The AR(1) process

Summary

AR(1) process

• We say that a time series process \(y_1,…,y_T\) follows a stationary AR(1) process if \(y_1,…,y_T\) is stationary and

\[y_t=β_1+ρy_{t-1}+ε_t , t=1,…,T\]

White noise

We say that the error process \(ε_1,…,ε_T\) is white noise if:

• \(ε_t\) is independent of all lagged values of \(y\) ( \(y_{t-1}, y_{t-2},…\) )
• Two distinct error terms are independent
• \(E\left( ε_t \right)=0\) and \(Var\left( ε_t \right)=σ^2\)

Moments for a stationary AR(1) process

• We say that an AR(1) process satisfy the stability condition if \(\left| ρ \right|<1\) . The stability condition is a necessary condition for stationarity.
• If \(y_t\) follows a stationary AR(1) process and the error process \(ε_1,…,ε_T\) is white noise with \(Var\left( ε_t \right)=σ^2\) then

\[E\left( y_t \right)= \frac{β_1}{1-ρ}\]

\[Var\left( y_t \right)= \frac{σ^2}{1-ρ^2}\]

\[Cov\left( y_t,y_{t-s} \right)=ρ^s \frac{σ^2}{1-ρ^2}\]

Estimating an AR(1) process

• Due to lagged dependent variables, the GM assumptions cannot hold
• However, if the process is stationary and the errors are white noise then the OLS estimator and the standard errors will be consistent.
• For \(0<ρ<1\) , the OLS estimator is biased downward.