We wish to develop simple estimators for the two parameters of an MA(1) model that can be used as starting values for computing the m.l.e. We consider two. First, use (6.44) to derive a method of moments estimator for

b, and, based on this, use (6.43) to get an estimate of ????.

The second way is to use the fact that an MA(1) model can be represented as an infinite AR model, which suggests estimating an AR(p) model via least squares and setting

̂

b = â 1. (6.57)

The choice of p will of course influence the quality of the estimator: If p is chosen too small, then the AR(p) model is “very” mis-specified, so that ̂

b will be quite biased, while if p is chosen too large, then the variance of ̂

b will be large.This tradeoff becomes acute as|b| approaches one. Koreisha and Pukkila (1990) recommend taking p to be √

T, rounded off to the nearest integer. Make a program to compute this.

Now compare via simulation the performance (in terms of bias and m.s.e.) of the two estimators for

b. Use a grid of b-values, T = 15 observations, ???? = 1, and 10,000 replications.