Econ 427 lecture 15 slides Forecasting with AR

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Econ 427 lecture 15 slides Forecasting with AR Models Byron Gangnes

Econ 427 lecture 15 slides Forecasting with AR Models Byron Gangnes

Optimal forecast • Remember, the best linear forecast is often the linear projection, Where

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 Byron Gangnes

Optimal forecast for AR • Consider the AR(1) process: • To get optimal fcst

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) Byron Gangnes

Optimal forecast for AR • For T+2: • Projecting this on the time T

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: Byron Gangnes

More complicated AR forecasts • What if we had a higher-order AR(p) time series?

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 Byron Gangnes

Uncertainty around optimal forecast • Again, we would like to know how much uncertainty

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: Byron Gangnes

Uncertainty around optimal forecast • In general: • Note that the errors are serially

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

Uncertainty around optimal forecast • forecast error variance is the variance of e. T+h,

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

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