Lag-1 autocorrelation of an MA(1) process

Question

A series follows the MA(1) model Xt = \varepsilont + 0.5\,\varepsilon{t-1}, where \varepsilont is white noise with variance \sigma^2. Compute the lag-1 autocorrelation \rho1, and state \rho2.

Accepted Answer

Write out the pieces of an MA(1) with coefficient 	heta. Variance: the two noise terms are uncorrelated, so \gamma0 = \operatorname{Var}(\varepsilont) + 	heta^2\operatorname{Var}(\varepsilon{t-1}) = \sigma^2(1 + 	heta^2). Lag-1 covariance: Xt and X{t-1} share exactly one noise term, \varepsilon{t-1}, so \gamma1 = 	heta\,\sigma^2. Therefore ho1 = \frac{\gamma1}{\gamma0} = \frac{	heta}{1+	heta^2}. With 	heta = 0.5: ho1 = \frac{0.5}{1 + 0.25} = \frac{0.5}{1.25} = 0.4. For lag 2 or more, Xt and X{t-2} share no common noise term, so \gamma2 = 0 and hence ho2 = 0. The MA(1) autocorrela…

Lag-1 autocorrelation of an MA(1) process

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