Temporal Aggregation in the Arima Process

Abstract

The effect of temporal aggregation on ARIMA models is investigated. The paper discusses the change of model form resulting from aggregation. For the IMA model it is noted that reduction of model order may occur, due to aggregation, which takes an arbitrarily high order IMA (d, q) process to an IMA ( 4 0) process for the aggregates. For the AR process, we derive the exact order for the aggregate model and show that aggregation of an AR ( p ) series does not necessarily produce an ARMA (p, q ) aggregate series as has been suggested in the literature. In particular, the AR order of AR (p) and ARIMA (p, d, q) can be reduced upon aggregation. The paper gives the general conditions under which this reduction of order may occur.

Modelling and forecasting time series sampled at different frequencies

“…Two important works connected with Theorem 1 are those of Stram & Wei (1986) and Wei & Stram (1990). Specifically, the loss of observability is related with the "hidden periodicity" defined by Stram & Wei (1986, Definition 4.1) for univariate models.…”

Random sampling of long-memory stationary processes

“…For example the effect of deterministic sampling on ARMA or ARIMA processes is studied in [5], [17] and [21] among others. The main result is that the ARMA structure is preserved, the order of the autoregressive part being never increased after sampling.…”

Beauty and Ugliness of Aggregation over Time: A Survey

“…Thus, a time series which is stationary (non stationary) at the disaggregated level remains so at the aggregated level. This property has been directly or indirectly derived among others by TELSER (1967), AMEMIYA and WU (1972), TIAO (1972), BREWER (1973), TIAO and WEI (1976), WEI (1981), HARVEY (1981), AHSANULLAH and WEI (1984), WEISS (1984), STRAM and WEI (1986), CHRISTIANO and EICHENBAUM (1987), ROSSANA andSEATER (1995) andPIERSE andSNELL (1995). Most of these studies resorted to analytical tools, accompanied by Monte Carlo experiments and empirical evidence to draw and back up their conclusions.…”

“…By the same token, a random walk process generally becomes an integrated moving average process of order one, IMA(1,1), under temporal aggregation but remains to be a random walk process under systematic sampling (AMEMIYA and WU, 1972;TELSER, 1967;TIAO, 1972, among others). The limiting result of an ARIMA(p,d,q) process and an IMA(d,q) process is an IMA(d,l) process with l ≤ d − 1 under systematic sampling (see WEI, 1978a) and an IMA(d,d) process under temporal aggregation (STRAM and WEI, 1986). ROSSANA and SEATER (1995) noted that the latter limiting process can become an IMA(d,d-1) process if the increase of standard error is bigger than the increase of the autocorrelation estimated coefficients.…”