Breusch–Godfrey Test

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A Breusch–Godfrey Test is a statistical test for serial correlation.



References

2016

Because the test is based on the idea of Lagrange multiplier testing, it is sometimes referred to as LM test for serial correlation.
A similar assessment can be also carried out with the Durbin–Watson test and the Ljung–Box test(...)
Procedure: Consider a linear regression of any form, for example
[math]\displaystyle{ Y_t = \alpha_0+ \alpha_1 X_{t,1} + \alpha_2 X_{t,2} + u_t \, }[/math]
where the residuals might follow an AR(p) autoregressive scheme, as follows:
[math]\displaystyle{ u_t = \rho_1 u_{t-1} + \rho_2 u_{t-2} + \cdots + \rho_p u_{t-p} + \varepsilon_t. \, }[/math]
The simple regression model is first fitted by ordinary least squares to obtain a set of sample residuals [math]\displaystyle{ \hat{u}_t }[/math].
Breusch and Godfrey proved that, if the following auxiliary regression model is fitted
[math]\displaystyle{ \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 \, }[/math]
and if the usual [math]\displaystyle{ R^2 }[/math] statistic is calculated for this model, then the following asymptotic approximation can be used for the distribution of the test statistic
[math]\displaystyle{ n R^2\,\sim\,\chi^2_p, \, }[/math]
when the null hypothesis [math]\displaystyle{ {H_0: \lbrace \rho_i = 0 \text{ for all } i \rbrace } }[/math] holds (that is, there is no serial correlation of any order up to p). Here n is the number of data-points available for the second regression, that for [math]\displaystyle{ \hat{u}_t }[/math],
[math]\displaystyle{ n=T-p, \, }[/math]
where T is the number of observations in the basic series. Note that the value of n depends on the number of lags of the error term (p).