Package 'vars'

Coefficient matrices of the lagged endogenous variables

Description

Returns the estimated coefficient matrices of the lagged endogenous variables as a list of matrices each with dimension $(K \times K)$.

Usage

Acoef(x)

Arguments

x	An object of class ‘varest’, generated by VAR().

Details

Given an estimated VAR(p) of the form:

$\hat{\bold{y}}_t = \hat{A}_1 \bold{y}_{t-1} + \ldots + \hat{A}_p \bold{y}_{t-p} + \hat{C}D_t$

the function returns the matrices $(\hat{A}_1, \ldots, \hat{A}_p)$ each with dimension $(K \times K)$ as a list object.

Value

A list object with coefficient matrices for the lagged endogenous variables.

Note

This function was named A in earlier versions of package vars; it is now deprecated. See vars-deprecated too.

Author(s)

Bernhard Pfaff

See Also

Bcoef, VAR

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
Acoef(var.2c)

---

ARCH-LM test

Description

This function computes univariate and multivariate ARCH-LM tests for a VAR(p).

Usage

arch.test(x, lags.single = 16, lags.multi = 5, multivariate.only = TRUE)

Arguments

x	Object of class ‘varest’; generated by VAR(), or an object of class ‘vec2var’; generated by vec2var().
lags.single	An integer specifying the lags to be used for the univariate ARCH statistics.
lags.multi	An integer specifying the lags to be used for the multivariate ARCH statistic.
multivariate.only	If TRUE (the default), only the multivariate test statistic is computed.

Details

The multivariate ARCH-LM test is based on the following regression (the univariate test can be considered as special case of the exhibtion below and is skipped):

$vech(\bold{\hat{u}}_t \bold{\hat{u}}_t') = \bold{\beta}_0 + B_1 vech(\bold{\hat{u}}_{t-1} \bold{\hat{u}}_{t-1}') + \ldots + B_q vech(\bold{\hat{u}}_{t-q} \bold{\hat{u}}_{t-q}' + \bold{v}_t)$

whereby $\bold{v}_t$ assigns a spherical error process and $vech$ is the column-stacking operator for symmetric matrices that stacks the columns from the main diagonal on downwards. The dimension of $\bold{\beta}_0$ is $\frac{1}{2}K(K +1)$ and for the coefficient matrices $B_i$ with $i=1, \ldots, q$, $\frac{1}{2}K(K +1) \times \frac{1}{2}K(K +1)$. The null hypothesis is: $H_0 := B_1 = B_2 = \ldots = B_q = 0$ and the alternative is: $H_1: B_1 \neq 0 or B_2 \neq 0 or \ldots B_q \neq 0$. The test statistic is:

$VARCH_{LM}(q) = \frac{1}{2}T K (K + 1)R_m^2 \quad ,$

with

$R_m^2 = 1 - \frac{2}{K(K+1)}tr(\hat{\Omega} \hat{\Omega}_0^{-1}) \quad ,$

and $\hat{\Omega}$ assigns the covariance matrix of the above defined regression model. This test statistic is distributed as $\chi^2(qK^2(K+1)^2/4)$.

Value

A list with class attribute ‘varcheck’ holding the following elements:

resid	A matrix with the residuals of the VAR.
arch.uni	A list with objects of class ‘htest’ containing the univariate ARCH-LM tests per equation. This element is only returned if multivariate.only = FALSE is set.
arch.mul	An object with class attribute ‘htest’ containing the multivariate ARCH-LM statistic.

Note

This function was named arch in earlier versions of package vars; it is now deprecated. See vars-deprecated too.

Author(s)

Bernhard Pfaff

References

Doornik, J. A. and D. F. Hendry (1997), Modelling Dynamic Systems Using PcFiml 9.0 for Windows, International Thomson Business Press, London.

Engle, R. F. (1982), Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50: 987-1007.

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

VAR, vec2var, resid, plot

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
arch.test(var.2c)

---

Coefficient matrix of an estimated VAR(p)

Description

Returns the estimated coefficients of a VAR(p) as a matrix.

Usage

Bcoef(x)

Arguments

x	An object of class ‘varest’, generated by VAR().

Details

Given an estimated VAR of the form:

$\hat{\bold{y}}_t = \hat{A}_1 \bold{y}_{t-1} + \ldots + \hat{A}_p \bold{y}_{t-p} + \hat{C}D_t$

the function returns the matrices $(\hat{A}_1 | \ldots | \hat{A}_p | \hat{C})$ as a matrix object.

Value

A matrix holding the estimated coefficients of a VAR.

Note

This function was named B in earlier versions of package vars; it is now deprecated. See vars-deprecated too.

Author(s)

Bernhard Pfaff

See Also

Acoef, VAR

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
Bcoef(var.2c)

---

Estimates a Blanchard-Quah type SVAR

Description

This function estimates a SVAR of type Blanchard and Quah. It returns a list object with class attribute ‘svarest’.

Usage

BQ(x)

Arguments

x	Object of class ‘varest’; generated by VAR().

Details

For a Blanchard-Quah model the matrix $A$ is set to be an identity matrix with dimension $K$. The matrix of the long-run effects is assumed to be lower-triangular and is defined as:

$(I_K - A_1 - \cdots - A_p)^{-1}B$

Hence, the residual of the second equation cannot exert a long-run influence on the first variable and likewise the third residual cannot impact the first and second variable. The estimation of the Blanchard-Quah model is achieved by a Choleski decomposition of:

$(I_K - \hat{A}_1 - \cdots - \hat{A}_p)^{-1}\hat{\Sigma}_u (I_K - \hat{A}_1' - \cdots - \hat{A}_p')^{-1}$

The matrices $\hat{A}_i$ for $i = 1, \ldots, p$ assign the reduced form estimates. The long-run impact matrix is the lower-triangular Choleski decomposition of the above matrix and the contemporaneous impact matrix is equal to:

$(I_K - A_1 - \cdots - A_p)Q$

where $Q$ assigns the lower-trinagular Choleski decomposition.

Value

A list of class ‘svarest’ with the following elements is returned:

A	An identity matrix.
Ase	NULL.
B	The estimated contemporaneous impact matrix.
Bse	NULL.
LRIM	The estimated long-run impact matrix.
Sigma.U	The variance-covariance matrix of the reduced form residuals times 100.
LR	NULL.
opt	NULL.
start	NULL.
type	Character: “Blanchard-Quah”.
var	The ‘varest’ object ‘x’.
call	The call to BQ().

Author(s)

Bernhard Pfaff

References

Blanchard, O. and D. Quah (1989), The Dynamic Effects of Aggregate Demand and Supply Disturbances, The American Economic Review, 79(4), 655-673.

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

SVAR, VAR

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
BQ(var.2c)

---

Canada: Macroeconomic time series

Description

The original time series are published by the OECD. The sample range is from the 1stQ 1980 until 4thQ 2000. The following series have been utilised in the construction of the series provided in Canada:

Main Economic Indicators:

Canadian unemployment rate in %	444113DSA
Canadian manufacturing real wage	444321KSA
Canadian consumer price index	445241K

Quarterly National Accounts:

Canadian nominal GDP	CAN1008S1

Labour Force Statistics:

Canadian civil employment in 1000 persons	445005DSA

The series in Canada are constructed as:

prod :=	100*(ln(CAN1008S1/445241K)-ln(445005DSA))
e :=	100*ln(445005DSA)
U :=	444113DSA
rw :=	100*ln(100*444321KSA)

Hence, prod is used as a measure of labour productivity; e is used for employment; U is the unemployment rate and rw assigns the real wage.

Usage

Canada

Format

An object with class attributes mts and ts containing four variables with 84 observations.

Source

OECD: https://www.oecd.org; data set is available for download at http://www.jmulti.org/data_atse.html, the official homepage of JMULTI is http://www.jmulti.com. The book resource for JMULTI is: Luetkepohl, H. and Kraetzig, M., Applied Time Series Econometrics, Cambridge University Press, Cambridge, 2004.

