This function computes the multivariate Portmanteau- and Breusch-Godfrey
test for serially correlated errors.
Usage
serial(x, lags.pt = 16, lags.bg = 5)
Arguments
x
Object of class varest; generated by
VAR().
lags.pt
An integer specifying the lags to be used for the
Portmanteau statistic.
lags.bg
An integer specifying the lags to be used for the
Breusch-Godfrey statistic.
Value
A list with class attribute varcheck holding the
following elements:
residA matrix with the residuals of the VAR.
pt.mulA list with objects of class attribute htest
containing the multivariate Portmanteau-statistic (asymptotic and
adjusted.
LMhAn object with class attribute htest
containing the Breusch-Godfrey LM-statistic.
LMFhAn object with class attribute htest
containing the Edgerton-Shukur F-statistic.
encoding
latin1
concept
VAR
Vector autoregressive model
Portmanteau
Breusch Godfrey
Serial Correlation
Serially correlated errors
Details
The Portmanteau statistic for testing the absence of up to the order $h$
serially correlated disturbances in a stable VAR(p) is defined as:
$$Q_h = T \sum_{j = 1}^h
tr(\hat{C}_j'\hat{C}_0^{-1}\hat{C}_j\hat{C}_0^{-1}) \quad ,$$
where $\hat{C}_i = \frac{1}{T}\sum_{t = i + 1}^T \bold{\hat{u}}_t
\bold{\hat{u}}_{t - i}'$. The test statistic is approximately
distributed as $\chi^2(K^2(h - p))$. For smaller sample sizes
and/or values of $h$ that are not sufficiently large, a corrected
test statistic is computed as:
$$Q_h^* = T^2 \sum_{j = 1}^h
\frac{1}{T - j}tr(\hat{C}_j'\hat{C}_0^{-1}\hat{C}_j\hat{C}_0^{-1}) \quad ,$$
The Breusch-Godfrey LM-statistic is based upon the following auxiliary
regressions:
$$\bold{\hat{u}}_t = A_1 \bold{y}_{t-1} + \ldots + A_p\bold{y}_{t-p} +
CD_t + B_1\bold{\hat{u}}_{t-1} + \ldots + B_h\bold{\hat{u}}_{t-h} +
\bold{\varepsilon}_t$$
The null hypothesis is: $H_0: B_1 = \ldots = B_h = 0$ and
correspondingly the alternative hypothesis is of the form $H_1:
\exists \; B_i \ne 0$ for $i = 1, 2, \ldots, h$. The test statistic
is defined as:
$$LM_h = T(K - tr(\tilde{\Sigma}_R^{-1}\tilde{\Sigma}_e)) \quad ,$$
where $\tilde{\Sigma}_R$ and $\tilde{\Sigma}_e$ assign the
residual covariance matrix of the restricted and unrestricted
model, respectively. The test statistic $LM_h$ is distributed as
$\chi^2(hK^2)$.
Edgerton and Shukur (1999) proposed a small sample correction, which
is defined as:
$$LMF_h = \frac{1 - (1 - R_r^2)^{1/r}}{(1 - R_r^2)^{1/r}} \frac{Nr -
q}{K m} \quad ,$$
with $R_r^2 = 1 - |\tilde{\Sigma}_e | / |\tilde{\Sigma}_R|$,
$r = ((K^2m^2 - 4)/(K^2 + m^2 - 5))^{1/2}$, $q = 1/2 K m - 1$
and $N = T - K - m - 1/2(K - m + 1)$, whereby $n$ is the
number of regressors in the original system and $m = Kh$. The
modified test statistic is distributed as $F(hK^2, int(Nr - q))$.
References
Breusch, T . S. (1978), Testing for autocorrelation in dynamic linear
models, Australian Economic Papers, 17: 334-355.
Edgerton, D. and Shukur, G. (1999), Testing autocorrelation in a
system perspective, Econometric Reviews, 18: 43-386.
Godfrey, L. G. (1978), Testing for higher order serial correlation in
regression equations when the regressors include lagged dependent
variables, Econometrica, 46: 1303-1313.
Hamilton, J. (1994), Time Series Analysis, Princeton
University Press, Princeton.
L�tkepohl, H. (2006), New Introduction to Multiple Time Series
Analysis, Springer, New York.