Breusch–Godfrey test

In

The test is named after Trevor S. Breusch and Leslie G. Godfrey.

Background

The Breusch–Godfrey test is a test for

Consider a linear regression of any form, for example

${\displaystyle Y_{t}=\beta _{1}+\beta _{2}X_{t,1}+\beta _{3}X_{t,2}+u_{t}\,}$

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

${\displaystyle u_{t}=\rho _{1}u_{t-1}+\rho _{2}u_{t-2}+\cdots +\rho _{p}u_{t-p}+\varepsilon _{t}.\,}$

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

Breusch and Godfrey^{[citation needed]} proved that, if the following auxiliary regression model is fitted

${\displaystyle {\hat {u}}_{t}=\alpha _{0}+\alpha _{1}X_{t,1}+\alpha _{2}X_{t,2}+\rho _{1}{\hat {u}}_{t-1}+\rho _{2}{\hat {u}}_{t-2}+\cdots +\rho _{p}{\hat {u}}_{t-p}+\varepsilon _{t}\,}$

and if the usual Coefficient of determination (${\displaystyle R^{2}}$ statistic) is calculated for this model:

${\displaystyle R^{2}:={\frac {\sum _{j=1}^{T-p}(u_{T-j}-{\hat {u}}_{T-j})^{2}}{\sum _{j=1}^{T-p}(u_{T-j}-{\bar {u}})^{2}}}}$,

where ${\displaystyle {\bar {u}}}$ stands for the arithmetic mean over the last ${\displaystyle n=T-p}$ samples, where ${\displaystyle T}$ is the total number of observations and ${\displaystyle p}$ is the number of error lags used in the auxiliary regression.

The following asymptotic approximation can be used for the distribution of the test statistic:

${\displaystyle nR^{2}\,\sim \,\chi _{p}^{2},\,}$

when the null hypothesis ${\displaystyle {H_{0}:\lbrace \rho _{i}=0{\text{ for all }}i\rbrace }}$ holds (that is, there is no serial correlation of any order up to p). Here n is

Software

• In R, this test is performed by function bgtest, available in package lmtest.^[5][6]
• In Stata, this test is performed by the command estat bgodfrey.^[7][8]
• In SAS, the GODFREY option of the MODEL statement in PROC AUTOREG provides a version of this test.
• In Python Statsmodels, the acorr_breusch_godfrey function in the module statsmodels.stats.diagnostic ^[9]
• In EViews, this test is already done after a regression, at "View" → "Residual Diagnostics" → "Serial Correlation LM Test".
• In Julia, the BreuschGodfreyTest function is available in the HypothesisTests package.^[10]
• In gretl, this test can be obtained via the modtest command, or under the "Test" → "Autocorrelation" menu entry in the GUI client.

See also

• Breusch–Pagan test
• Durbin–Watson test
• Ljung–Box test
• Autoregressive-moving-average model

References