MA(1) autocorrelation with a larger coefficient

Question

A series follows Xt = \varepsilont + 0.8\,\varepsilon{t-1} with white-noise shocks. Compute the lag-1 autocorrelation \rho1.

Accepted Answer

For an MA(1) with coefficient 	heta, the lag-1 autocorrelation is ho1 = \frac{	heta}{1 + 	heta^2}. With 	heta = 0.8: ho1 = \frac{0.8}{1 + 0.64} = \frac{0.8}{1.64} \approx 0.488. All autocorrelations at lag 2 and beyond are exactly 0, as always for an MA(1). Notice that even though 	heta jumped from a smaller value up to 0.8, ho1 only reached about 0.49: the formula is bounded, and no MA(1) can push its lag-1 correlation past 0.5. Takeaway. The MA(1) map ho1 = 	heta/(1+	heta^2) compresses large coefficients, so its lag-1 autocorrelation can never exceed 0.5 in magnitude no matte…

MA(1) autocorrelation with a larger coefficient

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