TS Module 12: MA(1) parameter estimation (Yule-Walker equations) practice problems


TS Module 12: MA(1) parameter estimation (Yule-Walker equations)...

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TS Module 12: MA(1) parameter estimation (Yule-Walker equations) practice problems

(The attached PDF file has better formatting.)

** Exercise 1.2: MA(1) model and Yule-Walker equations

If the autocorrelation of lag 1 for an MA(1) process is

ñ1,

What is the Yule-Walker initial estimate for

è?

What is the relation of the two roots of the Yule-Walker solution?

Part A:

An MA(1) model has

ñ1 = –è / (1 + è2).

We write this as a quadratic equation in

è, where ñ1 is a parameter:

ñ

1 è2 + è + ñ1 = 0 è = [ –1 ± (1 – 4 ñ12)0.5 ] / 2 ñ1

Part B:

The expression –

è / (1 + è2) has the same value 1/è as for è. Taking the reciprocal of è gives

–(1/

è) / (1 + 1/è2).

Multiplying the numerator and denominator of this fraction by

è2 gives the original expression.

Illustration:

We fit an MA(1) process with –1

è 1 to a time series. The sample autocorrelation of lag 1 is –0.400.

The Yule-Walker initial estimate for

è is [ –1 ± (1 – 4 × 0.16)0.5 ] / (2 × 0.4) = (–1 ± 0.6) / 0.8 = ±0.5.




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