ACtest: Test for Error Autocorrelation in VAR Models.
In VARtests: Bootstrap Tests for Cointegration and Autocorrelation in VARs
ACtest R Documentation
Test for Error Autocorrelation in VAR Models.
Description
Performs LM tests for error AC in VAR models. The code is based on Paul Catani's original R code used in the paper Wild Bootstrap Tests for Autocorrelation in Vector Autoregressive Models (Ahlgren and Catani, 2016).
Usage
ACtest(fit, h = 4, HCtype = c("LM", "HC0", "HC1", "HC2", "HC3"), univariate = FALSE)
## S3 method for class 'ACtest'
print(x, ...)
Arguments
fit
an object of class "VARfit" that was returned by the VARfit function, or an object of class "verest" from the function vars::VAR in the vars package.
h
the lag length of the alternative VAR(h) model for the errors.
HCtype
a vector containing some or all (default) of "LM", "HC0", "HC1", "HC2" and "HC3". The "LM" type assumes homoscedastic errors, while the "HC" types are different Heteroskedasticity-consistent tests (see 'Details').
univariate
either FALSE, TRUE, or "only". If TRUE, the test will also be performed in its univariate form, if "only", only the univariate version will be used.
x
Object with class attribute ‘ACtest’.
...
further arguments passed to or from other methods.
Details
To run the wild bootstrap version of the test, please use the output from this function with the function wildBoot.
Mathematical details
The tests for error AC are based on the least squares (LS) residuals from the
K-dimensional vector autoregressive (VAR) model with p lags (abstracting from
deterministic terms):
\mathbf{y}_{t}=\mathbf{\Pi }_{1}\mathbf{y}_{t-1}+\cdots +\mathbf{\Pi }_{p}
\mathbf{y}_{t-p}+\mathbf{u}_{t},\quad \text{E}(\mathbf{u}_{t})=\mathbf{0}
,\quad \text{E}(\mathbf{u}_{t}\mathbf{u}_{t}^{\prime })=\mathbf{\Omega},\ \ \ \ t=1,\ldots ,N.
The LS residuals are
\widehat{\mathbf{u}}_{t}=\mathbf{y}_{t}-\widehat{\mathbf{\Pi }}_{1}\mathbf{y}
_{t-1}-\cdots -\widehat{\mathbf{\Pi }}_{p}\mathbf{y}_{t-p},
where \widehat{\mathbf{\Pi }}_{1},\ldots ,\widehat{\mathbf{\Pi }}_{p} are
the LS estimates of the K\times K parameter matrices \mathbf{\Pi }
_{1},\ldots ,\mathbf{\Pi }_{p}.
The LM statistic is computed from the auxilary model
\widehat{\mathbf{u}}_{t}=\mathbf{\Pi }_{1}\mathbf{y}
_{t-1}+\cdots +\mathbf{\Pi }_{p}\mathbf{y}_{t-p} + \mathbf{D}_{1}\widehat{\mathbf{u}}_{t-1}+\cdots +%
\mathbf{D}_{h}\widehat{\mathbf{u}}_{t-h}+\mathbf{e}_{t}
=(\mathbf{Z}_{t-1}^{\prime }\otimes \mathbf{I}_{K})\boldsymbol{\phi }+(\widehat{%
\mathbf{U}}_{t-1}^{\prime }\otimes \mathbf{I}_{K})\boldsymbol{\psi }+\mathbf{e}%
_{t}, \
where \mathbf{Z}_{t-1}=(\mathbf{y}_{t-1}^{\prime },\ldots ,\mathbf{y}_{t-p}^{\prime })^{\prime }, \boldsymbol{\phi }
=\text{vec}(\mathbf{Pi}_{1},\ldots ,\mathbf{Pi}_{p})^{\prime }, \widehat{\mathbf{U}}%
_{t-1}=(\widehat{\mathbf{u}}_{t-1}^{\prime },\ldots ,\widehat{%
\mathbf{u}}_{t-h}^{\prime })^{\prime } and \boldsymbol{\psi } =\text{vec}(\mathbf{D}_{1},\ldots
,\mathbf{D}_{h})^{\prime }. The symbol \otimes denotes the Kronecker product and
the symbol vec denotes the column vectorisation operator. The first h
values of the residuals \widehat{\mathbf{u}}_{t} are set to zero in the
auxiliary model, so that the series length is equal to the series length in
the original VAR model.
