicating location of mean with 1 and 0 otherwise.
theta_sig_ind: vector indicating location of variance and covariances with 1 and 0 otherwise.
theta_var_ind: vector indicating location of variances with 1 and 0 otherwise.
theta_var_ind: vector indicating location of transition matrix elements with 1 and 0 otherwise.
n: number of observations (same as T).
q: number of series.
k: number of regimes in estimated model.
P: a (k x k) transition matrix.
pinf: a (k x 1) vector with limiting probabilities of each regime.
St: a (T x k) vector with smoothed probabilities of each regime at each time t.
deltath: double with maximum absolute difference in vector theta between last iteration.
iterations: number of EM iterations performed to achieve convergence (if less than maxit).
theta_0: vector of initial values used.
init_used: number of different initial values used to get a finite solution. See description of input maxit_converge.
msmu: Boolean. If TRUE model was estimated with switch in mean. If FALSE model was estimated with constant mean.
msvar: Boolean. If TRUE model was estimated with switch in variance. If FALSE model was estimated with constant variance.
control: List with model options used.
logLike: log-likelihood.
AIC: Akaike information criterion.
BIC: Bayesian (Schwarz) information criterion.
Hess: Hessian matrix. Approximated using hessian and only returned if getSE=TRUE.
info_mat: Information matrix. Computed as the inverse of -Hess. If matrix is not PD then nearest PD matrix is obtained using nearest_spd. Only returned if getSE=TRUE.
nearPD_used: Boolean determining whether nearPD function was used on info_mat if TRUE or not if FALSE. Only returned if getSE=TRUE.
theta_se: standard errors of parameters in theta. Only returned if getSE=TRUE.
trace: List with Lists of estimation output for each initial value used due to use_diff_init > 1.

References

Dempster, A. P., N. M. Laird, and D. B. Rubin. 1977. “Maximum Likelihood from Incomplete Data via the EM Algorithm.” Journal of the Royal Statistical Society. Series B 39 (1): 1–38..

Hamilton, James D. 1990. “Analysis of time series subject to changes in regime.” Journal of econometrics, 45 (1-2): 39–70.

Krolzig, Hans-Martin. 1997. “The markov-switching vector autoregressive model.”. Springer.

Monte Carlo Likelihood Ratio Test

Description

This function performs the Local Monte Carlo likelihood ratio test (LMC-LRT) proposed in Rodriguez-Rondon & Dufour (2022).

Usage

LMCLRTest(Y, p, k0, k1, control = list())
LMCLRTest(Y, p, k0, k1, control = list())

Arguments

`Y`	Series to be tested. Must be a (`T x q`) matrix.
`p`	Number of autoregressive lags. Must be greater than or equal to 0.
`k0`	Number of regimes under null hypothesis. Must be greater than or equal to 1.
`k1`	Number of regimes under alternative hypothesis. Must be greater than `k0`.
`control`	List with test procedure options including: N: Integer determining the number of Monte Carlo simulations. Default is set to `99` as in paper. burnin: Number of simulated observations to remove from beginning. Default is `100`. converge_check: String of NULL determining if convergence of model(s) should be verified. Allowed inputs are: "null", "alt", "both", or `NULL`. If `NULL` (default) no model convergence is verified. workers: Integer determining the number of workers to use for parallel computing version of test. Note that parallel pool must already be open. Default is `0`. mdl_h0_control: List with restricted model options. See `Nmdl`, `ARmdl`, `VARmdl`, `HMmdl`, `MSARmdl`, or `MSVARmdl` documentation for available and default values. mdl_h1_control: List with unrestricted model options. See `HMmdl`, `MSARmdl`, or `MSVARmdl` documentation for available and default values. use_diff_init_sim: Value which determines the number of initial values to use when estimating models for null distribution. Default is set to use the same as specified in `mdl_h0_control` and `mdl_h1_control`.

Value

List of class LMCLRTest (S3 object) with attributes including:

mdl_h0: List with restricted model attributes. See Nmdl, ARmdl, VARmdl, HMmdl, MSARmdl, or MSVARmdl documentation for return values.
mdl_h0: List with unrestricted model attributes. See HMmdl, MSARmdl, or MSVARmdl documentation for return values.
LRT_0: Value of test statistic from observed data.
LRN: A (N x 1) vector of test statistics from data simulated under the null hypothesis.
pval: P-value of Local Monte Carlo Likelihood Ratio Test.
LRN_cv: Vector with 90%, 95%, and 99% Monte Carlo critical values (from vector LRN).
control: List with test procedure options used.

References

Rodriguez-Rondon, Gabriel and Jean-Marie Dufour. 2022. "Simulation-Based Inference for Markov Switching Models” JSM Proceedings, Business and Economic Statistics Section: American Statistical Association.

Rodriguez-Rondon, Gabriel and Jean-Marie Dufour. 2023. “Monte Carlo Likelihood Ratio Tests for Markov Switching Models.” Unpublished manuscript.

Examples

set.seed(1234)
# Define DGP of MS AR process
mdl_ms2 <- list(n     = 200, 
                mu    = c(5,10),
                sigma = c(1,4),
                phi   = c(0.5),
                k     = 2,
                P     = rbind(c(0.90, 0.10),
                              c(0.10, 0.90)))

# Simulate process using simuMSAR() function
y_ms_simu <- simuMSAR(mdl_ms2)


# ------ MS-AR example ----- #
# Set test procedure options
lmc_control = list(N = 19,
                   burnin = 100,
                   converge_check = NULL,
                   mdl_h0_control = list(const  = TRUE, 
                                         getSE  = TRUE),
                   mdl_h1_control = list(msmu   = TRUE, 
                                         msvar  = TRUE,
                                         getSE  = TRUE,
                                         method = "EM",
                                         maxit  = 300,
                                         use_diff_init = 1))


  lmctest <- LMCLRTest(y_ms_simu$y, p = 1, k0 = 1 , k1 = 2, lmc_control)
  summary(lmctest)

set.seed(1234)
# Define DGP of MS AR process
mdl_ms2 <- list(n     = 200, 
                mu    = c(5,10),
                sigma = c(1,4),
                phi   = c(0.5),
                k     = 2,
                P     = rbind(c(0.90, 0.10),
                              c(0.10, 0.90)))

