TS module 12: Method of moments for ARMA(1,1) process (practice problem)
(The attached PDF file has better formatting.)
Know how to estimate ö and è for an ARMA(1,1) process by the method of moments. You solve a quadratic equation for è.
Exercise 1.2: ARMA(1,1) model and method of moments (Yule-Walker equations)
An ARMA(1,1) model is fit to a time series with sample autocorrelations for the first two lags of r1 = 0.880 and r2 = 0.704.
1. What is the method of moments estimate for ö?
2. What is the method of moments estimate for è?
Part A: For an ARMA(1,1) process, r2 = r1 × ö ➾ ö = 0.704 / 0.880 = 0.8
Part B: For an ARMA(1,1) process (for k ≥ 1):
We estimated ö as r2 / r1. We estimate è from
See Cryer and Chan, equation 7.1.6 on page 151.
In this exercise, 0.880 = (1 – 0.8 è) (0.8 – è) / (1 – 2(0.8 è) + è2).
This is a quadratic equation in θ, with roots of –0.4 and –2.5 (use the formula for roots of a quadratic).
The arithmetic is shown below; most final exam problems use simple numbers.
0.880 = (1 – 0.8 è) (0.8 – è) / (1 – 2(0.8 è) + è2)
0.880 × (1 – 2(0.8 è) + è2) = (1 – 0.8 è) (0.8 – è)
88 × (1 – 2(0.8 è) + è2) = (10 – 8 è) (8 – 10 è)
88 – 140.8 è + 88 è2 = 80 – 164 è + 80 è2
8 + 23.2 è + 8 è2 = 0
Using the formula for the roots of a quadratic equation gives
(82 ± (23.22 – 4 × 8 × 8) ) / (2 × 8) = -0.4 and -2.5