TS module 12 method of moments practice problem
Estimates of ì, ã0, and ñ do not depend on the ARIMA process.
A. Estimate ì from the mean of the observations.
B. Estimate ã0 from the variance of the observations.
C. Estimate ñ from the sample autocorrelations.
Exercise 12.1: The first five values of a stationary time series are 6, 5, 4, 6, and 4.
A. What is the estimate of ì?
B. What is the estimate of ã0?
C. What is the estimate of ñ1?
Part A: An unbiased estimator of ì is ∑ Yt / N = (6 + 5 + 4 + 6 + 4) / 5 = 25/5 = 5.
Part B: An unbiased estimator of ã0 is the variance of the observed values =
(12 + 02 + (–1)2 + 12 + (–1)2 ) / 4 = 1.
Part C: The numerator of the sample autocorrelation is
(1 × 0 + 0 × -1 + -1 × 1 + 1 × -1) = –2.
The denominator is ã0 = 4 (see Part B), so ñ1 = –2/4 = –½.
Jacob: Why doesn’t this exercise ask for ó2ε?
Rachel: The estimate of ó2ε depends on the type of model, such as AR(1), MA(1), or ARMA(1,1).
Jacob: Can’t we select the best model?
Rachel: With only five observed values, the standard error of the observed sample autocorrelations is high: 1/√5 = 0.44721. The width of the 95% confidence interval is 2 × 1.96 × 0.44721 = 1.75308. The data are too sp**** to select the best model.