---

Causality Analysis

Description

Computes the test statistics for Granger- and Instantaneous causality for a VAR(p).

Usage

causality(x, cause = NULL, vcov.=NULL, boot=FALSE, boot.runs=100)

Arguments

x	Object of class ‘varest’; generated by VAR().
cause	A character vector of the cause variable(s). If not set, then the variable in the first column of x$y is used as cause variable and a warning is printed.
vcov.	a specification of the covariance matrix of the estimated coefficients. This can be specified as a matrix or as a function yielding a matrix when applied to x.
boot	Logical. Whether a wild bootstrap procedure should be used to compute the critical values. Default is no
boot.runs	Number of bootstrap replications if boot=TRUE

Details

Two causality tests are implemented. The first is a F-type Granger-causality test and the second is a Wald-type test that is characterized by testing for nonzero correlation between the error processes of the cause and effect variables. For both tests the vector of endogenous variables $\bold{y}_t$ is split into two subvectors $\bold{y}_{1t}$ and $\bold{y}_{2t}$ with dimensions $(K_1 \times 1)$ and $(K_2 \times 1)$ with $K = K_1 + K_2$.
For the rewritten VAR(p):

$[\bold{y}_{1t} , \bold{y}_{2t}] = \sum_{i=1}^p [\bold{\alpha}_{11, i}' , \bold{\alpha}_{12, i}' | \bold{\alpha}_{21, i}' , \bold{\alpha}_{22, i}'][\bold{y}_{1,t-i}, \bold{y}_{2, t-i}] + CD_t + [\bold{u}_{1t}, \bold{u}_{2t}] \quad ,$

the null hypothesis that the subvector $\bold{y}_{1t}$ does not Granger-cause $\bold{y}_{2t}$, is defined as $\bold{\alpha}_{21, i} = 0$ for $i = 1, 2, \ldots, p$. The alternative is: $\exists \; \bold{\alpha}_{21,i} \ne 0$ for $i = 1, 2, \ldots, p$. The test statistic is distributed as $F(p K_1 K_2, KT - n^*)$, with $n^*$ equal to the total number of parameters in the above VAR(p) (including deterministic regressors).
The null hypothesis for instantaneous causality is defined as: $H_0: C \bold{\sigma} = 0$, where $C$ is a $(N \times K(K + 1)/2)$ matrix of rank $N$ selecting the relevant co-variances of $\bold{u}_{1t}$ and $\bold{u}_{2t}$; $\bold{\sigma} = vech(\Sigma_u)$. The Wald statistic is defined as:

$\lambda_W = T \tilde{\bold{\sigma}}'C'[2 C D_{K}^{+}(\tilde{\Sigma}_u \otimes \tilde{\Sigma}_u) D_{K}^{+'} C']^{-1} C \tilde{\bold{\sigma}} \quad ,$

hereby assigning the Moore-Penrose inverse of the duplication matrix $D_K$ with $D_{K}^{+}$ and $\tilde{\Sigma}_u = \frac{1}{T}\sum_{t=1}^T \hat{\bold{u}}_t \hat{\bold{u}}_t'$. The duplication matrix $D_K$ has dimension $(K^2 \times \frac{1}{2}K(K + 1))$ and is defined such that for any symmetric $(K \times K)$ matrix A, $vec(A) = D_K vech(A)$ holds. The test statistic $\lambda_W$ is asymptotically distributed as $\chi^2(N)$.

Fot the Granger causality test, a robust covariance-matrix estimator can be used in case of heteroskedasticity through argument vcov. It can be either a pre-computed matrix or a function for extracting the covariance matrix. See vcovHC from package sandwich for further details.

A wild bootstrap computation (imposing the restricted model as null) of the p values is available through argument boot and boot.runs following Hafner and Herwartz (2009).

Value

A list with elements of class ‘htest’:

Granger	The result of the Granger-causality test.
Instant	The result of the instantaneous causality test.

Note

The Moore-Penrose inverse matrix is computed with the function ginv contained in the package ‘MASS’.
The Granger-causality test is problematic if some of the variables are nonstationary. In that case the usual asymptotic distribution of the test statistic may not be valid under the null hypothesis.

Author(s)

Bernhard Pfaff

References

Granger, C. W. J. (1969), Investigating causal relations by econometric models and cross-spectral methods, Econometrica, 37: 424-438.

Hafner, C. M. and Herwartz, H. (2009) Testing for linear vector autoregressive dynamics under multivariate generalized autoregressive heteroskedasticity, Statistica Neerlandica, 63: 294-323

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

Venables, W. N. and B. D. Ripley (2002), Modern Applied Statistics with S, 4th edition, Springer, New York.

Zeileis, A. (2006) Object-Oriented Computation of Sandwich Estimators Journal of Statistical Software, 16, 1-16

See Also

VAR

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
causality(var.2c, cause = "e")

#use a robust HC variance-covariance matrix for the Granger test:
causality(var.2c, cause = "e", vcov.=vcovHC(var.2c))

#use a wild-bootstrap procedure to for the Granger test
## Not run: causality(var.2c, cause = "e", boot=TRUE, boot.runs=1000)

---

Coefficient method for objects of class varest

Description

Returns the coefficients of a VAR(p)-model for objects generated by VAR(). Thereby the coef-method is applied to the summary of the list element varresult, which is itself a list of summary.lm-objects.

Usage

## S3 method for class 'varest'
coef(object, ...)

Arguments

object	An object of class ‘varest’; generated by VAR()
...	Currently not used.

Author(s)

Bernhard Pfaff

References

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

VAR

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
coef(var.2c)

---

Fanchart plot for objects of class varprd

Description

Time Series plots of VAR forecasts with differently shaded confidence regions (fanchart) for each endogenous variable.

Usage

fanchart(x, colors = NULL, cis = NULL, names = NULL, main = NULL, ylab =
NULL, xlab = NULL, col.y = NULL, nc, plot.type = c("multiple",
"single"), mar = par("mar"), oma = par("oma"), ... )

Arguments

x	An object of class ‘varprd’; generated by predict().
colors	Character vector of colors to be used for shading. If unset, a gray color scheme is used.
cis	A numeric vector of confidence intervals. If unset the sequence from 0.1 to 0.9 is used (step size 0.1).
names	Character vector, names of variables for fancharts. If unset, all variables are plotted.
main	Character vector, title for fanchart plots.
ylab	Character, ylab for fanchart.
xlab	Character, xlab for fanchart.
col.y	Character, color for plotted time series.
plot.type	Character, if multiple all fancharts appear in one device.
nc	Integer, number of columns if plot.type is multiple.
mar	Vector, setting of margins.
oma	Vector, setting of outer margins.
...	Dot argument, passed to plot.ts.

Author(s)

Bernhard Pfaff

References

Britton, E., P.G. Fisher and J.D. Whitley (1998), Inflation Report projections: understanding the fan chart, Bank of England Quarterly Bulletin, February, Bank of England, pages 30-37.

See Also

VAR, predict, plot, par

Examples

## Not run: 
data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
var.2c.prd <- predict(var.2c, n.ahead = 8, ci = 0.95)
fanchart(var.2c.prd)

## End(Not run)

---

Forecast Error Variance Decomposition

Description

Computes the forecast error variance decomposition of a VAR(p) for n.ahead steps.

Usage

## S3 method for class 'varest'
fevd(x, n.ahead=10, ...)
## S3 method for class 'svarest'
fevd(x, n.ahead=10, ...)
## S3 method for class 'svecest'
fevd(x, n.ahead=10, ...)
## S3 method for class 'vec2var'
fevd(x, n.ahead=10, ...)

Arguments

x	Object of class ‘varest’; generated by VAR(), or an object of class ‘svarest’; generated by SVAR(), or an object of class ‘vec2var’; generated by vec2var(), or an object of class ‘svecest’; generated by SVEC().
n.ahead	Integer specifying the steps.
...	Currently not used.

