Solution to Exercise 3.5.

Problem Statement:

(a) Write down the Yule-Walker equations for Models 1 and 4.

(b) Solve these equations to obtain $\rho_1$ and $\rho_2$ for the two models.

(c) Obtain the partial autocorrelation function for the two models.

Solution:

Model 1: Since p=1, the Yule-Walker equations from (3.2.6), p. 57, reduce to

$$\rho_1 \; = \; \phi_1 .$$

Of course, we may use the more general recursive form (as discussed in lecture): starting with the AR(1) equation defining the model:

$$Z_t - 0.6 Z_{t-1} \; = \; a_t ,$$

we multiply by Z_t-k for k > 0 and use uncorrelatedness of Z_t-k and a_t to get

$$\rho_k - 0.6 \rho_{k-1} \; = \; 0 ,$$

or

$$\rho_k \; = \; 0.6 \rho_{k-1} ,$$

and hence $\rho_1 = 0.6$ ,

$$\rho_k \; = \; (0.6)^k , \quad k \gt 0 .$$

In particular,

$$\rho_2 \; = \; (0.6)^2 \; = \; 0.36 .$$

As in (3.2.33), p. 66,

$$\phi_{11} \; = \; \rho_1 \; = \; 0.6 .$$

Also, since this process is an AR(1), we have

$$\phi_{kk} \; = \; 0 , \quad k \gt 1 .$$

Model 4. From (3.2.25), p. 61, the Yule-Walker equations are

and the solution (as given in (3.2.27), p. 62) is

The formula for the PACF of on p. 66 yield