# Simulate process using simuMSAR() function
y_ms_simu <- simuMSAR(mdl_ms2)


# ------ MS-AR example ----- #
# Set test procedure options
lmc_control = list(N = 19,
                   burnin = 100,
                   converge_check = NULL,
                   mdl_h0_control = list(const  = TRUE, 
                                         getSE  = TRUE),
                   mdl_h1_control = list(msmu   = TRUE, 
                                         msvar  = TRUE,
                                         getSE  = TRUE,
                                         method = "EM",
                                         maxit  = 300,
                                         use_diff_init = 1))


  lmctest <- LMCLRTest(y_ms_simu$y, p = 1, k0 = 1 , k1 = 2, lmc_control)
  summary(lmctest)

Monte Carlo P-value

Description

This function computes the Monte Carlo P-value.

Usage

MCpval(test_stat, null_vec, type = "geq")
MCpval(test_stat, null_vec, type = "geq")

Arguments

`test_stat`	Test statistic under the alternative (e.g. `S_0`).
`null_vec`	A (`N x 1`) vector with test statistic under the null hypothesis.
`type`	String determining type of test. options are: "geq" for right-tail test, "leq" for left-tail test, "abs" for absolute value test and "two-tail" for two-tail test.

Value

MC p-value of test

References

Dufour, Jean-Marie 2006. "Monte Carlo tests with nuisance parameters: A general approach to finite-sample inference and nonstandard asymptotics". Journal of Econometrics, 133(2), 443-477.

Dufour, Jean-Marie, and Richard Luger. 2017. "Identification-robust moment-based tests for Markov switching in autoregressive models". Econometric Reviews, 36(6-9), 713-727.

Maximized Monte Carlo Likelihood Ratio Test

Description

This function performs the Maximized Monte Carlo likelihood ratio test (MMC-LRT) proposed in Rodriguez-Rondon & Dufour (2022).

Usage

MMCLRTest(Y, p, k0, k1, control = list())
MMCLRTest(Y, p, k0, k1, control = list())

Arguments

`Y`	Series to be tested. Must be a (`T x q`) matrix.
`p`	Number of autoregressive lags. Must be greater than or equal to 0.
`k0`	Number of regimes under null hypothesis. Must be greater than or equal to 1.
`k1`	Number of regimes under alternative hypothesis. Must be greater than `k0`.
`control`	List with test procedure options including: N: Integer determining the number of Monte Carlo simulations. Default is set to `99` as in paper. burnin: Number of simulated observations to remove from beginning. Default is `100`. converge_check: String of NULL determining if convergence of model(s) should be verified. Allowed inputs are: "null", "alt", "both", or `NULL`. If `NULL` (default) no model convergence is verified. workers: Integer determining the number of workers to use for parallel computing version of test. Note that parallel pool must already be open. Default is `0`. type: String that determines the type of optimization algorithm used. Arguments allowed are: `"pso"`, `"GenSA"`, and `"GA"`. Default is `"pso"`. eps: Double determining the constant value that defines a consistent set for search. Default is `0.1`. CI_union: Boolean determining if union of set determined by `eps` and confidence set should be used to define consistent set for search. Default is `TRUE`. lambda: Double determining penalty on nonlinear constraint. Default is `100`. stationary_constraint: Boolean determining if only stationary solutions are considered (if `TRUE`) or not (if `FALSE`). Default is `TRUE`. phi_low: Vector with lower bound for autoregressive parameters when optimizing. Default is `NULL`. phi_upp: Vector with upper bound for autoregressive parameters when optimizing. Default is `NULL`. P_low: Value with lower bound for transition probabilities when optimizing. Default is `0`. P_upp: Value with upper bound for transition probabilities when optimizing. Default is `1`. variance_constraint: Double used to determine the lower bound for variance in parameter set for search. Value should be between `0` and `1` as it is multiplied by consistent point estimates of variances. Default is `0.01` (i.e., `1%` of consistent point estimates. silence: Boolean determining if optimization steps should be silenced (if `TRUE`) or not (if `FALSE`). Default is `FALSE`. threshold_stop: Double determining the global optimum of function. Default is `1`. mdl_h0_control: List with restricted model options. See `Nmdl`, `ARmdl`, `VARmdl`, `HMmdl`, `MSARmdl`, or `MSVARmdl` documentation for available and default values. mdl_h1_control: List with unrestricted model options. See `HMmdl`, `MSARmdl`, or `MSVARmdl` documentation for available and default values. use_diff_init_sim: Value which determines the number of initial values to use when estimating models for null distribution. Default is set to use the same as specified in `mdl_h0_control` and `mdl_h1_control`. type_control: List with optimization algorithm options. See `psoptim`, `GenSA`, `ga`. Default is to set `list(maxit = 200)` so that maximum number of iterations is `200`.

Value

List of class LMCLRTest (S3 object) with attributes including:

mdl_h0: List with restricted model attributes. See Nmdl, ARmdl, VARmdl, HMmdl, MSARmdl, or MSVARmdl documentation for return values.
mdl_h0: List with unrestricted model attributes. See HMmdl, MSARmdl, or MSVARmdl documentation for return values.
LRT_0: Value of test statistic from observed data.
LRN: A (N x 1) vector of test statistics from data simulated under the null hypothesis.
pval: P-value of Local Monte Carlo Likelihood Ratio Test.
LRN_cv: Vector with 90%, 95%, and 99% Monte Carlo critical values (from vector LRN).
control: List with test procedure options used.

References

Rodriguez-Rondon, Gabriel and Jean-Marie Dufour. 2023. “Monte Carlo Likelihood Ratio Tests for Markov Switching Models.” Unpublished manuscript.