Details

The forecast error variance decomposition is based upon the orthogonalised impulse response coefficient matrices $\Psi_h$ and allow the user to analyse the contribution of variable $j$ to the h-step forecast error variance of variable $k$. If the orthogonalised impulse reponses are divided by the variance of the forecast error $\sigma_k^2(h)$, the resultant is a percentage figure. Formally:

$\sigma_k^2(h) = \sum_{n=0}^{h-1}(\psi_{k1, n}^2 + \ldots + \psi_{kK, n}^2)$

which can be written as:

$\sigma_k^2(h) = \sum_{j=1}^K(\psi_{kj, 0}^2 + \ldots + \psi_{kj, h-1}^2) \quad.$

Dividing the term $(\psi_{kj, 0}^2 + \ldots + \psi_{kj, h-1}^2)$ by $\sigma_k^2(h)$ yields the forecast error variance decompositions in percentage terms.

Value

A list with class attribute ‘varfevd’ of length K holding the forecast error variances as matrices.

Author(s)

Bernhard Pfaff

References

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

VAR, SVAR, vec2var, SVEC, Phi, Psi, plot

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
fevd(var.2c, n.ahead = 5)

---

Fit method for objects of class varest or vec2var

Description

Returns the fitted values of a VAR(p)-model for objects generated by VAR() or vec2var(). For objects of class varest the fitted.values-method is applied to the list element varresult, which is itself a list of lm-objects.

Usage

## S3 method for class 'varest'
fitted(object, ...)
## S3 method for class 'vec2var'
fitted(object, ...)

Arguments

object	An object of class ‘varest’; generated by VAR(), or an object of class ‘vec2var’; generated by vec2var().
...	Currently not used.

Author(s)

Bernhard Pfaff

References

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

VAR, vec2var

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
fitted(var.2c)

---

Impulse response function

Description

Computes the impulse response coefficients of a VAR(p) (or transformed VECM to VAR(p)) or a SVAR for n.ahead steps.

Usage

## S3 method for class 'varest'
irf(x, impulse = NULL, response = NULL, n.ahead = 10,
ortho = TRUE, cumulative = FALSE, boot = TRUE, ci = 0.95,
runs = 100, seed = NULL, ...)
## S3 method for class 'svarest'
irf(x, impulse = NULL, response = NULL, n.ahead = 10,
ortho = TRUE, cumulative = FALSE, boot = TRUE, ci = 0.95,
runs = 100, seed = NULL, ...)
## S3 method for class 'vec2var'
irf(x, impulse = NULL, response = NULL, n.ahead = 10,
ortho = TRUE, cumulative = FALSE, boot = TRUE, ci = 0.95,
runs = 100, seed = NULL, ...)
## S3 method for class 'svecest'
irf(x, impulse = NULL, response = NULL, n.ahead = 10,
ortho = TRUE, cumulative = FALSE, boot = TRUE, ci = 0.95,
runs = 100, seed = NULL, ...)

Arguments

x	Object of class ‘varest’; generated by VAR(), or object of class ‘svarest’; generated by SVAR(), or object of class ‘vec2var’; generated by vec2var(), or object of class ‘svecest’; generated by SVEC().
impulse	A character vector of the impulses, default is all variables.
response	A character vector of the responses, default is all variables.
n.ahead	Integer specifying the steps.
ortho	Logical, if TRUE (the default) the orthogonalised impulse response coefficients are computed (only for objects of class ‘varest’).
cumulative	Logical, if TRUE the cumulated impulse response coefficients are computed. The default value is false.
boot	Logical, if TRUE (the default) bootstrapped error bands for the imuplse response coefficients are computed.
ci	Numeric, the confidence interval for the bootstrapped errors bands.
runs	An integer, specifying the runs for the bootstrap.
seed	An integer, specifying the seed for the rng of the bootstrap.
...	Currently not used.

Details

The impulse response coefficients of a VAR(p) for n.ahead steps are computed by utilising either the function Phi() or Psi(). If boot = TRUE (the default), confidence bands for a given width specified by ci are derived from runs bootstrap. Hereby, it is at the users leisure to set a seed for the random number generator.
The standard percentile interval is defined as:

$CI_s = [s_{\alpha/2}^*, s_{1 - \alpha/2}^*] \quad ,$

with $s_{\alpha/2}^*$ and $s_{1 - \alpha/2}^*$ are the $\alpha/2$ and $1 - \alpha/2$ quantiles of the bootstrap distribution of $\Psi^*$ or $\Phi^*$ depending whether ortho = TRUE. In case cumulative = TRUE, the confidence bands are calculated from the cumulated impulse response coefficients.

Value

A list of class ‘varirf’ with the following elements is returned:

irf	A list with matrices for each of the impulse variables containing the impulse response coefficients.
Lower	If boot = TRUE, a list with matrices for each of the impulse variables containing the lower bands.
Upper	If boot = TRUE, a list with matrices for each of the impulse variables containing the upper bands.
response	Character vector holding the names of the response variables.
impulse	Character vector holding the names of the impulse variables.
ortho	Logical, if TRUE, orthogonalised impulse reponses have been computed.
cumulative	Logical, if TRUE, cumulated impulse reponses have been computed.
runs	An integer, specifying the number of bootstrap runs.
ci	Numeric, defining the confidence level.
boot	Logical, if TRUE bootstrapped error bands have been computed.
model	Character, containing ‘class(x)’.

Author(s)

Bernhard Pfaff

References

Efron, B. and R. J. Tibshirani (1993), An Introduction to the Bootstrap, Chapman and Hall, New York.

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

VAR, SVAR, vec2var, SVEC, Phi, Psi, plot

Examples

data(Canada)
## For VAR
var.2c <- VAR(Canada, p = 2, type = "const")
irf(var.2c, impulse = "e", response = c("prod", "rw", "U"), boot =
FALSE)
## For SVAR
amat <- diag(4)
diag(amat) <- NA
svar.a <- SVAR(var.2c, estmethod = "direct", Amat = amat)
irf(svar.a, impulse = "e", response = c("prod", "rw", "U"), boot =
FALSE)

---

Log-Likelihood method

Description

Returns the log-Likelihood of a VAR, level-VECM, SVAR or SVEC object.

Usage

## S3 method for class 'varest'
logLik(object, ...)
## S3 method for class 'vec2var'
logLik(object, ...)
## S3 method for class 'svarest'
logLik(object, ...)
## S3 method for class 'svecest'
logLik(object, ...)

Arguments

object	An object of class ‘varest’, generated by VAR(); or an object of class ‘vec2var’, generated by vec2var(); or an object of class ‘svarest’, generated by either SVAR() or an object of class ‘svecest’, generated by SVEC().
...	Currently not used.

Details

The log-likelihood of a VAR or level-VECM model is defined as:

$\log l = - \frac{K T}{2} \log 2 \pi - \frac{T}{2} \log |\Sigma_u| - \frac{1}{2} tr (U \Sigma^{-1}_u U')$

and for a SVAR / SVEC model the log-likelihood takes the form of:

$\log l = - \frac{K T}{2} \log 2 \pi + \frac{T}{2} \log |A|^2 - \frac{T}{2} \log |B|^2 - \frac{T}{2} tr (A'B'^{-1}B^{-1}A\Sigma_u)$

Value

An object with class attribute logLik.

Author(s)

Bernhard Pfaff

References

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

VAR, vec2var, SVAR, SVEC

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
logLik(var.2c)

---

Normality, multivariate skewness and kurtosis test

Description

This function computes univariate and multivariate Jarque-Bera tests and multivariate skewness and kurtosis tests for the residuals of a VAR(p) or of a VECM in levels.

Usage

normality.test(x, multivariate.only = TRUE)

Arguments

x	Object of class ‘varest’; generated by VAR(), or an object of class ‘vec2var’; generated by vec2var().
multivariate.only	If TRUE (the default), only multivariate test statistics are computed.

Details

Multivariate and univariate versions of the Jarque-Bera test are applied to the residuals of a VAR. The multivariate version of this test is computed by using the residuals that are standardized by a Choleski decomposition of the variance-covariance matrix for the centered residuals. Please note, that in this case the test result is dependant upon the ordering of the variables.