The LM statistic for error AC of order h is given by
Q_{\text{LM}}(h)
= N \widehat{\boldsymbol{\psi}}^{\prime}
\bigl(\widehat{\boldsymbol{\Sigma}}^{\psi\psi}\bigr)^{-1}
\widehat{\boldsymbol{\psi}}
where \widehat{\boldsymbol{\psi }} is the LS estimate of \boldsymbol{\psi }
and \widehat{\boldsymbol{\Sigma }}^{\psi \psi } is the block of
\left( N^{-1}\sum_{t=1}^{N}\left[
\begin{array}{c}
\mathbf{Z}_{t-1}\otimes \mathbf{I}_{K} \\
\widehat{\mathbf{U}}_{t-1}\otimes \mathbf{I}_{K}%
\end{array}%
\right] \widehat{\boldsymbol{\Sigma }}_{\mathbf{u}}^{-1}\left[
\begin{array}{cc}
\mathbf{Z}_{t-1}^{\prime }\otimes \mathbf{I}_{K} & \widehat{\mathbf{U}}%
_{t-1}^{\prime }\otimes \mathbf{I}_{K}%
\end{array}%
\right] \right) ^{-1}
corresponding to \boldsymbol{\psi }. Here \widehat{\boldsymbol{\Sigma }}_{\mathbf{u}%
}=N^{-1}\sum_{t=1}^{N}\widehat{\mathbf{u}}_{t}\widehat{\mathbf{u}}%
_{t}^{\prime } is the estimator of the error covariance matrix from the
VAR model.
The multivariate heteroskedasticity-consistent covariance matrix estimator (HCCME)
for the auxilary model is given by (Hafner and Herwartz 2009)
\mathbf{V}_{N}^{-1}\mathbf{W}_{N}\mathbf{V}_{N}^{-1}
= (\boldsymbol{\Gamma}_{N}\otimes \mathbf{I}_{K})^{-1}
\mathbf{W}_{N}
(\boldsymbol{\Gamma}_{N}\otimes \mathbf{I}_{K})^{-1}
where
\mathbf{V}_{N} =\boldsymbol{\Gamma }_{N}\otimes \mathbf{I}_{K},
\boldsymbol{\Gamma }_{N} =\frac{1}{N}\sum_{t=1}^{N}\left(
\begin{array}{c}
\widehat{\mathbf{U}}_{t-1} \\
\mathbf{Z}_{t-1}%
\end{array}%
\right) \left(
\begin{array}{cc}
\widehat{\mathbf{U}}_{t-1}^{\prime } & \mathbf{Z}_{t-1}^{\prime }%
\end{array}%
\right) ,
\mathbf{W}_{N} =\frac{1}{N}\sum_{t=1}^{N}\left(
\begin{array}{c}
\widehat{\mathbf{U}}_{t-1} \\
\mathbf{Z}_{t-1}%
\end{array}%
\right) \left(
\begin{array}{cc}
\widehat{\mathbf{U}}_{t-1}^{\prime } & \mathbf{Z}_{t-1}^{\prime }%
\end{array}%
\right) \otimes (\widehat{\mathbf{u}}_{t}\widehat{\mathbf{u}}_{t}^{\prime }).
The HCCME-based LM statistics for error AC are obtained from the expression for Q_{\text{LM}}(h)
by replacing \widehat{\boldsymbol{\Sigma }}^{\psi \psi } by the block of %
\mathbf{V}_{N}^{-1}\mathbf{W}_{N}\mathbf{V}_{N}^{-1}=(\Gamma _{N}\otimes \mathbf{I}_{K})^{-1}\mathbf{W}_{N}(\Gamma
_{N}\otimes \mathbf{I}_{K})^{-1} corresponding to \boldsymbol{\psi } and with %
\widehat{\mathbf{u}}_{t} defined by HC_{0}, HC_{1}, HC_{2} and HC_{3}
, respectively.
HC_{0} uses \widehat{\mathbf{u}}_{t}\widehat{\mathbf{u}}_{t}^{\prime }.
HC_{1} multiplies the elements of \widehat{\mathbf{u}}_{t}\widehat{\mathbf{u}}_{t}^{\prime } by
N/(N-Kp).