Examples


set.seed(1234)
# Define DGP of MS AR process
mdl_ms2 <- list(n     = 200, 
                mu    = c(5,10),
                sigma = c(1,4),
                phi   = c(0.5),
                k     = 2,
                P     = rbind(c(0.90, 0.10),
                              c(0.10, 0.90)))

# Simulate process using simuMSAR() function
y_ms_simu <- simuMSAR(mdl_ms2)


# Set test procedure options
mmc_control = list(N = 19,
                   burnin = 100,
                   converge_check = NULL,
                   eps = 0.1,
                   CI_union = TRUE,
                   silence = FALSE,
                   threshold_stop = 0.05 + 1e-6,
                   type = "pso",
                   maxit = 100,
                   mdl_h0_control = list(const  = TRUE, 
                                         getSE  = TRUE),
                   mdl_h1_control = list(msmu   = TRUE, 
                                         msvar  = TRUE,
                                         getSE  = TRUE,
                                         method = "EM",
                                         maxit  = 300,
                                         use_diff_init = 1))

  MMCtest <- MMCLRTest(y_ms_simu$y, p = 1 , k0 = 1 , k1 = 2, mmc_control)
  summary(MMCtest)




set.seed(1234)
# Define DGP of MS AR process
mdl_ms2 <- list(n     = 200, 
                mu    = c(5,10),
                sigma = c(1,4),
                phi   = c(0.5),
                k     = 2,
                P     = rbind(c(0.90, 0.10),
                              c(0.10, 0.90)))

# Simulate process using simuMSAR() function
y_ms_simu <- simuMSAR(mdl_ms2)


# Set test procedure options
mmc_control = list(N = 19,
                   burnin = 100,
                   converge_check = NULL,
                   eps = 0.1,
                   CI_union = TRUE,
                   silence = FALSE,
                   threshold_stop = 0.05 + 1e-6,
                   type = "pso",
                   maxit = 100,
                   mdl_h0_control = list(const  = TRUE, 
                                         getSE  = TRUE),
                   mdl_h1_control = list(msmu   = TRUE, 
                                         msvar  = TRUE,
                                         getSE  = TRUE,
                                         method = "EM",
                                         maxit  = 300,
                                         use_diff_init = 1))

  MMCtest <- MMCLRTest(y_ms_simu$y, p = 1 , k0 = 1 , k1 = 2, mmc_control)
  summary(MMCtest)

Markov-switching autoregressive model

Description

This function estimates a Markov-switching autoregressive model

Usage

MSARmdl(Y, p, k, control = list())
MSARmdl(Y, p, k, control = list())

Arguments

`Y`	(T x 1) vector with observational data.
`p`	integer for the number of lags to use in estimation. Must be greater than or equal to `1`.
`k`	integer for the number of regimes to use in estimation. Must be greater than or equal to `2`.
`control`	List with model options including: getSE: Boolean. If `TRUE` standard errors are computed and returned. If `FALSE` standard errors are not computed. Default is `TRUE`. msmu: Boolean. If `TRUE` model is estimated with switch in mean. If `FALSE` model is estimated with constant mean. Default is `TRUE`. msvar: Boolean. If `TRUE` model is estimated with switch in variance. If `FALSE` model is estimated with constant variance. Default is `TRUE`. init_theta: vector of initial values. vector must contain `(1 x q)` vector `mu`, `vech(sigma)`, and `vec(P)` where sigma is a `(q x q)` covariance matrix.This is optional. Default is `NULL`, in which case `initVals_MSARmdl` is used to generate initial values. method: string determining which method to use. Options are `'EM'` for EM algorithm or `'MLE'` for Maximum Likelihood Estimation. maxit: integer determining the maximum number of EM iterations. thtol: double determining the convergence criterion for the absolute difference in parameter estimates `theta` between iterations. Default is `1e-6`. maxit_converge: integer determining the maximum number of initial values attempted until solution is finite. For example, if parameters in `theta` or `logLike` are `NaN` another set of initial values (up to `maxit_converge`) is attempted until finite values are returned. This does not occur frequently for most types of data but may be useful in some cases. Once finite values are obtained, this counts as one iteration towards `use_diff_init`. Default is `500`. use_diff_init: integer determining how many different initial values to try (that do not return `NaN`; see `maxit_converge`). Default is `1`. mle_stationary_constraint: Boolean determining if only stationary solutions are considered (if `TRUE`) or not (if `FALSE`). Default is `TRUE`. mle_variance_constraint: Double used to determine the lower bound for variance in each regime. Value should be between `0` and `1` as it is multiplied by single regime variance. Default is `0.01` (i.e., `1%` of single regime variance. mle_theta_low: Vector with lower bounds on parameters (Used only if method = "MLE"). Default is `NULL`. mle_theta_upp: Vector with upper bounds on parameters (Used only if method = "MLE"). Default is `NULL`.

Value

List of class MSARmdl (S3 object) with model attributes including:

y: a (T x 1) matrix of observations.
X: a (T-p x p + const) matrix of lagged observations with a leading column of 1s.
x: a (T-p x p) matrix of lagged observations.
fitted: a (T x 1) matrix of fitted values.
resid: a (T x 1) matrix of residuals.
inter: a (k x 1) vector of estimated intercepts of each process.
mu: a (k x 1) vector of estimated means of each process.
phi: estimates of autoregressive coefficients.
stdev: a (k x 1) vector of estimated standard deviation of each process.
sigma: a (k x 1) estimated covariance matrix.
theta: vector containing: mu and vech(sigma).
theta_mu_ind: vector indicating location of mean with 1 and 0 otherwise.
theta_sig_ind: vector indicating location of variances with 1 and 0 otherwise.
theta_var_ind: vector indicating location of variances with 1 and 0 otherwise. This is the same as theta_sig_ind in MSARmdl.
theta_P_ind: vector indicating location of transition matrix elements with 1 and 0 otherwise.
stationary: Boolean indicating if process is stationary if TRUE or non-stationary if FALSE.
n: number of observations (same as T).
p: number of autoregressive lags.
q: number of series. This is always 1 in MSARmdl.
k: number of regimes in estimated model.
P: a (k x k) transition matrix.
pinf: a (k x 1) vector with limiting probabilities of each regime.
St: a (T x k) vector with smoothed probabilities of each regime at each time t.
deltath: double with maximum absolute difference in vector theta between last iteration.
iterations: number of EM iterations performed to achieve convergence (if less than maxit).
theta_0: vector of initial values used.
init_used: number of different initial values used to get a finite solution. See description of input maxit_converge.
msmu: Boolean. If TRUE model was estimated with switch in mean. If FALSE model was estimated with constant mean.
msvar: Boolean. If TRUE model was estimated with switch in variance. If FALSE model was estimated with constant variance.
control: List with model options used.
logLike: log-likelihood.
AIC: Akaike information criterion.
BIC: Bayesian (Schwarz) information criterion.
Hess: Hessian matrix. Approximated using hessian and only returned if getSE=TRUE.
info_mat: Information matrix. Computed as the inverse of -Hess. If matrix is not PD then nearest PD matrix is obtained using nearest_spd. Only returned if getSE=TRUE.
nearPD_used: Boolean determining whether nearPD function was used on info_mat if TRUE or not if FALSE. Only returned if getSE=TRUE.
theta_se: standard errors of parameters in theta. Only returned if getSE=TRUE.
trace: List with Lists of estimation output for each initial value used due to use_diff_init > 1.

