ACF cuts off at lag 1, PACF tails off

Question

For a stationary series, the sample ACF is significant at lag 1 and then falls inside the bands (essentially zero) for all higher lags. The sample PACF tails off gradually, decaying across several lags. Which model does this pattern indicate?

Accepted Answer

Apply the identification rules: - AR(p): ACF tails off, PACF cuts off after lag p. - MA(q): ACF cuts off after lag q, PACF tails off. Here the ACF cuts off sharply after lag 1 while the PACF tails off. That is exactly the MA signature with q = 1, so the model is MA(1). This makes sense from the theory: an MA(1) has ho1 = 	heta/(1+	heta^2) but hok = 0 for all k \ge 2 because observations more than one step apart share no common shock, giving the sharp ACF cutoff. Its PACF, by contrast, decays gradually. Takeaway. ACF cutting off after lag q with a gradually decaying PACF signals MA(q); a…

ACF cuts off at lag 1, PACF tails off

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