Byron Gangnes Econ 427 lecture 14 slides Forecasting with MA Models

Byron Gangnes Optimal forecast One which, given the available information, has the smallest average loss. –This will normally be the conditional mean (the mean given that we are a particular time period, t; i.e. given an “information set”): The best linear forecast will then be the linear approximation to this, called the linear projection, Remember the info set will generally be current and past values of y and innovations (epsilons).

Byron Gangnes Optimal forecast for MA Like before (Chs. 5 and 6), we calculate the optimal point forecast by writing down the process at period T+h and then “projecting it” on the information available at time T. –Book’s MA(2) example: 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 MA For period T+2: Projecting this on the time T info set, And so forth… So for periods beyond T+2, Why is that? an MA(q) process is not forecastable more than q steps ahead. Why? Recall Autocorr function drops to 0 after q steps

Byron Gangnes Uncertainty around optimal forecast 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, Why are errors serially correlated? Why can’t we use this info to improve forecast? Same for all forecasts T+h, h>2

Byron Gangnes Uncertainty around optimal forecast forecast error variance is the variance of e T+h,T We can use these conditional variances to construct confidence intervals. What will they look like? Notice that the error variance is less than the underlying variability of the series y t for h < q. Note that because of its MA(2) form, the variance of y t equals