References

Dempster, A. P., N. M. Laird, and D. B. Rubin. 1977. “Maximum Likelihood from Incomplete Data via the EM Algorithm.” Journal of the Royal Statistical Society. Series B 39 (1): 1–38..

Hamilton, James D. 1990. “Analysis of time series subject to changes in regime.” Journal of econometrics, 45 (1-2): 39–70.

Examples

# --------------------------- Use simulated process ----------------------------
set.seed(1234)
# Define DGP of MS AR process
mdl_ms2 <- list(n     = 200, 
                mu    = c(5,10),
                sigma = c(1,4),
                phi   = c(0.5),
                k     = 2,
                P     = rbind(c(0.90, 0.10),
                              c(0.10, 0.90)))

# Simulate process using simuMSAR() function
y_ms_simu <- simuMSAR(mdl_ms2)

# Set options for model estimation
control <- list(msmu   = TRUE, 
                msvar  = TRUE, 
                method = "EM",
                use_diff_init = 1)

# Estimate model

  ms_mdl <- MSARmdl(y_ms_simu$y, p = 1, k = 2, control)
  summary(ms_mdl)




# --------------------------- Use simulated process ----------------------------
set.seed(1234)
# Define DGP of MS AR process
mdl_ms2 <- list(n     = 200, 
                mu    = c(5,10),
                sigma = c(1,4),
                phi   = c(0.5),
                k     = 2,
                P     = rbind(c(0.90, 0.10),
                              c(0.10, 0.90)))

# Simulate process using simuMSAR() function
y_ms_simu <- simuMSAR(mdl_ms2)

# Set options for model estimation
control <- list(msmu   = TRUE, 
                msvar  = TRUE, 
                method = "EM",
                use_diff_init = 1)

# Estimate model

  ms_mdl <- MSARmdl(y_ms_simu$y, p = 1, k = 2, control)
  summary(ms_mdl)

Markov-switching vector autoregressive model

Description

This function estimates a Markov-switching vector autoregressive model

Usage

MSVARmdl(Y, p, k, control = list())
MSVARmdl(Y, p, k, control = list())

Arguments

`Y`	(`T x q`) vector with observational data.
`p`	integer for the number of lags to use in estimation. Must be greater than or equal to `0`.
`k`	integer for the number of regimes to use in estimation. Must be greater than or equal to `2`.
`control`	List with optimization options including: getSE: Boolean. If `TRUE` standard errors are computed and returned. If `FALSE` standard errors are not computed. Default is `TRUE`. msmu: Boolean. If `TRUE` model is estimated with switch in mean. If `FALSE` model is estimated with constant mean. Default is `TRUE`. msvar: Boolean. If `TRUE` model is estimated with switch in variance. If `FALSE` model is estimated with constant variance. Default is `TRUE`. init_theta: vector of initial values. vector must contain `(1 x q)` vector `mu`, `vech(sigma)`, and `vec(P)` where sigma is a `(q x q)` covariance matrix. This is optional. Default is `NULL`, in which case `initVals_MSARmdl` is used to generate initial values. method: string determining which method to use. Options are `'EM'` for EM algorithm or `'MLE'` for Maximum Likelihood Estimation. maxit: integer determining the maximum number of EM iterations. thtol: double determining the convergence criterion for the absolute difference in parameter estimates `theta` between iterations. Default is `1e-6`. maxit_converge: integer determining the maximum number of initial values attempted until solution is finite. For example, if parameters in `theta` or `logLike` are `NaN` another set of initial values (up to `maxit_converge`) is attempted until finite values are returned. This does not occur frequently for most types of data but may be useful in some cases. Once finite values are obtained, this counts as one iteration towards `use_diff_init`. Default is `500`. use_diff_init: integer determining how many different initial values to try (that do not return `NaN`; see `maxit_converge`). Default is `1`. mle_stationary_constraint: Boolean determining if only stationary solutions are considered (if `TRUE`) or not (if `FALSE`). Default is `TRUE`. mle_variance_constraint: double used to determine the lower bound on the smallest eigenvalue for the covariance matrix of each regime. Default is `1e-3`. mle_theta_low: Vector with lower bounds on parameters (Used only if method = "MLE"). Default is `NULL`. mle_theta_upp: Vector with upper bounds on parameters (Used only if method = "MLE"). Default is `NULL`.