HC_{2} replaces \widehat{%
\mathbf{u}}_{t} by \widehat{\mathbf{u}}_{t}/(1-h_{t})^{1/2}, where h_{t}=\mathbf{Z}_{t}(\mathbf{Z}^{\prime }\mathbf{Z})^{-1}\mathbf{Z}_{t}^{\prime } is the tth
diagonal element of \mathbf{Z}(\mathbf{Z}^{\prime }\mathbf{Z})^{-1}\mathbf{Z}^{\prime }, and \mathbf{%
Z}=(\mathbf{Z}_{0},\ldots ,\mathbf{Z}_{N-1}).
HC_{3} replaces \widehat{\mathbf{u}}_{t} by \widehat{\mathbf{u}}_{t}/(1-h_{t}).
See MacKinnon and White (1985) for details.
The recursive-design wild bootstrap (WB) tests for error AC are computed using
Algorithm 1 in Ahlgren and Catani (2016). The Fixed-design WB tests for error AC
are computed using Algorithm 2 in Ahlgren and Catani (2016).
Value
a list of class "ACtest".
fit
the fit argument object.
inputType
the type of object of fit.
HCtype
a vector of the HCtype's used.
h
the lag length of the alternative VAR(h) model for the errors.
pValues
a 1 x 5 matrix of the P. values of the tests.
Q
a 1 x 5 matrix of the Q statistics of the tests.
unipValues
a K x 5 matrix of the P. values of the univariate tests.
uniQ
a K x 5 matrix of the Q statistics of the univariate tests.
univariate
the 'univariate' argument.
description
who ran the test and when.
time
computation time taken to run the test.
call
how the function ACtest() was called.
References
Ahlgren, N. & Catani, P. (2016).
Wild bootstrap tests for autocorrelation in vector autoregressive models. Stat Papers, <doi:10.1007/s00362-016-0744-0>.
Hafner, C. M. and Herwartz, H., (2009).
Testing for Linear
Vector Autoregressive Dynamics under Multivariate Generalized Autoregressive Heteroskedasticity. Stat Neerl, 63, 294–323
MacKinnon, J. G. and White, H. (1985).
Some Heteroskedasticity Consistent Covariance Matrix Estimators with Improved Finite Sample Properties. J Econom, 29, 305–325
See Also
VARfit to estimate a VAR(p), and wildBoot to run the Wild Bootstrap versions of the tests.
Examples
fit <- VARfit(y = VodafoneCDS, p = 3, const = TRUE, trend = FALSE)
test <- ACtest(fit = fit, h = 1, HCtype = c("LM", "HC0", "HC1", "HC2", "HC3"))
test
VARtests documentation built on Aug. 8, 2025, 7:16 p.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
| ACtest | R Documentation |
Test for Error Autocorrelation in VAR Models.
Description
Performs LM tests for error AC in VAR models. The code is based on Paul Catani's original R code used in the paper Wild Bootstrap Tests for Autocorrelation in Vector Autoregressive Models (Ahlgren and Catani, 2016).
Usage
ACtest(fit, h = 4, HCtype = c("LM", "HC0", "HC1", "HC2", "HC3"), univariate = FALSE)
## S3 method for class 'ACtest'
print(x, ...)
Arguments
fit |
an object of class |
h |
the lag length of the alternative VAR(h) model for the errors. |
HCtype |
a vector containing some or all (default) of |
univariate |
either |
x |
Object with class attribute ‘ACtest’. |
... |
further arguments passed to or from other methods. |
Details
To run the wild bootstrap version of the test, please use the output from this function with the function wildBoot.
Mathematical details
The tests for error AC are based on the least squares (LS) residuals from the
K-dimensional vector autoregressive (VAR) model with p lags (abstracting from
deterministic terms):
\mathbf{y}_{t}=\mathbf{\Pi }_{1}\mathbf{y}_{t-1}+\cdots +\mathbf{\Pi }_{p}
\mathbf{y}_{t-p}+\mathbf{u}_{t},\quad \text{E}(\mathbf{u}_{t})=\mathbf{0}
,\quad \text{E}(\mathbf{u}_{t}\mathbf{u}_{t}^{\prime })=\mathbf{\Omega},\ \ \ \ t=1,\ldots ,N.
The LS residuals are
\widehat{\mathbf{u}}_{t}=\mathbf{y}_{t}-\widehat{\mathbf{\Pi }}_{1}\mathbf{y}
_{t-1}-\cdots -\widehat{\mathbf{\Pi }}_{p}\mathbf{y}_{t-p},
where \widehat{\mathbf{\Pi }}_{1},\ldots ,\widehat{\mathbf{\Pi }}_{p} are
the LS estimates of the K\times K parameter matrices \mathbf{\Pi }
_{1},\ldots ,\mathbf{\Pi }_{p}.
