General linear processes practice problems
Module 6: Autoregressive processes
(The attached PDF file has better formatting.)
**Exercise 6.1: Geometric decay
A time series has the form Yt = åt + ö × åt-1 + ö2 × åt-2 + ö3 × åt-3 + …
ö = 0.4 and ó2e = 4.
A. What is ã0, the variance of Yt?
B. What is ã1, the covariance of Yt and Yt-1?
C. What is ñ1, the autocorrelation of Yt and Yt-1?
D. What is ñ2, the correlation of Yt and Yt-2?
Part A: See Cryer and Chan, chapter 4, top of page 56:
ã0 = ó2 / (1 – ö2) = 4 / (1 – 0.16) = 4.762
Later modules refer to this process as AR(1), an autoregressive process of order 1. Final exam problems say: an AR(1) process with ö = 0.4 and ó2ε = 4.
Part B: See Cryer and Chan, chapter 4, middle of page 56:
ã1 = ö × ó2 / (1 – ö2) = 0.4 × 4 / (1 – 0.16) = 1.905
Part C: See Cryer and Chan, chapter 4, equation 4.1.3 at the bottom of page 56:
ñ1 = ö = 0.4
Part D: See Cryer and Chan, chapter 4, equation 4.1.3 at the bottom of page 56:
ã2 = ö2 = 0.42 = 0.160
This exercise is simple. Final exam problems are more complex. The autoregressive process may be of an order higher than 1, the ö parameters may be positive or negative, the process may have a moving average part, and the parameters may be estimated from the observed sample autocorrelations. The logic is the same for all the scenarios. This exercise is a good starting point for stationary time series.