Value

List of class MSVARmdl (S3 object) with model attributes including:

y: a (T-p x q) matrix of observations.
X: a (T-p x p*q + const) matrix of lagged observations with a leading column of 1s.
x: a (T-p x p*q) matrix of lagged observations.
resid: a (T-p x q) matrix of residuals.
inter: a (k x q) matrix of estimated intercepts of each process.
mu: a (k x q) matrix of estimated means of each process.
phi: estimates of autoregressive coefficients.
Fmat: Companion matrix containing autoregressive coefficients.
stdev: List with k (q x q) matrices with estimated standard deviation on the diagonal.
sigma: List with k (q x q) matrices with estimated covariance matrix.
theta: vector containing: mu and vech(sigma).
theta_mu_ind: vector indicating location of mean with 1 and 0 otherwise.
theta_sig_ind: vector indicating location of variance and covariances with 1 and 0 otherwise.
theta_var_ind: vector indicating location of variances with 1 and 0 otherwise.
theta_P_ind: vector indicating location of transition matrix elements with 1 and 0 otherwise.
stationary: Boolean indicating if process is stationary if TRUE or non-stationary if FALSE.
n: number of observations (same as T).
p: number of autoregressive lags.
q: number of series.
k: number of regimes in estimated model.
P: a (k x k) transition matrix.
pinf: a (k x 1) vector with limiting probabilities of each regime.
St: a (T x k) vector with smoothed probabilities of each regime at each time t.
deltath: double with maximum absolute difference in vector theta between last iteration.
iterations: number of EM iterations performed to achieve convergence (if less than maxit).
theta_0: vector of initial values used.
init_used: number of different initial values used to get a finite solution. See description of input maxit_converge.
msmu: Boolean. If TRUE model was estimated with switch in mean. If FALSE model was estimated with constant mean.
msvar: Boolean. If TRUE model was estimated with switch in variance. If FALSE model was estimated with constant variance.
control: List with model options used.
logLike: log-likelihood.
AIC: Akaike information criterion.
BIC: Bayesian (Schwarz) information criterion.
Hess: Hessian matrix. Approximated using hessian and only returned if getSE=TRUE.
info_mat: Information matrix. Computed as the inverse of -Hess. If matrix is not PD then nearest PD matrix is obtained using nearest_spd. Only returned if getSE=TRUE.
nearPD_used: Boolean determining whether nearPD function was used on info_mat if TRUE or not if FALSE. Only returned if getSE=TRUE.
theta_se: standard errors of parameters in theta. Only returned if getSE=TRUE.
trace: List with Lists of estimation output for each initial value used due to use_diff_init > 1.

List with model characteristics

References

Dempster, A. P., N. M. Laird, and D. B. Rubin. 1977. “Maximum Likelihood from Incomplete Data via the EM Algorithm.” Journal of the Royal Statistical Society. Series B 39 (1): 1–38..

Krolzig, Hans-Martin. 1997. “The markov-switching vector autoregressive model.”. Springer.

Examples

set.seed(123)
# Define DGP of MS VAR process
mdl_msvar2 <- list(n     = 200, 
                   p     = 1,
                   q     = 2,
                   mu    = rbind(c(5, -2),
                                 c(10, 2)),
                   sigma = list(rbind(c(5.0, 1.5),
                                      c(1.5, 1.0)),
                                rbind(c(7.0, 3.0),
                                      c(3.0, 2.0))),
                   phi   = rbind(c(0.50, 0.30),
                                 c(0.20, 0.70)),
                   k     = 2,
                   P     = rbind(c(0.90, 0.10),
                                 c(0.10, 0.90)))

# Simulate process using simuMSVAR() function
y_msvar_simu <- simuMSVAR(mdl_msvar2)

# Set options for model estimation
control <- list(msmu   = TRUE, 
                msvar  = TRUE,
                method = "EM",
                use_diff_init = 1)
                
# Estimate model

  y_msvar_mdl <- MSVARmdl(y_msvar_simu$y, p = 2, k = 2, control)
  summary(y_msvar_mdl)

set.seed(123)
# Define DGP of MS VAR process
mdl_msvar2 <- list(n     = 200, 
                   p     = 1,
                   q     = 2,
                   mu    = rbind(c(5, -2),
                                 c(10, 2)),
                   sigma = list(rbind(c(5.0, 1.5),
                                      c(1.5, 1.0)),
                                rbind(c(7.0, 3.0),
                                      c(3.0, 2.0))),
                   phi   = rbind(c(0.50, 0.30),
                                 c(0.20, 0.70)),
                   k     = 2,
                   P     = rbind(c(0.90, 0.10),
                                 c(0.10, 0.90)))

# Simulate process using simuMSVAR() function
y_msvar_simu <- simuMSVAR(mdl_msvar2)

# Set options for model estimation
control <- list(msmu   = TRUE, 
                msvar  = TRUE,
                method = "EM",
                use_diff_init = 1)
                
# Estimate model

  y_msvar_mdl <- MSVARmdl(y_msvar_simu$y, p = 2, k = 2, control)
  summary(y_msvar_mdl)

Normal distribution model

Description

This function estimates a univariate or multivariate normally distributed model. This can be used for the null hypothesis of a linear model against an alternative hypothesis of a HMM with k regimes.

Usage

Nmdl(Y, control = list())
Nmdl(Y, control = list())

Arguments

Y

a (T x q) matrix of observations.

control

List with model options including:

const: Boolean determining whether to estimate model with constant if TRUE or not if FALSE. Default is TRUE.
getSE: Boolean determining whether to compute standard errors of parameters if TRUE or not if FALSE. Default is TRUE.

Value

List of class Nmdl (S3 object) with model attributes including:

y: a (T x q) matrix of observations.
fitted: a (T x q) matrix of fitted values.
resid: a (T x q) matrix of residuals.
mu: a (1 x q) vector of estimated means of each process.
stdev: a (q x 1) vector of estimated standard deviation of each process.
sigma: a (q x q) estimated covariance matrix.
theta: vector containing: mu and vech(sigma).
theta_mu_ind: vector indicating location of mean with 1 and 0 otherwise.
theta_sig_ind: vector indicating location of variance and covariances with 1 and 0 otherwise.
theta_var_ind: vector indicating location of variances with 1 and 0 otherwise.
n: number of observations (same as T).
q: number of series.
k: number of regimes. This is always 1 in Nmdl.
control: List with model options used.
logLike: log-likelihood.
AIC: Akaike information criterion.
BIC: Bayesian (Schwarz) information criterion.
Hess: Hessian matrix. Approximated using hessian and only returned if getSE=TRUE.
info_mat: Information matrix. Computed as the inverse of -Hess. If matrix is not PD then nearest PD matrix is obtained using nearest_spd. Only returned if getSE=TRUE.
nearPD_used: Boolean determining whether nearPD function was used on info_mat if TRUE or not if FALSE. Only returned if getSE=TRUE.
theta_se: standard errors of parameters in theta. Only returned if getSE=TRUE.

