Postscript version of this assignment

STAT 804: 99-3

Assignment 2

1.

Consider the ARIMA(1,0,1) process

$$X_t-\phi X_{t-1} = \epsilon_t-\psi\epsilon_{t-1}\, .$$

Show that the autocorrelation function is

$$\rho(1) = {(1-\psi\phi)(\phi-\psi)\over{1+\psi^2-2\psi\phi}}$$

and

$$\rho(k) = \phi^{k-1}\rho(1)\qquad k=2,3,\ldots$$

Plot the autocorrelation functions for the ARMA(1,1) process above, the AR(1) process with

$$X_t=\phi X_{t-1}+\epsilon_t$$

and the MA(1) process

$$X_t = \epsilon_t-\psi\epsilon_{t-1}$$

on the same plot when $\phi=0.6$ and $\theta = -0.9$. Compute and plot the partial autocorrelation functions up to lag 30. Comment on the usefulness of these plots in distinguishing the three models. Explain what goes wrong when $\phi$is close to $\psi$.

2.

Suppose $\Phi$ is a Uniform$[0,2\pi ]$ random variable. Define

$$X_t=\cos(\omega t + \Phi) \, .$$

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

3.

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

$$X_t=(2-\lambda^2)X_{t-1}-X_{t-2}$$

for some value of $\lambda$. Show that the roots of the characteristic polynomial lie on the boundary of the unit circle in the complex plain. (Hint: show that $e^{i\theta}$ is a root if $\theta$ 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.)

4.

Suppose that X_t is an ARMA(1,1) process

$$X_t-\rho X_{t-1} = \epsilon_t - \theta \epsilon_{t-1}$$

(a)

Suppose we mistakenly fit an AR(1) model (mean 0) to Xusing the Yule-Walker estimate

$${\hat\rho} = \left(\sum_1^{T-1}X_tX_{t-1}\right)/\left(\sum_0^{T-1}X_t^2\right)$$

In terms of $\theta$, $\rho$ and $\sigma$ what is $\hat\rho$ close to?

(b)

If we use this AR(1) estimate $\hat\rho$ and calculate residuals using ${\hat\epsilon}_t = X_t - {\hat \rho} X_{t-1}$ what kind of time series is $\hat\epsilon$? What will plots of the Autocorrelation and Partial Autocorrelation functions of this residual series look like?