Both ACF and PACF tail off gradually

Question

For a stationary series, the sample ACF tails off gradually and the sample PACF also tails off gradually. Neither plot cuts off sharply at any lag. Which model does this pattern indicate (give the simplest one)?

Accepted Answer

Recall the rules: - AR(p): ACF tails off, PACF cuts off after lag p. - MA(q): ACF cuts off after lag q, PACF tails off. - ARMA(p,q): both the ACF and the PACF tail off gradually, with neither cutting off. Since neither plot has a clean cutoff here, this is a mixed ARMA model. The simplest model that produces two gradually decaying plots is ARMA(1,1), one autoregressive term and one moving-average term. The lesson is that ACF and PACF diagnostics identify a pure AR or pure MA cleanly (one plot cuts off, the other tails off), but a mixture shows no cutoff in either. In practice you then compare…

Both ACF and PACF tail off gradually

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