ACF cuts off at lag 2, PACF tails off

Question

For a stationary series, the sample ACF is significant at lags 1 and 2 and then cuts off (falls inside the bands) from lag 3 onward. The sample PACF tails off gradually. Which model does this pattern indicate?

Accepted Answer

Match against the 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 after lag 2 while the PACF tails off, so q = 2 and the model is MA(2). An MA(2) is Xt = \varepsilont + 	heta1\varepsilon{t-1} + 	heta2\varepsilon{t-2}. Observations up to two steps apart share at least one common shock, so the ACF is nonzero at lags 1 and 2, but observations three or more steps apart share nothing, forcing hok = 0 for k \ge 3. That is the sharp cutoff you see. The PACF of a moving-average model always decays gradually rather …

ACF cuts off at lag 2, PACF tails off

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