| serial.test {vars} | R Documentation |
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") )
x |
Object of class ‘ |
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. |
type |
Character, the type of test. The default is an asymptotic Portmanteau test. |
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 ∑_{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}∑_{t = i + 1}^T \bold{\hat{u}}_t
\bold{\hat{u}}_{t - i}'. The test statistic is approximately
distributed as χ^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 ∑_{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} + … + A_p\bold{y}_{t-p} + CD_t + B_1\bold{\hat{u}}_{t-1} + … + B_h\bold{\hat{u}}_{t-h} + \bold{\varepsilon}_t
The null hypothesis is: H_0: B_1 = … = B_h = 0 and correspondingly the alternative hypothesis is of the form H_1: \exists \; B_i \ne 0 for i = 1, 2, …, h. The test statistic is defined as:
LM_h = T(K - tr(\tilde{Σ}_R^{-1}\tilde{Σ}_e)) \quad ,
where \tilde{Σ}_R and \tilde{Σ}_e assign the
residual covariance matrix of the restricted and unrestricted
model, respectively. The test statistic LM_h is distributed as
χ^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{Σ}_e | / |\tilde{Σ}_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().
A list with class attribute ‘varcheck’ holding the
following elements:
resid |
A matrix with the residuals of the VAR. |
pt.mul |
A list with objects of class attribute ‘ |
LMh |
An object with class attribute ‘ |
LMFh |
An object with class attribute ‘ |
This function was named serial in earlier versions of package
vars; it is now deprecated. See vars-deprecated too.
Bernhard Pfaff
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.
data(Canada) var.2c <- VAR(Canada, p = 2, type = "const") serial.test(var.2c, lags.pt = 16, type = "PT.adjusted")