Value

A list of class ‘varcheck’ with the following elements is returned:

resid	A matrix of the residuals.
jb.uni	A list of elements with class attribute ‘htest’ containing the univariate Jarque-Bera tests. This element is only returned if multivariate.only = FALSE is set.
jb.mul	A list of elements with class attribute ‘htest’.

containing the mutlivariate Jarque-Bera test, the multivariate Skewness and Kurtosis tests.

Note

This function was named normality in earlier versions of package vars; it is now deprecated. See vars-deprecated too.

Author(s)

Bernhard Pfaff

References

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Jarque, C. M. and A. K. Bera (1987), A test for normality of observations and regression residuals, International Statistical Review, 55: 163-172.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

VAR, vec2var, plot

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
normality.test(var.2c)

---

Coefficient matrices of the MA represention

Description

Returns the estimated coefficient matrices of the moving average representation of a stable VAR(p), of an SVAR as an array or a converted VECM to VAR.

Usage

## S3 method for class 'varest'
Phi(x, nstep=10, ...)
## S3 method for class 'svarest'
Phi(x, nstep=10, ...)
## S3 method for class 'svecest'
Phi(x, nstep=10, ...)
## S3 method for class 'vec2var'
Phi(x, nstep=10, ...)

Arguments

x	An object of class ‘varest’, generated by VAR(), or an object of class ‘svarest’, generated by SVAR(), or an object of class ‘svecest’, generated by SVEC(), or an object of class ‘vec2var’, generated by vec2var().
nstep	An integer specifying the number of moving error coefficient matrices to be calculated.
...	Currently not used.

Details

If the process $\bold{y}_t$ is stationary (i.e. $I(0)$, it has a Wold moving average representation in the form of:

$\bold{y}_t = \Phi_0 \bold{u}_t + \Phi_1 \bold{u}_{t-1} + \Phi \bold{u}_{t-2} + \ldots ,$

whith $\Phi_0 = I_k$ and the matrices $\Phi_s$ can be computed recursively according to:

$\Phi_s = \sum_{j=1}^s \Phi_{s-j} A_j \quad s = 1, 2, \ldots ,$

whereby $A_j$ are set to zero for $j > p$. The matrix elements represent the impulse responses of the components of $\bold{y}_t$ with respect to the shocks $\bold{u}_t$. More precisely, the $(i, j)$th element of the matrix $\Phi_s$ mirrors the expected response of $y_{i, t+s}$ to a unit change of the variable $y_{jt}$.
In case of a SVAR, the impulse response matrices are given by:

$\Theta_i = \Phi_i A^{-1} B \quad .$

Albeit the fact, that the Wold decomposition does not exist for nonstationary processes, it is however still possible to compute the $\Phi_i$ matrices likewise with integrated variables or for the level version of a VECM. However, a convergence to zero of $\Phi_i$ as i tends to infinity is not ensured; hence some shocks may have a permanent effect.

Value

An array with dimension $(K \times K \times nstep + 1)$ holding the estimated coefficients of the moving average representation.

Note

The first returned array element is the starting value, i.e., $\Phi_0$.

Author(s)

Bernhard Pfaff

References

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

Psi, VAR, SVAR, vec2var, SVEC

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
Phi(var.2c, nstep=4)

---

Plot methods for objects in vars

Description

Plot method for objects with class attribute varest, vec2var, varcheck, varfevd, varirf, varprd, varstabil.

Usage

## S3 method for class 'varcheck'
plot(x, names = NULL, main.resid = NULL, main.hist =
NULL, main.acf = NULL, main.pacf = NULL, main.acf2 = NULL, main.pacf2 =
NULL, ylim.resid = NULL, ylim.hist = NULL, ylab.resid = NULL, xlab.resid
= NULL, xlab.acf = NULL, lty.resid = NULL, lwd.resid = NULL, col.resid =
NULL, col.edf = NULL, lag.acf = NULL, lag.pacf = NULL, lag.acf2 = NULL,
lag.pacf2 = NULL, mar = par("mar"), oma = par("oma"), ...)
## S3 method for class 'varest'
plot(x, names = NULL, main.fit = NULL, main.acf = NULL,
main.pacf = NULL, ylim.fit = NULL, ylim.resid = NULL, lty.fit = NULL,
lty.resid = NULL, lwd.fit = NULL, lwd.resid = NULL, lag.acf = NULL,
lag.pacf = NULL, col.fit = NULL, col.resid = NULL, ylab.fit = NULL,
ylab.resid = NULL, ylab.acf = NULL, ylab.pacf = NULL, xlab.fit = NULL,
xlab.resid = NULL, nc, mar = par("mar"), oma = par("oma"), adj.mtext =
NA, padj.mtext = NA, col.mtext = NA, ...)
## S3 method for class 'vec2var'
plot(x, ...)
## S3 method for class 'varfevd'
plot(x, plot.type = c("multiple", "single"),
names = NULL, main = NULL, col = NULL, ylim = NULL, ylab = NULL,
xlab = NULL, legend = NULL, names.arg = NULL, nc,
mar = par("mar"), oma = par("oma"), addbars = 1, ...)
## S3 method for class 'varirf'
plot(x, plot.type = c("multiple", "single"), names =
NULL, main = NULL, sub = NULL, lty = NULL, lwd = NULL, col = NULL, ylim
= NULL, ylab = NULL, xlab = NULL, nc, mar.multi = c(0, 4, 0, 4),
oma.multi = c(6, 4, 6, 4), adj.mtext = NA, padj.mtext = NA, col.mtext =
NA, ...)  
## S3 method for class 'varprd'
plot(x, plot.type = c("multiple", "single"),
names = NULL, main = NULL, col = NULL, lty = NULL, lwd = NULL,
ylim = NULL, ylab = NULL, xlab = NULL, nc, mar = par("mar"),
oma = par("oma"), ...)
## S3 method for class 'varstabil'
plot(x, plot.type = c("multiple", "single"), names =
NULL, main = NULL, nc, mar = par("mar"), oma = par("oma"), ...)

Arguments

addbars	Integer, number of empty bars in barplot to reserve space for legend. If set to zero, no legend will be returned.
adj.mtext	Adjustment for mtext(), only applicable if plot.type = "multiple".
col	Character vector, colors to use in plot.
col.edf	Character, color of residuals' EDF.
col.fit	Character vector, colors for diagram of fit.
col.mtext	Character, color for mtext(), only applicable if plot.type = "multiple".
col.resid	Character vector, colors for residual plot.
lag.acf	Integer, lag.max for ACF of residuals.
lag.acf2	Integer, lag.max for ACF of squared residuals.
lag.pacf	Integer, lag.max for PACF of residuals.
lag.pacf2	Integer, lag.max for PACF of squared residuals.
legend	Character vector of names in legend.
lty	Integer/Character, the line types.
lty.fit	Vector, lty for diagram of fit.
lty.resid	Vector, lty for residual plot.
lwd	The width of the lines.
lwd.fit	Vector, lwd for diagram of fit.
lwd.resid	Vector, lwd for residual plot.
main	Character vector, the titles of the plot.
main.acf	Character vector, main for residuals' ACF.
main.acf2	Character vector, main for squared residuals' ACF.
main.fit	Character vector, main for diagram of fit.
main.hist	Character vector, main for histogram of residuals.
main.pacf	Character vector, main for residuals' PACF.
main.pacf2	Character vector, main for squared residuals' PACF.
main.resid	Character vector, main for residual plot.
mar	Setting of margins.
mar.multi	Setting of margins, if plot.type = "multiple".
names	Character vector, the variables names to be plotted. If left NULL, all variables are plotted.
names.arg	Character vector, names for x-axis of barplot.
nc	Integer, number of columns for multiple plot.
oma	Setting of outer margins.
oma.multi	Setting of margins, if plot.type = "multiple".
padj.mtext	Adjustment for mtext(), only applicable if plot.type = "multiple".
plot.type	Character, if multiple all plots are drawn in a single device, otherwise the plots are shown consecutively.
sub	Character, sub title in plot.
x	An object of one of the above classes.
xlab	Character vector signifying the labels for the x-axis.
xlab.acf	Character, xlab for ACF and PACF of residuals and their squares in plot.varcheck.
xlab.fit	Character vector, xlab for diagram of fit.
xlab.resid	Character vector, xlab for residual plot.
ylab	Character vector signifying the labels for the y-axis.
ylab.acf	Character, ylab for ACF.
ylab.fit	Character vector, ylab for diagram of fit.
ylab.pacf	Character, ylab for PACF
ylab.resid	Character vector, ylab for residual plot.
ylim	Vector, the limits of the y-axis.
ylim.fit	Vector, ylim for diagram of fit.
ylim.hist	Vector, ylim for histogram of residuals.
ylim.resid	Vector, ylim for residual plot.
...	Passed to internal plot function.

