The AR(p) process

Summary

• We say that a time series process $y_1,…,y_T$ follows a stationary AR( $p$ ) process if $y_1,…,y_T$ is stationary and

$$y_t=β_1+ρ_1y_{t-1}+ρ_2y_{t-2}+…+ρ_py_{t-p}+ε_t , t=1,…,T$$

• The white noise definition of the error process from the AR(1) process carries over.
• It is possible to define stability conditions for an AR( $p$ ) process in terms of $ρ_1,…,ρ_p$ as necessary conditions for stationarity (not given here).
• If $y_t$ follows a stationary AR( $p$ ) process and the error process is white noise with $Var\left( ε_t \right)=σ^2$ then it is possible to find $E\left( y_t \right), Var\left( y_t \right)$ and $Cov\left( y_t,y_{t-s} \right)$ as a function of the unknown parameters $β_1,ρ_1,…,ρ_p,σ^2$ (not given here).
• If the process is stationary and the errors are white noise then OLS will be consistent.