Breusch–Godfrey Test

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

• AKA: Breusch-Godfrey LM Test.
• See: Unit Root Test, Dickey–Fuller Test, Phillips–Perron Test, KPSS Test, Ljung–Box Test, Durbin–Watson Test, Augmented Dickey–Fuller Test, Time Series Analysis.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Breusch–Godfrey_test Retrieved 2016-08-07
  • In statistics, the Breusch–Godfrey test, named after Trevor S. Breusch and Leslie G. Godfrey, is used to assess the validity of some of the modelling assumptions inherent in applying regression-like models to observed data series. In particular, it tests for the presence of serial correlation that has not been included in a proposed model structure and which, if present, would mean that incorrect conclusions would be drawn from other tests, or that sub-optimal estimates of model parameters are obtained if it is not taken into account. The regression models to which the test can be applied include cases where lagged values of the dependent variables are used as independent variables in the model's representation for later observations. This type of structure is common in econometric models.

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).

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