Details

The plot-method for objects with class attribute vec2var is the same as for objects with class attribute varest. Hence, the same arguments can be utilised.

Author(s)

Bernhard Pfaff

References

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

Zeileis, A., F. Leisch, K. Hornik and C. Kleiber (2002), strucchange: An R Package for Testing for Structural Change in Linear Regression Models, Journal of Statistical Software, 7(2): 1-38, https://www.jstatsoft.org/v07/i02/

See Also

VAR, vec2var, fevd, irf, predict, fanchart, stability, arch.test, normality.test, serial.test

Examples

## Not run: 
data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
plot(var.2c)
## Diagnostic Testing
## ARCH test
archtest <- arch.test(var.2c)
plot(archtest)
## Normality test
normalitytest <- normality.test(var.2c)
plot(normalitytest)
## serial correlation test
serialtest <- serial.test(var.2c)
plot(serialtest)
## FEVD
var.2c.fevd <- fevd(var.2c, n.ahead = 5)
plot(var.2c.fevd)
## IRF
var.2c.irf <- irf(var.2c, impulse = "e",
response = c("prod", "rw", "U"), boot = FALSE)
plot(var.2c.irf)
## Prediction
var.2c.prd <- predict(var.2c, n.ahead = 8, ci = 0.95)
plot(var.2c.prd)
## Stability
var.2c.stabil <- stability(var.2c, type = "Rec-CUSUM")
plot(var.2c.stabil)

## End(Not run)

---

Predict method for objects of class varest and vec2var

Description

Forecating a VAR object of class ‘varest’ or of class ‘vec2var’ with confidence bands.

Usage

## S3 method for class 'varest'
predict(object, ..., n.ahead = 10, ci = 0.95, dumvar = NULL)
## S3 method for class 'vec2var'
predict(object, ..., n.ahead = 10, ci = 0.95, dumvar = NULL)

Arguments

object	An object of class ‘varest’; generated by VAR(), or an object of class ‘vec2var’; generated by vec2var().
n.ahead	An integer specifying the number of forecast steps.
ci	The forecast confidence interval
dumvar	Matrix for objects of class ‘vec2var’ or ‘varest’, if the dumvar argument in ca.jo() has been used or if the exogen argument in VAR() has been used, respectively. The matrix should have the same column dimension as in the call to ca.jo() or to VAR() and row dimension equal to n.ahead.
...	Currently not used.

Details

The n.ahead forecasts are computed recursively for the estimated VAR, beginning with $h = 1, 2, \ldots, n.ahead$:

$\bold{y}_{T+1 | T} = A_1 \bold{y}_T + \ldots + A_p \bold{y}_{T+1-p} + C D_{T+1}$

The variance-covariance matrix of the forecast errors is a function of $\Sigma_u$ and $\Phi_s$.

Value

A list with class attribute ‘varprd’ holding the following elements:

fcst	A list of matrices per endogenous variable containing the forecasted values with lower and upper bounds as well as the confidence interval.
endog	Matrix of the in-sample endogenous variables.
model	The estimated VAR object.
exo.fcst	If applicable provided values of exogenous variables, otherwise NULL.

Author(s)

Bernhard Pfaff

References

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

VAR, vec2var, plot, fanchart

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
predict(var.2c, n.ahead = 8, ci = 0.95)

---

Coefficient matrices of the orthogonalised MA represention

Description

Returns the estimated orthogonalised coefficient matrices of the moving average representation of a stable VAR(p) as an array.

Usage

## S3 method for class 'varest'
Psi(x, nstep=10, ...)
## S3 method for class 'vec2var'
Psi(x, nstep=10, ...)

Arguments

x	An object of class ‘varest’, generated by VAR(), or an object of class ‘vec2var’, generated by vec2var().
nstep	An integer specifying the number of othogonalised moving error coefficient matrices to be calculated.
...	Dots currently not used.

Details

In case that the components of the error process are instantaneously correlated with each other, that is: the off-diagonal elements of the variance-covariance matrix $\Sigma_u$ are not null, the impulses measured by the $\Phi_s$ matrices, would also reflect disturbances from the other variables. Therefore, in practice a Choleski decomposition has been propagated by considering $\Sigma_u = PP'$ and the orthogonalised shocks $\bold{\epsilon}_t = P^{-1}\bold{u}_t$. The moving average representation is then in the form of:

$\bold{y}_t = \Psi_0 \bold{\epsilon}_t + \Psi_1 \bold{\epsilon}_{t-1} + \Psi \bold{\epsilon}_{t-2} + \ldots ,$

whith $\Psi_0 = P$ and the matrices $\Psi_s$ are computed as $\Psi_s = \Phi_s P$ for $s = 1, 2, 3, \ldots$.

Value

An array with dimension $(K \times K \times nstep + 1)$ holding the estimated orthogonalised coefficients of the moving average representation.

Note

The first returned array element is the starting value, i.e., $\Psi_0$. Due to the utilisation of the Choleski decomposition, the impulse are now dependent on the ordering of the vector elements in $\bold{y}_t$.

Author(s)

Bernhard Pfaff

References

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

Phi, VAR, SVAR, vec2var

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
Psi(var.2c, nstep=4)

---

Residuals method for objects of class varest and vec2var

Description

Returns the residuals of a VAR(p)-model or for a VECM in levels. For the former class the residuals-method is applied to the list element varresult, which is itself a list of lm-objects.

Usage

## S3 method for class 'varest'
residuals(object, ...)
## S3 method for class 'vec2var'
residuals(object, ...)

Arguments

object	An object of class ‘varest’; generated by VAR(), or an object of class ‘vec2var’; generated by vec2var()
...	Currently not used.

Author(s)

Bernhard Pfaff

References

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

VAR, vec2var

Examples

## Not run: 
data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
resid(var.2c)

## End(Not run)

---

Restricted VAR

Description

Estimation of a VAR, by imposing zero restrictions manually or by significance.

Usage

restrict(x, method = c("ser", "manual"), thresh = 2.0, resmat = NULL)

Arguments

x	An object of class ‘varest’ generated by VAR().
method	A character, choosing the method
thresh	If method ser: the threshhold value for the t-statistics.
resmat	If method manual: The restriction matrix.

Details

Given an estimated VAR object of class ‘varest’, a restricted VAR can be obtained by either choosing method ser or manual. In the former case, each equation is re-estimated separately as long as there are t-values that are in absolut value below the threshhold value set by the function's argument thresh. In the latter case, a restriction matrix has to be provided that consists of 0/1 values, thereby selecting the coefficients to be retained in the model.

Value

A list with class attribute ‘varest’ holding the following elements:

varresult	list of ‘lm’ objects.
datamat	The data matrix of the endogenous and explanatory variables.
y	The data matrix of the endogenous variables
type	A character, specifying the deterministic regressors.
p	An integer specifying the lag order.
K	An integer specifying the dimension of the VAR.
obs	An integer specifying the number of used observations.
totobs	An integer specifying the total number of observations.
restrictions	The matrix object containing the zero restrictions provided as argument resmat.
call	The call to VAR().

Note

Currently, the restricted VAR is estimated by OLS and not by an efficient EGLS-method.

