Time series mod 5: MA(2) autocorrelations practice problems
For AR(1), AR(2), MA(1), MA(2), and ARMA(1,1) processes, know how to calculate ã0, ã1, ã2, ñ1, and ñ2 from ö1, ö2, è1, è2, and óε. Know the formulas for these five simple processes. For more complex processes, such as ARMA(p, q) for p > 1 and q > 1, or AR(p) for p > 2, or MA(q) for q > 2, know the principles, such as the shapes of the sample autocorrelation and partial autocorrelation functions.
Exercise 1.1: MA(2) process
An MA(2) process has è1 = 0.7, è2 = 0.5, and óε = 2.
A. What is ã0?
B. What is ã1?
C. What is ã2?
D. What is ñ1?
E. What is ñ2?
Solution 1.1: For an MA(2) process:
ã0 =(1 + è12 + è22) × ó2
ã1 = (–è1 + è1 × è2) × ó2
ã2 = (–è2) × ó2
ñ1 = (–è1 + è1 × è2) / (1 + è12 + è22)
ñ2 = (–è2) / (1 + è12 + è22)
ñk = 0 for k = 3, 4, …
Cryer and Chan, page 63 (equation 4.2.3)
Part A: ã0 = (1 + è12 + è22) × ó2 = (1 + 0.49 + 0.25 ) × 22 = 6.960
Part B: ã1 = (–è1 + è1 × è2) × ó2 = (–0.7 + 0.7 × 0.5) × 22 = -1.400
Part C: ã2 = (–è2) × ó2 = –0.5 × 22 = -2.000
Part D: ñ1 = (–è1 + è1 × è2) / (1 + è12 + è22) = -0.201
Part E: ñ2 = (–è2) / (1 + è12 + è22) = -0.287