Breusch–Godfrey test

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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.

[edit] 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 = \rho_1{u_{t-1}} + \rho_2{u_{t-2}} + \cdots + \rho_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}} + \rho_1{\hat{u}_{t-1}} + \rho_2{\hat{u}_{t-2}} + \cdots + \rho_p{\hat{u}_{t-p}} + \varepsilon_t

and if the usual R2 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 H0:{ρi = 0 for all i} holds. (That is, there is no serial correlation of any order up to p.) Here n is the number of observations.