Breusch–Godfrey test

In statistics, the Breusch-Godfrey serial correlation LM test is a robust test for autocorrelation in the residuals from a regression analysis and is considered more general than the standard Durbin–Watson statistic (or Durbin's h statistic).

While the Durbin–Watson d statistic is only valid for stochastic regressors and first order autoregressive schemes (eg AR(1)), the BG test has none of these restrictions, and is statistically more powerful than Durbin's h statistic.

Procedure

Consider a linear regression of any form but, for example,

:$Y\_t\; =\; alpha\_0+\; alpha\_1\{X\_\{t,1\; +\; alpha\_2\{X\_\{t,2\; +\{u\_\{t\; ,$

where the residuals might follow an AR("p") autoregressive scheme, as follows:

:$u\_t\; =\; ho\_1\{u\_\{t-1\; +\; ho\_2\{u\_\{t-2\; +\; cdots\; +\; ho\_p\{u\_\{t-p\; +\; varepsilon\_t\; .$

The simple regression model is first fitted by least squares to obtain a set of sample residuals $hat\{u\}\_\{t\}$.

Breusch and Godfrey proved that, if the following the auxiliary regression model is fitted

: $hat\{u\}\_t\; =\; alpha\_0+\; alpha\_1\{X\_\{t,1\; +\; alpha\_2\{X\_\{t,2\; +\; ho\_1\{hat\{u\}\_\{t-1\; +\; ho\_2\{hat\{u\}\_\{t-2\; +\; cdots\; +\; ho\_p\{hat\{u\}\_\{t-p\; +\; varepsilon\_t$

and if the usual $R^2$ statistic is calculated for this model, then the following asymptotic approximation can be used for the distribution of the test statistic: $(n-p)R^2,sim,chi^2\_p\; ,$when the null hypothosis $\{H\_0:\; lbrace\; ho\_i\; =\; 0\; ext\{\; for\; all\; \}\; i\; brace\; \}$ holds. (That is, there is no serial correlation of any order up to "p".) Here "n" is the number of observations.