vars (version 1.6-1)

arch.test: 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)

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.

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.

Author

Bernhard Pfaff

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 \(\chi^2(qK^2(K+1)^2/4)\).

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)