Assignment 4

Suppose that is an ARMA(1,1) process

- Suppose we mistakenly fit an AR(1) model (mean 0) to
**X**using the Yule-Walker estimateIn terms of , and what is close to?

**Solution:**This is a ratio of two sample covariances. Each converges, as to the corresponding theoretical covariance so that . In the previous assignment we computed and found that the lag 1 autocorrelation isThis is the limit of .

- If we use this AR(1) estimate and calculate
residuals using what kind of
time series is ? What will plots of the Autocorrelation and
Partial Autocorrelation functions of this residual series look like?
**Solution:**Let denote the autocovariance at lag 1. For large values of**T**we may write approximatelyor or just

which makes an ARMA(1,2) process. By way of answer about the plots I was merely looking for the knowledge that the plots will match those of an ARMA(1,2) with autoregressive parameter and MA parameters and . The model identification problem may well be somewhat harder. It is a useful exercise to generate some data with

`ar.sim`from an ARIMA(1,0,1) and try to model fitting process. Look at what happens if you fit an AR(1) and then look at the residuals; you don't see anything helpful in general.

Mon Dec 8 11:07:38 PST 1997