Recover the MA(1) coefficient from its autocorrelation

Question

An MA(1) process Xt = \varepsilont + 	heta\,\varepsilon{t-1} has an observed lag-1 autocorrelation of ho1 = 0.4. Find the invertible value of 	heta.

Accepted Answer

Set the MA(1) formula equal to the observed value: \frac{	heta}{1 + 	heta^2} = 0.4. Cross-multiplying: 	heta = 0.4(1 + 	heta^2), which rearranges to 0.4\,	heta^2 - 	heta + 0.4 = 0 \quad\Longrightarrow\quad 	heta^2 - 2.5\,	heta + 1 = 0. The quadratic formula gives 	heta = \frac{2.5 \pm \sqrt{6.25 - 4}}{2} = \frac{2.5 \pm 1.5}{2} = 2 \ 	ext{ or } \ 0.5. Two different coefficients produce the same autocorrelation. The convention is to pick the invertible one, which requires 	heta < 1. That selects 	heta = 0.5. Invertibility matters because it makes the model identifiable and lets you…

Recover the MA(1) coefficient from its autocorrelation

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