TS Module 12: Parameter estimation Yule-Walker equations

(The attached PDF file has better formatting.)

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 also use Yule-Walker equations for MA(2) and ARMA models.

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

Exercise 1.1: 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 1.1: An MA(1) model has .

We invert the equation to get

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

The final exam uses multiple choice questions. To avoid arithmetic errors, after solving the problem, check that it gives the correct autocorrelation.

The table below shows selected MA(1) values for ñ₁ and è₁. Note several items:

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 have opposite signs, reflecting 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   zero