The autocorrelation of an MA(1) process

Question

Consider the process Xt = \varepsilont + 	heta\varepsilon{t-1} where \varepsilont is white noise with variance \sigma^2 and 	heta = 0.5. Derive the autocorrelation function hok. Why does it cut off sharply, and what is the largest ho1 an MA(1) can produce?

Accepted Answer

Since the \varepsilont are uncorrelated with variance \sigma^2: Variance: \gamma0 = \operatorname{Var}(\varepsilont + 	heta\varepsilon{t-1}) = \sigma^2 + 	heta^2\sigma^2 = (1 + 	heta^2)\sigma^2. Lag 1: \gamma1 = \operatorname{Cov}(\varepsilont + 	heta\varepsilon{t-1},\ \varepsilon{t-1} + 	heta\varepsilon{t-2}) = 	heta\sigma^2 (only the shared \varepsilon{t-1} term survives). Lag 2 and beyond: Xt and X{t-2} share no common innovation, so \gammak = 0 for k \geq 2. Therefore ho1 = \frac{	heta}{1 + 	heta^2} = \frac{0.5}{1.25} = 0.4, \qquad hok = 0 \ 	ext{ for } k \geq 2. The ACF cuts…

The autocorrelation of an MA(1) process

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