archBootTest: Combined LM test for ARCH errors in VAR models.

In VARtests: Bootstrap Tests for Cointegration and Autocorrelation in VARs

Combined LM test for ARCH errors in VAR models.

Description

Performs the bootstrap combined Lagrange multiplier (LM) test for autoregressive conditional heteroskedastic (ARCH) errors in vector autoregressive (VAR) models of Catani and Ahlgren (2016).

The tests of Eklund and Teräsvirta (2007), as well as the Multivariate LM test for ARCH as described for example in Lütkepohl (2006, sect. 16.5), are also included if the arguments ET respectively MARCH are set to TRUE. The bootstrap procedure for those are the same as in Catani and Ahlgren (2016).

Usage

archBootTest(fit, h = 2, B = 499, CA = TRUE, ET = TRUE, MARCH = TRUE,
             dist = "norm", skT.param = c(0, 1, 0, 5), verbose = TRUE)
## S3 method for class 'archBootTest'
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.
B	the number of bootstrap simulations.
CA	if TRUE, the Catani and Ahlgren (2017) test will run.
ET	if TRUE, the Eklund and Teräsvirta (2007) test will run.
MARCH	if TRUE, the Multivariate LM test for ARCH will run. See e.g. Lütkepohl (2006, sect. 16.5).
dist	the error distribution. Either "norm" for the standard normal distribution, or "skT" for the skew-t distribution. The parameters of the skew-t distribution can be set with the skT.param argument. Can also be a function that returns random draws as an (N-p) x K matrix or a vector of length (N-p) * K.
skT.param	a vector of four parameters for the skew-t distribution in case "skT" was used for the dist argument. The function rmst is used to draw the errors and the parameters are passed as skT.param = c(xi, Omega, alpha, nu)
verbose	logical; if TRUE, prints progress messages and an estimated completion time during the bootstrap simulation.
x	Object with class attribute ‘archBootTest’.
...	further arguments passed to or from other methods.

Details

All tests for ARCH are based on Cholesky-standardised 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 multivariate LS residuals are \widehat{\mathbf{U}}=(\widehat{\mathbf{u}}_{1},\ldots ,\widehat{\mathbf{u}} _{K}), which is an N\times K matrix. The Cholesky-standardised LS residuals are

\widetilde{\mathbf{w}}_{t}=(\mathbf{S}_{\widehat{\mathbf{U}}}^{-1})^{\prime } \widehat{\mathbf{u}}_{t},

where \mathbf{S}_{\widehat{\mathbf{U}}} is the Cholesky factor of N^{-1} \widehat{\mathbf{U}}^{\prime }\widehat{\mathbf{U}}, i.e. \mathbf{S}_{ \widehat{\mathbf{U}}} is the (unique) upper triangular matrix such that

\widehat{\mathbf{\Omega }}=\mathbf{S}_{\widehat{\mathbf{U}}}^{\prime } \mathbf{S}_{\widehat{\mathbf{U}}},\quad \widehat{\mathbf{\Omega }} ^{-1}=(N^{-1}\widehat{\mathbf{U}}^{\prime }\widehat{\mathbf{U}})^{-1}= \mathbf{S}_{\widehat{\mathbf{U}}}^{-1}(\mathbf{S}_{\widehat{\mathbf{U}} }^{-1})^{\prime }.

The LM test for ARCH of order h (Engle 1982) in equation i, i=1,\ldots ,K, is a test of H_{0}:b_{1}=\cdots =b_{h} against H_{1}:b_{j}\neq 0 for at least one j\in \{1,\ldots ,h\} in the auxiliary regression

\widetilde{w}_{it}^{2}=b_{0}+b_{1}\widetilde{w}_{i,t-1}^{2}+\cdots +b_{h} \widetilde{w}_{i,t-h}^{2}+e_{it}.

The LM statistic has the form

LM_{i}=(N-p)R_{i}^{2},

where R_{i}^{2} is R^{2} from the auxiliary regression for equation i.