The LM statistic is computed from the auxilary model
\widehat{\mathbf{u}}_{t}=\mathbf{\Pi }_{1}\mathbf{y}
_{t-1}+\cdots +\mathbf{\Pi }_{p}\mathbf{y}_{t-p} + \mathbf{D}_{1}\widehat{\mathbf{u}}_{t-1}+\cdots +%
\mathbf{D}_{h}\widehat{\mathbf{u}}_{t-h}+\mathbf{e}_{t}
=(\mathbf{Z}_{t-1}^{\prime }\otimes \mathbf{I}_{K})\boldsymbol{\phi }+(\widehat{%
\mathbf{U}}_{t-1}^{\prime }\otimes \mathbf{I}_{K})\boldsymbol{\psi }+\mathbf{e}%
_{t}, \
where \mathbf{Z}_{t-1}=(\mathbf{y}_{t-1}^{\prime },\ldots ,\mathbf{y}_{t-p}^{\prime })^{\prime }, \boldsymbol{\phi }
=\text{vec}(\mathbf{Pi}_{1},\ldots ,\mathbf{Pi}_{p})^{\prime }, \widehat{\mathbf{U}}%
_{t-1}=(\widehat{\mathbf{u}}_{t-1}^{\prime },\ldots ,\widehat{%
\mathbf{u}}_{t-h}^{\prime })^{\prime } and \boldsymbol{\psi } =\text{vec}(\mathbf{D}_{1},\ldots
,\mathbf{D}_{h})^{\prime }. The symbol \otimes denotes the Kronecker product and
the symbol vec denotes the column vectorisation operator. The first h
values of the residuals \widehat{\mathbf{u}}_{t} are set to zero in the
auxiliary model, so that the series length is equal to the series length in
the original VAR model.
The LM statistic for error AC of order h is given by
Q_{\text{LM}}(h)
= N \widehat{\boldsymbol{\psi}}^{\prime}
\bigl(\widehat{\boldsymbol{\Sigma}}^{\psi\psi}\bigr)^{-1}
\widehat{\boldsymbol{\psi}}
where \widehat{\boldsymbol{\psi }} is the LS estimate of \boldsymbol{\psi }
and \widehat{\boldsymbol{\Sigma }}^{\psi \psi } is the block of
\left( N^{-1}\sum_{t=1}^{N}\left[
\begin{array}{c}
\mathbf{Z}_{t-1}\otimes \mathbf{I}_{K} \\
\widehat{\mathbf{U}}_{t-1}\otimes \mathbf{I}_{K}%
\end{array}%
\right] \widehat{\boldsymbol{\Sigma }}_{\mathbf{u}}^{-1}\left[
\begin{array}{cc}
\mathbf{Z}_{t-1}^{\prime }\otimes \mathbf{I}_{K} & \widehat{\mathbf{U}}%
_{t-1}^{\prime }\otimes \mathbf{I}_{K}%
\end{array}%
\right] \right) ^{-1}
corresponding to \boldsymbol{\psi }. Here \widehat{\boldsymbol{\Sigma }}_{\mathbf{u}%
}=N^{-1}\sum_{t=1}^{N}\widehat{\mathbf{u}}_{t}\widehat{\mathbf{u}}%
_{t}^{\prime } is the estimator of the error covariance matrix from the
VAR model.
The multivariate heteroskedasticity-consistent covariance matrix estimator (HCCME) for the auxilary model is given by (Hafner and Herwartz 2009)
\mathbf{V}_{N}^{-1}\mathbf{W}_{N}\mathbf{V}_{N}^{-1}
= (\boldsymbol{\Gamma}_{N}\otimes \mathbf{I}_{K})^{-1}
\mathbf{W}_{N}
(\boldsymbol{\Gamma}_{N}\otimes \mathbf{I}_{K})^{-1}
where
\mathbf{V}_{N} =\boldsymbol{\Gamma }_{N}\otimes \mathbf{I}_{K},
\boldsymbol{\Gamma }_{N} =\frac{1}{N}\sum_{t=1}^{N}\left(
\begin{array}{c}
\widehat{\mathbf{U}}_{t-1} \\
\mathbf{Z}_{t-1}%
\end{array}%
\right) \left(
\begin{array}{cc}
\widehat{\mathbf{U}}_{t-1}^{\prime } & \mathbf{Z}_{t-1}^{\prime }%
\end{array}%
\right) ,
\mathbf{W}_{N} =\frac{1}{N}\sum_{t=1}^{N}\left(
\begin{array}{c}
\widehat{\mathbf{U}}_{t-1} \\
\mathbf{Z}_{t-1}%
\end{array}%
\right) \left(
\begin{array}{cc}
\widehat{\mathbf{U}}_{t-1}^{\prime } & \mathbf{Z}_{t-1}^{\prime }%
\end{array}%
\right) \otimes (\widehat{\mathbf{u}}_{t}\widehat{\mathbf{u}}_{t}^{\prime }).
