This function computes univariate and multivariate ARCH-LM tests for a VAR(p).
Usage
arch(x, lags.single = 16, lags.multi = 5)
Arguments
x
Object of class varest; generated by VAR().
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.
Value
A list with class attribute varcheck holding the
following elements:
residA matrix with the residuals of the VAR.
arch.uniA list with objects of class htest
containing the univariate ARCH-LM tests per equation.
arch.mulAn object with class attribute htest
containing the multivariate ARCH-LM statistic.
encoding
latin1
concept
VAR
Vector autoregressive model
ARCH
Heteroskedasticity
Autoregressive Conditional Heteroskedasticity
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.
L�tkepohl, H. (2006), New Introduction to Multiple Time Series
Analysis, Springer, New York.