1. Byron Gangnes Econ 427 lecture 15 slides Forecasting with AR Models

2. Byron Gangnes Optimal forecast Remember, the best linear forecast is often the linear projection, Where the info set will generally be current and past values of y and innovations (epsilons). For forecasting AR processes, we will proceed as we did for MA: Write out the process at time T+1 Projecting this on the time T info set We could rewrite the cov. Stationary AR in MA form But there is a simpler way—the chain rule of forecasting

3. Byron Gangnes Optimal forecast for AR Consider the AR(1) process: To get optimal fcst for t=T+1, write out the process at time T+1: Projecting this on the time T info set, (remember that expectations of future innovs are zero)

4. Byron Gangnes Optimal forecast for AR For T+2: Projecting this on the time T info set, But we already have an optimal fcst of y T+1. Substituting: Similarly, for a 3-step-ahead forecast, we would get: Generally:

5. Byron Gangnes More complicated AR forecasts What if we had a higher-order AR(p) time series? –There would be p terms in each time period What if we had both MA and AR terms? –We would combine the two methods—see pp. 178-79 in the book

6. Byron Gangnes Uncertainty around optimal forecast Again, we would like to know how much uncertainty there will be around point estimates of forecasts. To see that, let’s look at the forecast errors, Can you show that the error for a 3-step-ahead fcst is:

7. Byron Gangnes Uncertainty around optimal forecast In general: Note that the errors are serially correlated but don’t drop off

8. Byron Gangnes Uncertainty around optimal forecast forecast error variance is the variance of e T+h,T And we can use these conditional variances to construct confidence intervals. What will they look like? Generally,