Author(s)

Bernhard Pfaff

References

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

VAR

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
## Restrictions determined by thresh
restrict(var.2c, method = "ser")
## Restrictions set manually
restrict <- matrix(c(1, 1, 1, 1, 1, 1, 0, 0, 0, 
                     1, 0, 1, 0, 0, 1, 0, 1, 1,
                     0, 0, 1, 1, 0, 1, 0, 0, 1,
                     1, 1, 1, 0, 1, 1, 0, 1, 0),
                   nrow=4, ncol=9, byrow=TRUE)
restrict(var.2c, method = "man", resmat = restrict)

---

Eigenvalues of the companion coefficient matrix of a VAR(p)-process

Description

Returns a vector of the eigenvalues of the companion coefficient matrix.

Usage

roots(x, modulus = TRUE)

Arguments

x	An object of class ‘varest’, generated by VAR().
modulus	Logical, set to TRUE for returning the modulus.

Details

Any VAR(p)-process can be written in a first-order vector autoregressive form: the companion form. A VAR(p)-process is stable, if its reverse characteristic polynomial:

$\det(I_K - A_1 z - \cdots - A_p z^p) \neq 0 \; \hbox{for} \; |z| \le 1 \; ,$

has no roots in or on the complex circle. This is equivalent to the condition that all eigenvalues of the companion matrix $A$ have modulus less than 1. The function roots(), does compute the eigen values of the companion matrix $A$ and returns by default their moduli.

Value

A vector object with the eigen values of the companion matrix, or their moduli (default).

Author(s)

Bernhard Pfaff

References

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

VAR

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
roots(var.2c)

---

Test for serially correlated errors

Description

This function computes the multivariate Portmanteau- and Breusch-Godfrey test for serially correlated errors.

Usage

serial.test(x, lags.pt = 16, lags.bg = 5, type = c("PT.asymptotic",
"PT.adjusted", "BG", "ES") )

Arguments

x	Object of class ‘varest’; generated by VAR(), or an object of class ‘vec2var’; generated by vec2var().
lags.pt	An integer specifying the lags to be used for the Portmanteau statistic.
lags.bg	An integer specifying the lags to be used for the Breusch-Godfrey statistic.
type	Character, the type of test. The default is an asymptotic Portmanteau test.

Details

The Portmanteau statistic for testing the absence of up to the order $h$ serially correlated disturbances in a stable VAR(p) is defined as:

$Q_h = T \sum_{j = 1}^h tr(\hat{C}_j'\hat{C}_0^{-1}\hat{C}_j\hat{C}_0^{-1}) \quad ,$

where $\hat{C}_i = \frac{1}{T}\sum_{t = i + 1}^T \bold{\hat{u}}_t \bold{\hat{u}}_{t - i}'$. The test statistic is approximately distributed as $\chi^2(K^2(h - p))$. This test statistic is choosen by setting type = "PT.asymptotic". For smaller sample sizes and/or values of $h$ that are not sufficiently large, a corrected test statistic is computed as:

$Q_h^* = T^2 \sum_{j = 1}^h \frac{1}{T - j}tr(\hat{C}_j'\hat{C}_0^{-1}\hat{C}_j\hat{C}_0^{-1}) \quad ,$

This test statistic can be accessed, if type = "PT.adjusted" is set.

The Breusch-Godfrey LM-statistic is based upon the following auxiliary regressions:

$\bold{\hat{u}}_t = A_1 \bold{y}_{t-1} + \ldots + A_p\bold{y}_{t-p} + CD_t + B_1\bold{\hat{u}}_{t-1} + \ldots + B_h\bold{\hat{u}}_{t-h} + \bold{\varepsilon}_t$

The null hypothesis is: $H_0: B_1 = \ldots = B_h = 0$ and correspondingly the alternative hypothesis is of the form $H_1: \exists \; B_i \ne 0$ for $i = 1, 2, \ldots, h$. The test statistic is defined as:

$LM_h = T(K - tr(\tilde{\Sigma}_R^{-1}\tilde{\Sigma}_e)) \quad ,$

where $\tilde{\Sigma}_R$ and $\tilde{\Sigma}_e$ assign the residual covariance matrix of the restricted and unrestricted model, respectively. The test statistic $LM_h$ is distributed as $\chi^2(hK^2)$. This test statistic is calculated if type = "BG" is used.

Edgerton and Shukur (1999) proposed a small sample correction, which is defined as:

$LMF_h = \frac{1 - (1 - R_r^2)^{1/r}}{(1 - R_r^2)^{1/r}} \frac{Nr - q}{K m} \quad ,$

with $R_r^2 = 1 - |\tilde{\Sigma}_e | / |\tilde{\Sigma}_R|$, $r = ((K^2m^2 - 4)/(K^2 + m^2 - 5))^{1/2}$, $q = 1/2 K m - 1$ and $N = T - K - m - 1/2(K - m + 1)$, whereby $n$ is the number of regressors in the original system and $m = Kh$. The modified test statistic is distributed as $F(hK^2, int(Nr - q))$. This modified statistic will be returned, if type = "ES" is provided in the call to serial().

Value

A list with class attribute ‘varcheck’ holding the following elements:

resid	A matrix with the residuals of the VAR.
pt.mul	A list with objects of class attribute ‘htest’ containing the multivariate Portmanteau-statistic (asymptotic and adjusted.
LMh	An object with class attribute ‘htest’ containing the Breusch-Godfrey LM-statistic.
LMFh	An object with class attribute ‘htest’ containing the Edgerton-Shukur F-statistic.

Note

This function was named serial in earlier versions of package vars; it is now deprecated. See vars-deprecated too.

Author(s)

Bernhard Pfaff

References

Breusch, T . S. (1978), Testing for autocorrelation in dynamic linear models, Australian Economic Papers, 17: 334-355.

Edgerton, D. and Shukur, G. (1999), Testing autocorrelation in a system perspective, Econometric Reviews, 18: 43-386.

Godfrey, L. G. (1978), Testing for higher order serial correlation in regression equations when the regressors include lagged dependent variables, Econometrica, 46: 1303-1313.

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

VAR, vec2var, plot

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
serial.test(var.2c, lags.pt = 16, type = "PT.adjusted")

---

Structural stability of a VAR(p)

Description

Computes an empirical fluctuation process according to a specified method from the generalised fluctuation test framework. The test utilises the function efp() and its methods from package ‘strucchange’.

Usage

## Default S3 method:
stability(x, type = c("OLS-CUSUM", "Rec-CUSUM",
"Rec-MOSUM", "OLS-MOSUM", "RE", "ME", "Score-CUSUM", "Score-MOSUM",
"fluctuation"), h = 0.15, dynamic = FALSE, rescale = TRUE, ...)
## S3 method for class 'varest'
stability(x, type = c("OLS-CUSUM", "Rec-CUSUM",
"Rec-MOSUM", "OLS-MOSUM", "RE", "ME", "Score-CUSUM", "Score-MOSUM",
"fluctuation"), h = 0.15, dynamic = FALSE, rescale = TRUE, ...)

Arguments

x	Object of class ‘varest’; generated by VAR().
type	Specifies which type of fluctuation process will be computed, the default is ‘OLS-CUSUM’. For details see: efp.
h	A numeric from interval (0,1) sepcifying the bandwidth. Determins the size of the data window relative to sample size (for ‘MOSUM’ and ‘ME’ processes only).
dynamic	Logical. If ‘TRUE’ the lagged observations are included as a regressor.
rescale	Logical. If ‘TRUE’ the estimates will be standardized by the regressor matrix of the corresponding subsample; if ‘FALSE’ the whole regressor matrix will be used. (only if ‘type’ is either ‘RE’ or ‘E’).
...	Ellipsis, is passed to strucchange::sctest(), as default.

Details

For details, please refer to documentation efp.

Value

A list with class attribute ‘varstabil’ holding the following elements:

stability	A list with objects of class ‘efp’; length is equal to the dimension of the VAR.
names	Character vector containing the names of the endogenous variables.
K	An integer of the VAR dimension.