The combined LM statistic (Dufour et al. 2010, Catani and Ahlgren 2016) is given by

\widetilde{LM}=1-\min_{1\leq i\leq K}(p(LM_{i})),

where p(LM_{i}) is the p-value of the LM_{i} statistic, derived from the asymptotic \chi ^{2}(h) distibution. The test is only available as a bootstrap test. The bootstrap p-value is simulated using Bootstrap Algorithm 1 of Catani and Ahlgren (2016) if the errors are normal,

w_{i1},\ldots ,w_{iT}\sim \text{N}(0,1),

and Bootstrap Algorithm 2 if the errors are skew-t (by setting the function argument dist = "skT"),

w_{i1},\ldots ,w_{iT}\sim \text{skT}(0,1;\lambda ,v),

where \lambda is the skewness parameter and v is the degrees-of-freedom parameter of the skew-t distribution. These parameters can be set with the skT.param argument.

The multivariate LM test for ARCH of order h is a generalisation of the univariate test, and is based on the auxiliary regression

\text{vech}(\widetilde{\mathbf{u}}_{t}\widetilde{\mathbf{u}}_{t}^{\prime })=\mathbf{b} _{0}+\mathbf{B}_{1} \text{vech}(\widetilde{\mathbf{u}}_{t-1}\widetilde{\mathbf{u}}_{t-1}^{\prime }) +\cdots+\mathbf{B}_{h} \text{vech}(\widetilde{\mathbf{u}}_{t-h}\widetilde{\mathbf{u}}_{t-h}^{\prime }) +\mathbf{e}_{t},

where \text{vech} is the half-vectorisation operator. The null hypothesis is H_{0}:\mathbf{B }_{1}=\cdots =\mathbf{B}_{h}=\mathbf{0} against H_{1}:\mathbf{B}_{j}\neq \mathbf{0\!} for at least one j\in \{1,\ldots ,h\}. The multivariate LM statistic has the form

MLM=\frac{1}{2}(N-p)K(K+1)-(N-p)\text{tr}(\widehat{\mathbf{\Omega}}_{\text{vech}}\widehat{\mathbf{\Omega}}^{-1}),

where \widehat{\mathbf{\Omega }}_{\text{vech}} is the estimator of the error covariance matrix from the auxiliary regression and \widehat{\mathbf{ \Omega }} =N^{-1}\sum_{t=1}^{N}\widetilde{\mathbf{u}}_{t}\widetilde{ \mathbf{u}}_{t}^{\prime } is the estimator of the error covariance matrix from the VAR model (see Lütkepohl 2006, sect. 16.5). The MLM statistic is asymptotically distributed as \chi ^{2}(K^{2}(K+1)^{2}h/4). The test is available as an asymptotic test using the asymptotic \chi ^{2}(K^{2}(K+1)^{2}h/4) distribution to derive the p-value, and as a bootstrap test. The bootstrap p-value is simulated using Bootstrap Algorithms 1 and 2 of Catani and Ahlgren (2016). The asymptotic validity of the bootstrap multivariate LM test has not been established.

The Eklund and Teräsvirta (2007) LM test of constant error covariance matrix assumes the alternative hypothesis is a constant conditional correlation autoregressive conditional heteroskedasticity (CCC-ARCH) process of order h: \mathbf{H}_{t}=\mathbf{D}_{t}\mathbf{PD}_{t}, where \mathbf{ D}_{t}=\text{diag}(h_{1t}^{1/2},\ldots ,h_{Kt}^{1/2}) is a diagonal matrix of conditional standard deviations of the errors \{\mathbf{u}_{t}\} and \mathbf{P}=(\rho _{ij}), i,j=1,\ldots ,K, is a positive definite matrix of conditional correlations. The conditional variance \mathbf{h}_{t}=(h_{1t},\ldots ,h_{Kt})^{\prime } is assumed to follow a CCC-ARCH(h) process:

\mathbf{h}_{t}=\mathbf{a}_{0}+\sum_{j=1}^{h}\mathbf{A}_{j}\boldsymbol{u} _{t-j}^{(2)},

where \mathbf{a}_{0}=(a_{01},\ldots ,a_{0K})^{\prime } is a K-dimensional vector of positive constants, \mathbf{A}_{1},\ldots ,\mathbf{A} _{h} are K\times K diagonal matrices and \boldsymbol{u} _{t}^{(2)}=(u_{1t}^{2},\ldots ,u_{Kt}^{2})^{\prime }.

