STAT 804: 97-3

Assignment 3

1. Suppose is a Uniform random variable. Define

   

   Show that X is weakly stationary. (In fact it is strongly stationary so show that if you can.) Compute the autocorrelation function of X.

2. Show that X of the previous question satisfies the AR(2) model

   

   for some value of . Show that the roots of the characteristic polynomial lie on the boundary of the unit circle in the complex plain. (Hint: show that is a root if is chosen correctly. Do not spend too much time on this question; the point is to illustrate that AR(2) models can be found whose behaviour is much like a sinusoid.)

3. For the data set raindiff in the same directory as earlier datasets fit the model

   

   where the are iid in each of the following ways

   1. By full maximum likelihood. (Maximize the joint density of the X's over the parameters , and .)

   2. By estimating by the sample mean and then doing full maximum likelihood for the model

      

      where .

   3. By estimating by the sample mean and then doing conditional maximum likelihood for the previous model.

   Remarks: I do not want you to use ar or other built-in S time-series functions to do these but you may want to use them to get starting values for estimates. It is probably easier to do the methods in reverse order, using the results as starting points for iterative solutions.

4. Derive formulas for the L-step ahead forecast standard error for the AR(1) model and for the ARIMA(1,1,0) model . Compute the limits of the forecast standard errors for these two models as La tends to infinity.

5. Delete the last 4 values for the earnings data set and use the model you selected in the previous assignment to re-estimate the parameters and forecast the deleted values. Compare the actual errors with the forecasts and the forecast standard errors.

6. For the dataset faketrend in the usual directory remove a trend by ordinary least squares. Then fit the model you used with fake in the previous assignment to the residuals. Use the autocovariance of the fitted to re-estimate the trend by generalized squares. Does the fit change much?

DUE: Friday, 21 November.