Econ 427 lecture 15 slides

1. Econ 427 lecture 15 slides Forecasting with AR Models

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

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

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