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.