The HCCME-based LM statistics for error AC are obtained from the expression for Q_{\text{LM}}(h)
by replacing \widehat{\boldsymbol{\Sigma }}^{\psi \psi } by the block of %
\mathbf{V}_{N}^{-1}\mathbf{W}_{N}\mathbf{V}_{N}^{-1}=(\Gamma _{N}\otimes \mathbf{I}_{K})^{-1}\mathbf{W}_{N}(\Gamma
_{N}\otimes \mathbf{I}_{K})^{-1} corresponding to \boldsymbol{\psi } and with %
\widehat{\mathbf{u}}_{t} defined by HC_{0}, HC_{1}, HC_{2} and HC_{3}
, respectively.
HC_{0} uses \widehat{\mathbf{u}}_{t}\widehat{\mathbf{u}}_{t}^{\prime }.
HC_{1} multiplies the elements of \widehat{\mathbf{u}}_{t}\widehat{\mathbf{u}}_{t}^{\prime } by
N/(N-Kp).
HC_{2} replaces \widehat{%
\mathbf{u}}_{t} by \widehat{\mathbf{u}}_{t}/(1-h_{t})^{1/2}, where h_{t}=\mathbf{Z}_{t}(\mathbf{Z}^{\prime }\mathbf{Z})^{-1}\mathbf{Z}_{t}^{\prime } is the tth
diagonal element of \mathbf{Z}(\mathbf{Z}^{\prime }\mathbf{Z})^{-1}\mathbf{Z}^{\prime }, and \mathbf{%
Z}=(\mathbf{Z}_{0},\ldots ,\mathbf{Z}_{N-1}).
HC_{3} replaces \widehat{\mathbf{u}}_{t} by \widehat{\mathbf{u}}_{t}/(1-h_{t}).
See MacKinnon and White (1985) for details.
The recursive-design wild bootstrap (WB) tests for error AC are computed using Algorithm 1 in Ahlgren and Catani (2016). The Fixed-design WB tests for error AC are computed using Algorithm 2 in Ahlgren and Catani (2016).
Value
a list of class "ACtest".
fit |
the |
inputType |
the type of object of |
HCtype |
a vector of the |
h |
the lag length of the alternative VAR(h) model for the errors. |
pValues |
a 1 x 5 matrix of the P. values of the tests. |
Q |
a 1 x 5 matrix of the Q statistics of the tests. |
unipValues |
a K x 5 matrix of the P. values of the univariate tests. |
uniQ |
a K x 5 matrix of the Q statistics of the univariate tests. |
univariate |
the 'univariate' argument. |
description |
who ran the test and when. |
time |
computation time taken to run the test. |
call |
how the function |
References
Ahlgren, N. & Catani, P. (2016). Wild bootstrap tests for autocorrelation in vector autoregressive models. Stat Papers, <doi:10.1007/s00362-016-0744-0>.
Hafner, C. M. and Herwartz, H., (2009). Testing for Linear Vector Autoregressive Dynamics under Multivariate Generalized Autoregressive Heteroskedasticity. Stat Neerl, 63, 294–323
MacKinnon, J. G. and White, H. (1985). Some Heteroskedasticity Consistent Covariance Matrix Estimators with Improved Finite Sample Properties. J Econom, 29, 305–325
See Also
VARfit to estimate a VAR(p), and wildBoot to run the Wild Bootstrap versions of the tests.
Examples
fit <- VARfit(y = VodafoneCDS, p = 3, const = TRUE, trend = FALSE)
test <- ACtest(fit = fit, h = 1, HCtype = c("LM", "HC0", "HC1", "HC2", "HC3"))
test
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.