Partial Autocorrelation Properties For Non‐Stationary Autoregressive Moving‐Average Models

. We are primarily interested in relating the partial autocorrelation behaviour of an autoregressive integrated moving‐average process of order (p, d, q), {Zi} say, with those of its D‐differenced processes {(1 ‐ B)DZi} (D= 1, …, d). To this end, we evaluate the early partial correlations corresponding to serial correlations which initially follow a slow linear decline from unity. These partials, to a first approximation, take a small constant negative value from lag 2 onwards. We also demonstrate a relationship between the theoretical partials πk and πk(Δ) for a once‐integrated process {Zi} and its first‐differenced process {(1 ‐ B)Zi} respectively. These results carry over to cases where the non‐stationary zeros are at ‐1, and differencing is replaced by the corresponding ‘simplifying’ transformation implicit in the operator 1 +B.