MA(1) autocorrelation with a negative coefficient

Question

A series follows Xt = \varepsilont - 0.5\,\varepsilon{t-1} with white-noise shocks. Compute the lag-1 autocorrelation \rho1.

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The MA(1) lag-1 autocorrelation formula carries the sign of 	heta: ho1 = \frac{	heta}{1 + 	heta^2}. Here 	heta = -0.5, and note 	heta^2 = 0.25 is unchanged by the sign: ho1 = \frac{-0.5}{1 + 0.25} = \frac{-0.5}{1.25} = -0.4. The negative value means neighboring observations tend to move in opposite directions: a value above the mean is typically followed by one below it. As always for an MA(1), ho2 and all higher lags are exactly 0. Takeaway. A negative MA(1) coefficient gives a negative lag-1 autocorrelation of the same magnitude as its positive twin, producing an alternating, mea…

MA(1) autocorrelation with a negative coefficient

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