Test for autocorrelation, Breusch-Godfrey test

Summary

Setup

• The linear regression model

$$y_t=β_1+β_2x_{t,2}+...+β_kx_{t,k}+ε_t , t=1,…,T$$

• All data is stationary
• The explanatory variables are exogenous
• The error terms follow a stationary AR(1) process

$$ε_t=ρε_{t-1}+ν_t , t=1,…,T$$

• where $ν_t$ is white noise .
• We want to test the null hypothesis “the errors are not autocorrelated” which is the same as

$$H_0:ρ=0$$

Procedure

• Estimate the LRM using OLS
• Save the OLS residuals
• Run an auxiliary regression

$$e_t=ρe_{t-1}+β_1+β_2x_{t,2}+...+ β_kx_{t,k}+ν_t , t=2,…,T$$

• where $e_{t-1}$ are lagged residuals.
• Test the null hypothesis $H_0:ρ=0$ . The test statistic is the $t$ -value which is approximately standard normal as $n$ is large.

Extension

• If the error terms follow an AR( $p$ ) process

$$ε_t=ρ_1ε_{t-1}+…ρ_pε_{t-p}+ν_t , t=1,…,T$$

• then the auxiliary regression becomes

$$e_t=ρ_1e_{t-1}+…+ρ_pe_{t-p}+β_1+β_2x_{t,2}+...+ β_kx_{t,k}+ν_t , t=2,…,T$$

• and we test $H_0:ρ_1=0,…,ρ_p=0$ using an $F$ -test or an LM test.