Examples

set.seed(1234)

# ----- Univariate ----- # 
# Define DGP 
mdl_norm <- list(n     = 1000, 
                 q     = 1,
                 mu    = as.matrix(5),
                 sigma = as.matrix(5.0))

# Simulate process using simuNorm() function
y_norm_simu <- simuNorm(mdl_norm)

# estimate parameters
y_norm_mdl <- Nmdl(y_norm_simu$y)
summary(y_norm_mdl)


# ----- Multivariate ----- # 
# Define DGP 
mdl_norm <- list(n     = 1000, 
                 q     = 2,
                 mu    = c(5, -2),
                 sigma = rbind(c(5.0, 1.5),
                               c(1.5, 1.0)))

# Simulate process using simuNorm() function
y_norm_simu <- simuNorm(mdl_norm)

# estimate parameters
y_norm_mdl <- Nmdl(y_norm_simu$y)
summary(y_norm_mdl)
set.seed(1234)

# ----- Univariate ----- # 
# Define DGP 
mdl_norm <- list(n     = 1000, 
                 q     = 1,
                 mu    = as.matrix(5),
                 sigma = as.matrix(5.0))

# Simulate process using simuNorm() function
y_norm_simu <- simuNorm(mdl_norm)

# estimate parameters
y_norm_mdl <- Nmdl(y_norm_simu$y)
summary(y_norm_mdl)


# ----- Multivariate ----- # 
# Define DGP 
mdl_norm <- list(n     = 1000, 
                 q     = 2,
                 mu    = c(5, -2),
                 sigma = rbind(c(5.0, 1.5),
                               c(1.5, 1.0)))

# Simulate process using simuNorm() function
y_norm_simu <- simuNorm(mdl_norm)

# estimate parameters
y_norm_mdl <- Nmdl(y_norm_simu$y)
summary(y_norm_mdl)

Simulate autoregressive process

Description

This function simulates an autoregresive process.

Usage

simuAR(mdl_h0, burnin = 100)
simuAR(mdl_h0, burnin = 100)

Arguments

mdl_h0

List containing the following DGP parameters

n: Length of series.
mu: Mean of process.
sigma: Standard deviation of process.
phi: Vector of autoregressive coefficients.
eps: An optional (T+burnin x q) matrix with standard normal errors to be used. Errors will be generated if not provided.

burnin

Number of simulated observations to remove from beginning. Default is 100.

Value

List with simulated autoregressive series and its DGP parameters.

Examples

set.seed(1234)
# Define DGP of AR process
mdl_ar <- list(n     = 500, 
               mu    = 5,
               sigma = 2,
               phi   = c(0.5,0.2))

# Simulate process using simuAR() function
y_simu <- simuAR(mdl_ar)

plot(y_simu)
set.seed(1234)
# Define DGP of AR process
mdl_ar <- list(n     = 500, 
               mu    = 5,
               sigma = 2,
               phi   = c(0.5,0.2))

# Simulate process using simuAR() function
y_simu <- simuAR(mdl_ar)

plot(y_simu)

Simulate Hidden Markov model with normally distributed errors

Description

This function simulates a Hidden Markov Model process.

Usage

simuHMM(mdl_h0, burnin = 100)
simuHMM(mdl_h0, burnin = 100)

Arguments

mdl_h0

List containing the following DGP parameters

n: Length of series.
k: Number of regimes.
mu: A (k x q) vector of means.
sigma: A (q x q) covariance matrix.
q: Number of series.
P: A (k x k) transition matrix (columns must sum to one).
eps: An optional (T+burnin x q) matrix with standard normal errors to be used. Errors will be generated if not provided.

burnin

Number of simulated observations to remove from beginning. Default is 100.

Value

List with simulated series and its DGP parameters.

Examples

set.seed(1234)

# ----- Univariate ----- #
# Define DGP 
mdl_hmm <- list(n     = 1000, 
                q     = 1,
                mu    = as.matrix(c(5,
                                   -2)),
                sigma = list(as.matrix(5.0),
                             as.matrix(7.0)),
                k     = 2,
                P     = rbind(c(0.90, 0.10),
                              c(0.10, 0.90)))

# Simulate process using simuHMM() function
y_hmm_simu <- simuHMM(mdl_hmm)

plot(y_hmm_simu)

# ----- Multivariate ----- #
# Define DGP 
mdl_hmm <- list(n     = 1000, 
                q     = 2,
                mu    = rbind(c(5, -2),
                              c(10, 2)),
                sigma = list(rbind(c(5.0, 1.5),
                                   c(1.5, 1.0)),
                             rbind(c(7.0, 3.0),
                                   c(3.0, 2.0))),
                k     = 2,
                P     = rbind(c(0.90, 0.10),
                              c(0.10, 0.90)))

# Simulate process using simuHMM() function
y_hmm_simu <- simuHMM(mdl_hmm)


plot(y_hmm_simu)
set.seed(1234)

# ----- Univariate ----- #
# Define DGP 
mdl_hmm <- list(n     = 1000, 
                q     = 1,
                mu    = as.matrix(c(5,
                                   -2)),
                sigma = list(as.matrix(5.0),
                             as.matrix(7.0)),
                k     = 2,
                P     = rbind(c(0.90, 0.10),
                              c(0.10, 0.90)))

# Simulate process using simuHMM() function
y_hmm_simu <- simuHMM(mdl_hmm)

plot(y_hmm_simu)

# ----- Multivariate ----- #
# Define DGP 
mdl_hmm <- list(n     = 1000, 
                q     = 2,
                mu    = rbind(c(5, -2),
                              c(10, 2)),
                sigma = list(rbind(c(5.0, 1.5),
                                   c(1.5, 1.0)),
                             rbind(c(7.0, 3.0),
                                   c(3.0, 2.0))),
                k     = 2,
                P     = rbind(c(0.90, 0.10),
                              c(0.10, 0.90)))

# Simulate process using simuHMM() function
y_hmm_simu <- simuHMM(mdl_hmm)


plot(y_hmm_simu)

Simulate Markov-switching autoregressive process

Description

This function simulates a Markov-switching autoregressive process.

