serial {vars} | R Documentation |
This function computes the multivariate Portmanteau- and Breusch-Godfrey test for serially correlated errors.
serial(x, lags.pt = 16, lags.bg = 5)
x |
Object of class ‘varest ’; generated by
VAR() , or an object of class ‘vec2var ’;
generated by vec2var() . |
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. |
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}_jhat{C}_0^{-1}) quad ,
where hat{C}_i = frac{1}{T}sum_{t = i + 1}^T hat{u}_t 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}_jhat{C}_0^{-1}) quad ,
The Breusch-Godfrey LM-statistic is based upon the following auxiliary regressions:
hat{u}_t = A_1 y_{t-1} + ... + A_py_{t-p} + CD_t + B_1hat{u}_{t-1} + ... + B_hhat{u}_{t-h} + 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 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{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)).
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 ‘htest ’
containing the multivariate Portmanteau-statistic (asymptotic and
adjusted. |
LMh |
An object with class attribute ‘htest ’
containing the Breusch-Godfrey LM-statistic. |
LMFh |
An object with class attribute ‘htest ’
containing the Edgerton-Shukur F-statistic. |
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(var.2c, lags.pt = 16, lags.bg = 5)