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.