Usage

simuMSAR(mdl_h0, burnin = 100)
simuMSAR(mdl_h0, burnin = 100)

Arguments

mdl_h0

List containing the following DGP parameters

n: Length of series.
k: Number of regimes.
mu: A (k x 1) vector with mean of process in each regime.
sigma: A (k x 1) vector with standard deviation of process in each regime.
phi: Vector of autoregressive coefficients.
P: A (k x k) transition matrix (columns must sum to one).
eps: An optional (T+burnin x q) matrix with standard normal errors to be used. Errors will be generated if not provided.

burnin

Number of simulated observations to remove from beginning. Default is 100.

Value

List with simulated Markov-switching autoregressive process and its DGP properties.

Examples

set.seed(1234)
# Define DGP of MS AR process
mdl_ms2 <- list(n     = 500, 
                mu    = c(5,10),
                sigma = c(1,2),
                phi   = c(0.5, 0.2),
                k     = 2,
                P     = rbind(c(0.90, 0.10),
                              c(0.10, 0.90)))

# Simulate process using simuMSAR() function
y_ms_simu <- simuMSAR(mdl_ms2)

plot(y_ms_simu)
set.seed(1234)
# Define DGP of MS AR process
mdl_ms2 <- list(n     = 500, 
                mu    = c(5,10),
                sigma = c(1,2),
                phi   = c(0.5, 0.2),
                k     = 2,
                P     = rbind(c(0.90, 0.10),
                              c(0.10, 0.90)))

# Simulate process using simuMSAR() function
y_ms_simu <- simuMSAR(mdl_ms2)

plot(y_ms_simu)

Simulate Markov-switching vector autoregressive process

Description

This function simulates a Markov-switching vector autoregressive process.

Usage

simuMSVAR(mdl_h0, burnin = 100)
simuMSVAR(mdl_h0, burnin = 100)

Arguments

mdl_h0

List containing the following DGP parameters

n: Length of series.
k: Number of regimes.
mu: A (k x q) matrix of means.
sigma: List with k (q x q) covariance matrices.
phi: A (q x qp) matrix of autoregressive coefficients.
p: Number of autoregressive lags.
q: Number of series.
P: A (k x k) transition matrix (columns must sum to one).
eps: An optional (T+burnin x q) matrix with standard normal errors to be used. Errors will be generated if not provided.

burnin

Number of simulated observations to remove from beginning. Default is 100.

Value

List with simulated vector autoregressive series and its DGP parameters.

Examples

set.seed(1234)
# Define DGP of MS VAR process
mdl_msvar2 <- list(n     = 1000, 
                   p     = 1,
                   q     = 2,
                   mu    = rbind(c(5, -2),
                                 c(10, 2)),
                   sigma = list(rbind(c(5.0, 1.5),
                                      c(1.5, 1.0)),
                                rbind(c(7.0, 3.0),
                                      c(3.0, 2.0))),
                   phi   = rbind(c(0.50, 0.30),
                                 c(0.20, 0.70)),
                   k     = 2,
                   P     = rbind(c(0.90, 0.10),
                                 c(0.10, 0.90)))

# Simulate process using simuMSVAR() function
y_msvar_simu <- simuMSVAR(mdl_msvar2)

plot(y_msvar_simu)
set.seed(1234)
# Define DGP of MS VAR process
mdl_msvar2 <- list(n     = 1000, 
                   p     = 1,
                   q     = 2,
                   mu    = rbind(c(5, -2),
                                 c(10, 2)),
                   sigma = list(rbind(c(5.0, 1.5),
                                      c(1.5, 1.0)),
                                rbind(c(7.0, 3.0),
                                      c(3.0, 2.0))),
                   phi   = rbind(c(0.50, 0.30),
                                 c(0.20, 0.70)),
                   k     = 2,
                   P     = rbind(c(0.90, 0.10),
                                 c(0.10, 0.90)))

# Simulate process using simuMSVAR() function
y_msvar_simu <- simuMSVAR(mdl_msvar2)

plot(y_msvar_simu)

Simulate normally distributed process

Description

This function simulates a normally distributed process.

Usage

simuNorm(mdl_h0, burnin = 0)
simuNorm(mdl_h0, burnin = 0)

Arguments

mdl_h0

List containing the following DGP parameters

n: Length of series.
mu: A (q x 1) vector of means.
sigma: A (q x q) covariance matrix.
q: Number of series.
eps: An optional (T+burnin x q) matrix with standard normal errors to be used. Errors will be generated if not provided.

burnin

Number of simulated observations to remove from beginning. Default is 100.

Value

List with simulated series and its DGP parameters.

Examples

set.seed(1234)
# Define DGP 
mdl_norm <- list(n     = 1000, 
                 q     = 2,
                 mu    = c(5, -2),
                 sigma = rbind(c(5.0, 1.5),
                               c(1.5, 1.0)))

# Simulate process using simuNorm() function
y_norm_simu <- simuNorm(mdl_norm)

plot(y_norm_simu)
set.seed(1234)
# Define DGP 
mdl_norm <- list(n     = 1000, 
                 q     = 2,
                 mu    = c(5, -2),
                 sigma = rbind(c(5.0, 1.5),
                               c(1.5, 1.0)))

# Simulate process using simuNorm() function
y_norm_simu <- simuNorm(mdl_norm)

plot(y_norm_simu)

Simulate VAR process

Description

This function simulates a vector autoregresive process.

Usage

simuVAR(mdl_h0, burnin = 100)
simuVAR(mdl_h0, burnin = 100)

Arguments

mdl_h0

List containing the following DGP parameters

n: Length of series.
mu: A (q x 1) vector of means.
sigma: A (q x q) covariance matrix.
phi: A (q x qp) matrix of autoregressive coefficients.
p: Number of autoregressive lags.
q: Number of series.
eps: An optional (T+burnin x q) matrix with standard normal errors to be used. Errors will be generated if not provided.

burnin

Number of simulated observations to remove from beginning. Default is 100.

Value

List with simulated vector autoregressive series and its DGP parameters.