Author(s)

Bernhard Pfaff

References

Zeileis, A., F. Leisch, K. Hornik and C. Kleiber (2002), strucchange: An R Package for Testing for Structural Change in Linear Regression Models, Journal of Statistical Software, 7(2): 1-38, https://www.jstatsoft.org/v07/i02/

and see the references provided in the reference section of efp, too.

See Also

VAR, plot, efp

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
var.2c.stabil <- stability(var.2c, type = "OLS-CUSUM")
var.2c.stabil
## Not run: 
plot(var.2c.stabil)

## End(Not run)

---

Summary method for objects of class varest, svarest and svecest

Description

'summary' methods for class '"varest"', '"svarest"' and '"svecest"'.

Usage

## S3 method for class 'varest'
summary(object, equations = NULL, ...)
## S3 method for class 'varsum'
print(x, digits = max(3, getOption("digits") - 3),
signif.stars = getOption("show.signif.stars"), ...)
## S3 method for class 'svarest'
summary(object,  ...)
## S3 method for class 'svarsum'
print(x, digits = max(3, getOption("digits") - 3), ...)
## S3 method for class 'svecest'
summary(object,  ...)
## S3 method for class 'svecsum'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

object	Object of class ‘varest’, usually, a result of a call to VAR, or object of class ‘svarest’, usually, a result of a call to SVAR, or object of class ‘svecest’, usually, a result of a call to SVEC.
equations	Character vector of endogenous variable names for which summary results should be returned. The default is NULL and results are returned for all equations in the VAR.
x	Object with class attribute ‘varsum’, ‘svarsum’.
digits	the number of significant digits to use when printing.
signif.stars	logical. If 'TRUE', ‘significance stars’ are printed for each coefficient.
...	further arguments passed to or from other methods.

Value

Returns either a list with class attribute varsum which contains the following elements:

names	Character vector with the names of the endogenous correlation matrix of VAR residuals.
logLik	Numeric, value of log Likelihood.
obs	Integer, sample size.
roots	Vector, roots of the characteristic polynomial.
type	Character vector, deterministic regressors included in VAR:
call	Call, the initial call to VAR.

Or a list with class attribute svarsum which contains the following elements:

type	Character, the type of SVAR-model.
A	Matrix, estimated coefficients for A matrix.
B	Matrix, estimated coefficients for B matrix.
Ase	Matrix, standard errors for A matrix.
Bse	Matrix, standard errors for B matrix.
LRIM	Matrix, long-run impact coefficients for BQ.
Sigma.U	Matrix, variance/covariance of reduced form residuals.
logLik	Numeric, value of log-Likelihood.
LR	htest, LR result of over-identification test.
obs	Integer, number of observations used.
opt	List, result of optim().
iter	Integer, the count of iterations.
call	Call, the call to SVAR().

Or a list with class attribute svecsum which contains the following elements:

type	Character, the type of SVEC-model.
SR	Matrix, contemporaneous impact matrix.
LR	Matrix, long-run impact matrix.
SRse	Matrix, standard errors for SR matrix.
LRse	Matrix, standard errors for LR matrix.
Sigma.U	Matrix, variance/covariance of reduced form residuals.
logLik	Numeric, value of log-Likelihood.
LRover	htest, LR result of over-identification test.
obs	Integer, number of observations used.
r	Integer, co-integration rank of VECM.
iter	Integer, the count of iterations.
call	Call, the call to SVEC().

Author(s)

Bernhard Pfaff

See Also

VAR, SVAR, SVEC

Examples

data(Canada)
## summary-method for varest
var.2c <- VAR(Canada, p = 2 , type = "const")
summary(var.2c)
## summary-method for svarest
amat <- diag(4)
diag(amat) <- NA
amat[2, 1] <- NA
amat[4, 1] <- NA
## Estimation method scoring
svar.a <- SVAR(x = var.2c, estmethod = "scoring", Amat = amat, Bmat = NULL,
max.iter = 100, maxls = 1000, conv.crit = 1.0e-8)
summary(svar.a)
## summary-method for svecest
vecm <- ca.jo(Canada[, c("prod", "e", "U", "rw")], type = "trace",
              ecdet = "trend", K = 3, spec = "transitory")
SR <- matrix(NA, nrow = 4, ncol = 4)
SR[4, 2] <- 0
LR <- matrix(NA, nrow = 4, ncol = 4)
LR[1, 2:4] <- 0
LR[2:4, 4] <- 0
svec.b <- SVEC(vecm, LR = LR, SR = SR, r = 1, lrtest = FALSE, boot =
FALSE)
summary(svec.b)

---

Estimation of a SVAR

Description

Estimates an SVAR (either ‘A-model’, ‘B-model’ or ‘AB-model’) by using a scoring algorithm or by directly minimising the negative log-likelihood with optim().

Usage

SVAR(x, estmethod = c("scoring", "direct"), Amat = NULL, Bmat = NULL,
start = NULL, max.iter = 100, conv.crit = 0.1e-6, maxls = 1.0,
lrtest = TRUE, ...)
## S3 method for class 'svarest'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

x	Object of class ‘varest’; generated by VAR().
estmethod	Character, either scoring for estimating the SVAR-model with the scoring algorithm (default), or directly minimizing the negative log-likelihood.
start	Vector with starting values for the parameters to be optimised.
lrtest	Logical, over-identification LR test, the result is set to NULL for just-identified system.
max.iter	Integer, maximum number of iteration (used if estmethod = "scoring").
conv.crit	Real, convergence value of algorithm (used if estmethod = "scoring").
maxls	Real, maximum movement of the parameters between two iterations of the scoring algorithm (used if estmethod = "scoring").
Amat	Matrix with dimension $(K \times K)$ for A- or AB-model.
Bmat	Matrix with dimension $(K \times K)$ for B- or AB-model.
digits	the number of significant digits to use when printing.
...	further arguments passed to or from other methods.

Details

Consider the following structural form of a k-dimensional vector autoregressive model:

$A \bold{y}_t = A_1^*\bold{y}_{t-1} + \ldots + A_p^*\bold{y}_{t-p} + C^*D_t + B\bold{\varepsilon}_t$

The coefficient matrices $(A_1^* | \ldots | A_p^* | C^*)$ might now differ from the ones of a VAR (see ?VAR). One can now impose restrictions on ‘A’ and/or ‘B’, resulting in an ‘A-model’ or ‘B-model’ or if the restrictions are placed on both matrices, an ‘AB-model’. In case of a SVAR ‘A-model’, $B = I_K$ and conversely for a SVAR ‘B-model’. Please note that for either an ‘A-model’ or ‘B-model’, $K(K-1)/2$ restrictions have to be imposed, such that the models' coefficients are identified. For an ‘AB-model’ the number of restrictions amounts to: $K^2 + K(K-1)/2$.
For an ‘A-model’ a $(K \times K)$ matrix has to be provided for the functional argument ‘Amat’ and the functional argument ‘Bmat’ must be set to ‘NULL’ (the default). Hereby, the to be estimated elements of ‘Amat’ have to be set as ‘NA’. Conversely, for a ‘B-model’ a matrix object with dimension $(K \times K)$ with elements set to ‘NA’ at the positions of the to be estimated parameters has to be provided and the functional argument ‘Amat’ is ‘NULL’ (the default). Finally, for an ‘AB-model’ both arguments, ‘Amat’ and ‘Bmat’, have to be set as matrix objects containing desired restrictions and ‘NA’ values. The parameters are estimated by minimising the negative of the concentrated log-likelihood function:

$\ln L_c(A, B) = - \frac{KT}{2}\ln(2\pi) + \frac{T}{2}\ln|A|^2 - \frac{T}{2}\ln|B|^2 - \frac{T}{2}tr(A'B'^{-1}B^{-1}A\tilde{\Sigma}_u)$

Two alternatives are implemented for this: a scoring algorithm or direct minimization with optim(). If the latter is chosen, the standard errors are returned if SVAR() is called with ‘hessian = TRUE’.

If ‘start’ is not set, then 0.1 is used as starting values for the unknown coefficients.

