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.