US GNP data 1947Q2 - 2023Q4

Description

US GNP data 1947Q2 - 2023Q4

Usage

USGNP
USGNP

Format

This data is used in Rodriguez-Rondon & Dufour (2024). The series ranges from 1947Q2 to 2023Q4.

DATE: Vector of dates
GNP_logdiff: log difference of US GDP series
GNP: US GNP series

Source

https://fred.stlouisfed.org/graph/?g=UKDQ

References

Rodriguez-Rondon, Gabriel and Jean-Marie Dufour. 2024. “Monte Carlo Likelihood Ratio Tests for Markov Switching Models.” Unpublished manuscript.

Vector autoregressive model

Description

This function estimates a vector autoregresive model with p lags. This can be used for the null hypothesis of a linear model against an alternative hypothesis of a Markov switching vector autoregressive model with k regimes.

Usage

VARmdl(Y, p, control = list())
VARmdl(Y, p, control = list())

Arguments

Y

a (T x q) matrix of observations.

p

integer determining the number of autoregressive lags.

control

List with model options including:

const: Boolean determining whether to estimate model with constant if TRUE or not if FALSE. Default is TRUE.
getSE: Boolean determining whether to compute standard errors of parameters if TRUE or not if FALSE. Default is TRUE.

Value

List of class VARmdl (S3 object) with model attributes including:

y: a (T-p x q) matrix of observations.
X: a (T-p x p*q + const) matrix of lagged observations with a leading column of 1s if const=TRUE or not if const=FALSE.
x: a (T-p x p*q) matrix of lagged observations.
fitted: a (T-p x q) matrix of fitted values.
resid: a (T-p x q) matrix of residuals.
inter: a (1 x q) vector of estimated intercepts of each process.
mu: a (1 x q) vector of estimated means of each process.
coef: coefficient estimates. First row are the intercept (i.e., not mu) if const=TRUE. This is the same as t(phi) if const=FALSE.
intercept: estimate of intercepts.
phi: a (q x p*q) matrix of estimated autoregressive coefficients.
Fmat: Companion matrix containing autoregressive coefficients.
stdev: a (q x 1) vector of estimated standard deviation of each process.
sigma: a (q x q) estimated covariance matrix.
theta: vector containing: mu, vech(sigma), and vec(t(phi)).
theta_mu_ind: vector indicating location of mean with 1 and 0 otherwise.
theta_sig_ind: vector indicating location of variance and covariances with 1 and 0 otherwise.
theta_var_ind: vector indicating location of variances with 1 and 0 otherwise.
theta_phi_ind: vector indicating location of autoregressive coefficients with 1 and 0 otherwise.
stationary: Boolean indicating if process is stationary if TRUE or non-stationary if FALSE.
n: number of observations after lag transformation (i.e., n = T-p).
p: number of autoregressive lags.
q: number of series.
k: number of regimes. This is always 1 in VARmdl.
Fmat: matrix from companion form. Used to determine is process is stationary.
control: List with model options used.
logLike: log-likelihood.
AIC: Akaike information criterion.
BIC: Bayesian (Schwarz) information criterion.
Hess: Hessian matrix. Approximated using hessian and only returned if getSE=TRUE.
info_mat: Information matrix. Computed as the inverse of -Hess. If matrix is not PD then nearest PD matrix is obtained using nearest_spd. Only returned if getSE=TRUE.
nearPD_used: Boolean determining whether nearPD function was used on info_mat if TRUE or not if FALSE. Only returned if getSE=TRUE.
theta_se: standard errors of parameters in theta. Only returned if getSE=TRUE.

Examples

# ----- Bivariate VAR(1) process ----- #
set.seed(1234)
# Define DGP of VAR process
mdl_var <- list(n     = 1000, 
                p     = 1,
                q     = 2,
                mu    = c(5,-2),
                sigma = rbind(c(5.0, 1.5),
                              c(1.5, 1.0)),
                phi   = rbind(c(0.50, 0.30),
                              c(0.20, 0.70)))

# Simulate process using simuVAR() function
y_simu <- simuVAR(mdl_var)

# Set options for model estimation
control <- list(const  = TRUE, 
                getSE  = TRUE)

# Estimate model
y_var_mdl <- VARmdl(y_simu$y, p = 2, control)
summary(y_var_mdl)
# ----- Bivariate VAR(1) process ----- #
set.seed(1234)
# Define DGP of VAR process
mdl_var <- list(n     = 1000, 
                p     = 1,
                q     = 2,
                mu    = c(5,-2),
                sigma = rbind(c(5.0, 1.5),
                              c(1.5, 1.0)),
                phi   = rbind(c(0.50, 0.30),
                              c(0.20, 0.70)))

# Simulate process using simuVAR() function
y_simu <- simuVAR(mdl_var)

# Set options for model estimation
control <- list(const  = TRUE, 
                getSE  = TRUE)

# Estimate model
y_var_mdl <- VARmdl(y_simu$y, p = 2, control)
summary(y_var_mdl)

Package 'MSTest'

Help Index

Testing Markov Switching Models

Description

Author(s)

References

Autoregressive Model

Description

Usage

Arguments

Value

See Also

Examples

Bootstrap Likelihood Ratio Test

Description

Usage

Arguments

Value

References

Carrasco, Hu, & Ploberger 2010 GNP data

Description

Usage

Format

Source

References

Carrasco, Hu, and Ploberger (2014) parameter stability test

Description

Usage

Arguments

Value

References

Examples

Monte Carlo moment-based test for Markov switching model

Description

Usage

Arguments

Value

References

Examples

Maximized Monte Carlo moment-based test for Markov switching model

Description

Usage

Arguments

Value

References

Examples

Hamilton 1984 & Hansen 1992 GNP data

Description

Usage

Format

Source

References

Hansen (1992) likelihood ratio test

Description

Usage

Arguments

Value

References

Examples

Hidden Markov model

Description

Usage

Arguments

Value

References

See Also

Monte Carlo Likelihood Ratio Test

Description

Usage

Arguments

Value

References

Examples

Monte Carlo P-value

Description

Usage

Arguments

Value

References

Maximized Monte Carlo Likelihood Ratio Test