The reduced form residuals can be obtained from the above equation via the relation: $\bold{u}_t = A^{-1}B\bold{\varepsilon}_t$, with variance-covariance matrix $\Sigma_U = A^{-1}BB'A^{-1'}$.

Finally, in case of an overidentified SVAR, a likelihood ratio statistic is computed according to:

$LR = T(\ln\det(\tilde{\Sigma}_u^r) - \ln\det(\tilde{\Sigma}_u)) \quad ,$

with $\tilde{\Sigma}_u^r$ being the restricted variance-covariance matrix and $\tilde{\Sigma}_u$ being the variance covariance matrix of the reduced form residuals. The test statistic is distributed as $\chi^2(nr - 2K^2 - \frac{1}{2}K(K + 1))$, where $nr$ is equal to the number of restrictions.

Value

A list of class ‘svarest’ with the following elements is returned:

A	If A- or AB-model, the matrix of estimated coefficients.
Ase	The standard errors of ‘A’.
B	If A- or AB-model, the matrix of estimated coefficients.
Bse	The standard errors of ‘B’.
LRIM	For Blanchard-Quah estimation LRIM is the estimated long-run impact matrix; for all other SVAR models LRIM is NULL.
Sigma.U	The variance-covariance matrix of the reduced form residuals times 100, i.e., $\Sigma_U = A^{-1}BB'A^{-1'} \times 100$.
LR	Object of class ‘htest’, holding the Likelihood ratio overidentification test.
opt	List object returned by optim().
start	Vector of starting values.
type	SVAR-type, character, either ‘A-model’, ‘B-model’ or ‘AB-model’.
var	The ‘varest’ object ‘x’.
iter	Integer, the count of iterations.
call	The call to SVAR().

Author(s)

Bernhard Pfaff

References

Amisano, G. and C. Giannini (1997), Topics in Structural VAR Econometrics, 2nd edition, Springer, Berlin.

Breitung, J., R. Brüggemann and H. Lütkepohl (2004), Structural vector autoregressive modeling and impulse responses, in H. Lütkepohl and M. Krätzig (editors), Applied Time Series Econometrics, Cambridge University Press, Cambridge.

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

VAR, SVEC, logLik, irf, fevd

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
amat <- diag(4)
diag(amat) <- NA
amat[2, 1] <- NA
amat[4, 1] <- NA
## Estimation method scoring
SVAR(x = var.2c, estmethod = "scoring", Amat = amat, Bmat = NULL,
max.iter = 100, maxls = 1000, conv.crit = 1.0e-8) 
## Estimation method direct
SVAR(x = var.2c, estmethod = "direct", Amat = amat, Bmat = NULL,
hessian = TRUE, method="BFGS")

---

Estimation of a SVEC

Description

Estimates an SVEC by utilising a scoring algorithm.

Usage

SVEC(x, LR = NULL, SR = NULL, r = 1, start = NULL, max.iter = 100,
conv.crit = 1e-07, maxls = 1.0, lrtest = TRUE, boot = FALSE, runs = 100)
## S3 method for class 'svecest'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

x	Object of class ‘ca.jo’; generated by ca.jo() contained in urca.
LR	Matrix of the restricted long run impact matrix.
SR	Matrix of the restricted contemporaneous impact matrix.
r	Integer, the cointegration rank of x.
start	Vector of starting values for $\gamma$.
max.iter	Integer, maximum number of iteration.
conv.crit	Real, convergence value of algorithm..
maxls	Real, maximum movement of the parameters between two iterations of the scoring algorithm.
lrtest	Logical, over-identification LR test, the result is set to NULL for just-identified system.
boot	Logical, if TRUE, standard errors of the parameters are computed by bootstrapping. Default is FALSE.
runs	Integer, number of bootstrap replications.
digits	the number of significant digits to use when printing.
...	further arguments passed to or from other methods.

Details

Consider the following reduced form of a k-dimensional vector error correction model:

$A \Delta \bold{y}_t = \Pi \bold{y}_{t-1} + \Gamma_1 \Delta \bold{y}_{t-1} + \ldots + \Gamma_p \Delta \bold{y}_{t-p + 1} + \bold{u}_t \quad .$

This VECM has the following MA representation:

$\bold{y}_t = \Xi \sum_{i=1}^t \bold{u}_i + \Xi^*(L)\bold{u}_t + \bold{y}_0^* \quad ,$

with $\Xi = \beta_{\perp} (\alpha_{\perp}'(I_K - \sum_{i=1}^{p-1}\Gamma_i)\beta_{\perp} )^{-1}\alpha_{\perp}'$ and $\Xi^*(L)$ signifies an infinite-order polynomial in the lag operator with coefficient matrices $\Xi^*_j$ that tends to zero with increasing size of $j$.

Contemporaneous restrictions on the impact matrix $B$ must be supplied as zero entries in SR and free parameters as NA entries. Restrictions on the long run impact matrix $\Xi B$ have to be supplied likewise. The unknown parameters are estimated by maximising the concentrated log-likelihood subject to the imposed restrictions by utilising a scoring algorithm on:

$\ln L_c(A, B) = - \frac{KT}{2}\ln(2\pi) + \frac{T}{2}\ln|A|^2 - \frac{T}{2}\ln|B|^2 - \frac{T}{2}tr(A'B'^{-1}B^{-1}A\tilde{\Sigma}_u)$

with $\tilde{\Sigma}_u$ signifies the reduced form variance-covariance matrix and $A$ is set equal to the identity matrix $I_K$.

If ‘start’ is not set, then normal random numbers are used as starting values for the unknown coefficients. In case of an overidentified SVEC, a likelihood ratio statistic is computed according to:

$LR = T(\ln\det(\tilde{\Sigma}_u^r) - \ln\det(\tilde{\Sigma}_u)) \quad ,$

with $\tilde{\Sigma}_u^r$ being the restricted variance-covariance matrix and $\tilde{\Sigma}_u$ being the variance covariance matrix of the reduced form residuals. The test statistic is distributed as $\chi^2(K*(K+1)/2 - nr)$, where $nr$ is equal to the number of restrictions.

Value

A list of class ‘svecest’ with the following elements is returned:

SR	The estimated contemporaneous impact matrix.
SRse	The standard errors of the contemporaneous impact matrix, if boot = TRUE.
LR	The estimated long run impact matrix.
LRse	The standard errors of the long run impact matrix, if boot = TRUE.
Sigma.U	The variance-covariance matrix of the reduced form residuals times 100, i.e., $\Sigma_U = A^{-1}BB'A^{-1'} \times 100$.
Restrictions	Vector, containing the ranks of the restricted long run and contemporaneous impact matrices.
LRover	Object of class ‘htest’, holding the Likelihood ratio overidentification test.
start	Vector of used starting values.
type	Character, type of the SVEC-model.
var	The ‘ca.jo’ object ‘x’.
LRorig	The supplied long run impact matrix.
SRorig	The supplied contemporaneous impact matrix.
r	Integer, the supplied cointegration rank.
iter	Integer, the count of iterations.
call	The call to SVEC().

Author(s)

Bernhard Pfaff

References

Amisano, G. and C. Giannini (1997), Topics in Structural VAR Econometrics, 2nd edition, Springer, Berlin.

Breitung, J., R. Brüggemann and H. Lütkepohl (2004), Structural vector autoregressive modeling and impulse responses, in H. Lütkepohl and M. Krätzig (editors), Applied Time Series Econometrics, Cambridge University Press, Cambridge.

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

SVAR, irf, fevd

Examples

data(Canada)
vecm <- ca.jo(Canada[, c("prod", "e", "U", "rw")], type = "trace",
              ecdet = "trend", K = 3, spec = "transitory")
SR <- matrix(NA, nrow = 4, ncol = 4)
SR[4, 2] <- 0
SR
LR <- matrix(NA, nrow = 4, ncol = 4)
LR[1, 2:4] <- 0
LR[2:4, 4] <- 0
LR
SVEC(vecm, LR = LR, SR = SR, r = 1, lrtest = FALSE, boot = FALSE)

---