MVARCHTEST Procedure

@MVARCHTest is a procedure for testing a set of series for multivariate ARCH effects. The null is that the series are mean zero, not serially correlated and with a fixed covariance matrix. It performs an LM test by regressing the crossproducts of the series (that is u(i,t) x u(j,t) for all combinations of i and j) on a constant and its lag(s) and testing the coefficients on the lags. The number of degrees of freedom is

${\left( {n(n + 1)/2} \right)^2}L$

since it is including all crossproducts on all crossproducts. This test is described in Hacker & Hatemi-J(2005).

To use this for diagnostics from a GARCH model, the list of series need to be standardized to remove the estimated GARCH variances.

@MVARCHTest( options ) start end

# list of series

Parameters

start end

range to use. By default, the common range of list of series

Options

LAGS=number of ARCH lags to test [1]

This generally should be a relatively small number (no more than 5)—it's unlikely for a relationship to only be apparent when longer lags are used.

[PRINT]/NOPRINT

TITLE="title of report" ["Multivariate ARCH Test"]

Variables Defined

%CDSTAT	test statistic (REAL)
%SIGNIF	significance level treating statistic as chi-squared (REAL)
%NDFTEST	degrees of freedom of test (INTEGER)

Example

This is taken from the VECMGARCH.RPF example. The first @MVARCHTest is done using the residuals from a standard VECM (Vector Error Correction Model). The second is done using jointly standardized residuals from a GARCH estimation. The joint standardization is needed to take out the "GARCH" effects to see if there is any remaining effect not explained by the GARCH model.

system(model=basevecm)

variables logdjia logrut

lags 1 to nlags

det constant

ect ecteq

end(system)

estimate(resids=vecmresids)

@mvarchtest(lags=5)

# vecmresids

garch(model=basevecm,mv=bekk,asymmetric,stdresids=stdu,hmatrices=hh,rvectors=rr,$

pmethod=simplex,piters=10,method=bfgs,iters=500,vechmat=vechcomps)

@mvarchtest(lags=5)

# stdu

Output

This is the output from the two tests. There are 45 degrees of freedom (5 lags × 3 × 3 for all combinations of cross products on cross products). The first test is strongly significant, the second (the post-GARCH diagnostic) is not.

Multivariate ARCH Test

Statistic Degrees Signif

459.73 45 0.00000

Multivariate ARCH Test

Statistic Degrees Signif

46.50 45 0.41044