Efficient estimation with AR(1) errors

Summary

Setup

• The linear regression model

\[y_t=β_1+β_2x_t+ε_t , t=1,…,T\]

• All data is stationary
• The explanatory variable is 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.

The transformed model

• Calculating \(y_t-ρy_{t-1}\) :

\[y_t-ρy_{t-1}=β_1+β_2x_t+ε_t-ρ(β_1+β_2x_{t-1}+ε_{t-1})\]

• or

\[y_t=β_1(1-ρ)+β_2x_t-ρβ_2x_{t-1}+ρy_{t-1}+ν_t\]

• This is a nonlinear ADL(1,1) model where the error terms are white noise .
• All the parameters can be consistently estimated using NLS.