The null hypothesis is H_{0}:\text{diag}(\mathbf{A}_{1})=\cdots =\text{diag}(\mathbf{A}_{h})=\mathbf{0} against H_{1}: \text{diag}(\mathbf{A}_{j})\neq \mathbf{0\!} for at least one j\in \{1,\ldots ,h\}. The LM statistic has the form

LM_{CCC}=(N-p)\mathbf{s}(\widehat{\boldsymbol{\theta }})^{\prime }\mathbf{I}( \widehat{\boldsymbol{\theta }})^{-1}\mathbf{s}(\widehat{\boldsymbol{\theta }} ),

where \mathbf{s}(\widehat{\boldsymbol{\theta }}) and \mathbf{I}(\widehat{ \boldsymbol{\theta }}) are the score vector and information matrix, respectively, estimated under the null hypothesis (see Eklund and Teräsvirta 2007 for details). The asymptotic distribution of the LM_{CCC} statistic is \chi ^{2}(Kh). The test is available as an asymptotic test using the asymptotic \chi ^{2}(Kh) distribution to derive the p-value, and as a bootstrap test. The bootstrap p-value is simulated using Bootstrap Algorithms 1 and 2 of Catani and Ahlgren (2016). The asymptotic validity of the bootstrap LM_{CCC} test has not been established.

Value

a list of class "ACtest".

fit	the fit argument object.
inputType	the type of object of fit.
h	the lag length h of the alternative VAR(h) model for the errors.
B	the number of bootstrap simulations.
K	the number of series/equations in the fitted VAR model.
CA	the CA input argument.
ET	the ET input argument.
MARCH	the MARCH input argument.
dist	the dist argument object.
standardizedResi	the Cholesky-standardized residuals.
CA_LM	the combined LM statistic of Catani and Ahlgren (2016), computed as 1 - min(P(CA_LMi)).
CA_bootPV	the bootstrap P. value of the combined LM test of Catani and Ahlgren (2016).
CA_LMi	the LM statistics of Catani and Ahlgren (2016) for each time series.
CA_LMijStar	an (N-p) x K matrix of the bootstrap LM statistics for each time series (columns) and bootstrap sample (rows), for the Catani and Ahlgren (2016) test.
CA_uniBootPV	a vector of length K with the univariate bootstrap P. values for each time series, for the Catani and Ahlgren (2016) test.
ET_LM	the LM statistic of the Eklund and Teräsvirta (2007) test.
ET_PV	the P.value of the Eklund and Teräsvirta (2007) LM test statistic.
ET_bootPV	the bootstrap P.value of the Eklund and Teräsvirta (2007) test.
ET_LMStar	the bootstrap LM test statistics for the Eklund and Teräsvirta (2007) test.
MARCH_LM	the LM statistic of the Multivariate LM test for ARCH. See e.g. Lütkepohl (2006, sect. 16.5).
MARCH_PV	the P.value of the MARCH LM test statistic.
MARCH_bootPV	the bootstrap P.value of the MARCH test.
MARCH_LMStar	the bootstrap LM test statistics for the MARCH test.
description	who ran the test and when.
time	computation time taken to run the test.
call	how the function ACtest() was called.

References

Catani, P. and Ahlgren, N. (2016). Combined Lagrange multiplier test for ARCH in vector autoregressive models, Economics and Statistics, <doi:10.1016/j.ecosta.2016.10.006>.

Dufour, J.-M., Khalaf, L., and Beaulieu, M.-C. (2010). Multivariate residual-based finite-sample tests for serial dependence and arch effects with applications to asset pricing models, Journal of Applied Econometrics, 25 (2010) 263–285.

Eklund, B. and Teräsvirta, T. (2007). Testing constancy of the error covariance matrix in vector models, Journal of Econometrics, 140, 753-780.

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

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

See Also

VARfit to estimate a VAR(p).

Examples

fit <- VARfit(y = VodafoneCDS, p = 3, const = TRUE, trend = FALSE)
test <- archBootTest(fit = fit, h = 5, B = 199, CA = TRUE, ET = TRUE, MARCH = TRUE, dist = "norm")
test

VARtests documentation built on Aug. 8, 2025, 7:16 p.m.