vars (version 1.1-9)

serial: Test for serially correlated errors

Description

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(), 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.

Value

• 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.

encoding

latin1

concept

• VAR
• VECM
• 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.

See Also

VAR, vec2var, plot

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
serial(var.2c, lags.pt = 16, lags.bg = 5)