arch.test {vars} | R Documentation |
This function computes univariate and multivariate ARCH-LM tests for a VAR(p).
arch.test(x, lags.single = 16, lags.multi = 5, multivariate.only = TRUE)
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. |
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(hat{u}_t hat{u}_t') = β_0 + B_1 vech(hat{u}_{t-1} hat{u}_{t-1}') + ... + B_q vech(hat{u}_{t-q} hat{u}_{t-q}' + v_t)
whereby 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 β_0 is frac{1}{2}K(K +1) and for the coefficient matrices B_i with i=1, ..., q, frac{1}{2}K(K +1) times frac{1}{2}K(K +1). The null hypothesis is: H_0 := B_1 = B_2 = ... = B_q = 0 and the alternative is: H_1: B_1 neq 0 or B_2 neq 0 or ... 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).
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. |
This function was named arch
in earlier versions of package
vars; it is now deprecated. See vars-deprecated
too.
Bernhard Pfaff
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.
data(Canada) var.2c <- VAR(Canada, p = 2, type = "const") arch.test(var.2c)