Series A has an autocorrelation function decaying geometrically and a partial autocorrelation function that cuts off after lag 2. Series B shows the reverse pattern. Name the model for each with the reasoning, and say what pattern would tell you to difference before fitting anything.

Series A has an autocorrelation function decaying geometrically and a partial autocorrelation function that cuts off after lag 2. Series B shows the reverse pattern. Name the model for each with the reasoning, and say what pattern would tell you to difference before fitting anything.

Approach: Recall what each correlogram measures for a pure autoregression and for a pure moving average, then read the cut-off against the decay.

Series A is an AR(2) and series B is an MA(2). The partial autocorrelation at lag k is the coefficient on the k-th lag in a regression of the series on all lags up to k, so for an AR(p) it is exactly zero beyond p while the autocorrelation decays geometrically as powers of the inverse roots. For an MA(q) the reverse holds: the autocorrelation is exactly zero beyond q, since observations more than q apart share no shocks, while the partial autocorrelation decays because the invertible representation is an infinite autoregression. This identification rule is the whole content of reading a correlogram pair. If instead the autocorrelation starts near 1 and falls off very slowly and almost linearly over many lags, that is the signature of a unit root, and you difference first, because the sample autocorrelations of a non-stationary series are not estimating any fixed population quantity.

Follow-up: Both correlograms decay geometrically with no cut-off anywhere. What model does that suggest, and how do you choose its two orders?

Key concepts: partial autocorrelation, model identification, unit root, differencing.