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