This function computes the multivariate Portmanteau- and Breusch-Godfrey test for serially correlated errors.
serial.test(x, lags.pt = 16, lags.bg = 5, type = c("PT.asymptotic",
"PT.adjusted", "BG", "ES") )A list with class attribute ‘varcheck’ holding the
following elements:
A matrix with the residuals of the VAR.
A list with objects of class attribute ‘htest’
containing the multivariate Portmanteau-statistic (asymptotic and
adjusted.
An object with class attribute ‘htest’
containing the Breusch-Godfrey LM-statistic.
An object with class attribute ‘htest’
containing the Edgerton-Shukur F-statistic.
Object of class ‘varest’; generated by
VAR(), or an object of class ‘vec2var’;
generated by vec2var().
An integer specifying the lags to be used for the Portmanteau statistic.
An integer specifying the lags to be used for the Breusch-Godfrey statistic.
Character, the type of test. The default is an asymptotic Portmanteau test.
Bernhard Pfaff
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 \(\chi^2(K^2(h - p))\). This test statistic is
choosen by setting type = "PT.asymptotic". For smaller sample sizes
and/or values of \(h\) that are not sufficiently large, a corrected
test statistic is computed as:
$$
Q_h^* = T^2 \sum_{j = 1}^h
\frac{1}{T - j}tr(\hat{C}_j'\hat{C}_0^{-1}\hat{C}_j\hat{C}_0^{-1}) \quad ,
$$
This test statistic can be accessed, if type = "PT.adjusted" is
set.
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
\(\chi^2(hK^2)\). This test statistic is calculated if type =
"BG" is used.
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 \(F(hK^2, int(Nr -
q))\). This modified statistic will be returned, if type =
"ES" is provided in the call to serial().
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.
Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.
VAR, vec2var, plot
data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
serial.test(var.2c, lags.pt = 16, type = "PT.adjusted")
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