TS Module 12: Parameter estimation method of moments practice problems

(The attached PDF file has better formatting.)

Time series practice problems on Yule-Walker equations

We use the Yule-Walker equations to derive initial estimates of the ARMA coefficients. Know how to solve the Yule-Walker equations for AR(1), AR(2), and MA(1) processes.

        A student project might use the Yule-Walker equations for MA(1) and MA(2) models.

        For the final exam, focus on the equations for AR(1), AR(2), and MA(1) models.

The Cryer and Chan textbook shows the efficiency of the Yule-Walker estimates for various processes. Know when these equations are good estimators and when they are not. (They are not efficient for moving average and mixed models.)

Many exam problems specify that the time series parameters lie between –1 and +1. The term used in the textbook is that the time series is invertible.

*Exercise 12.1: MA(1) model and Yule-Walker equations

An MA(1) model has an estimated ñ₁ of –0.35.  What is the Yule-Walker initial estimate for è₁ if it lies between –1 and +1?

Solution 12.1: An MA(1) model has .

We invert the equation to get

We compute (–1 + (1 – 4 × 035²)^0.5 ) / (2 × –0.35) =  0.408

Note: For an MA(1) process, Cryer and Chan use è instead of è₁.

For a given value of ñ₁, two values of è₁ may solve the Yule-Walker equation. The exam problem may specify bounds for è₁, such as an absolute value less than one. The textbook expresses this as the MA(1) model is invertible.

For an invertible MA(1) model, ñ₁ and è₁ has opposite signs. This reflects the sign convention for the moving average parameter.

Know several limiting cases.

        As è₁   zero, ñ₁   zero

        As è₁   one, ñ₁   negative one half (–0.5)

        As è₁   infinity, ñ₁   negative one (–1)