The maximum autocorrelation of an MA(1)

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A series follows Xt = \varepsilont + 1.0\,\varepsilon{t-1} with white-noise shocks. Compute the lag-1 autocorrelation \rho1.

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For an MA(1) with coefficient 	heta: ho1 = \frac{	heta}{1 + 	heta^2}. With 	heta = 1: ho1 = \frac{1}{1 + 1} = \frac{1}{2} = 0.5. This is the largest lag-1 autocorrelation any MA(1) can produce. To see why, treat ho1(	heta) = 	heta/(1+	heta^2) as a function and set its derivative to zero: the maximum occurs at 	heta = 1, where ho1 = 0.5 (and the minimum at 	heta = -1, where ho1 = -0.5). So if you ever estimate a lag-1 autocorrelation above 0.5 from data, an MA(1) cannot be the right model. Takeaway. The MA(1) lag-1 autocorrelation peaks at exactly 0.5 (at 	heta = 1); a samp…

The maximum autocorrelation of an MA(1)

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