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.