Order determination is related to the problem of finding the subset of nonzero coefficients of an ARMA model with sufficiently high ARMA orders. A subset ARMA(p,q) model is an ARMA(p,q) model with a subset of its coefficients known to be zero. For example, the model
Yt = 0.8Yt−12 + et + 0.7et−12 (6.5.4)
is a subset ARMA(12,12) model useful for modeling some monthly seasonal time series. For ARMA models of very high orders, such as the preceding ARMA(12,12) model, finding a subset ARMA model that adequately approximates the underlying process is more important from a practical standpoint than simply determining the ARMA orders. The method of Hannan and Rissanen (1982) for estimating the ARMA orders can be extended to solving the problem of finding an optimal subset ARMA model
Indeed, several model selection criteria (including AIC and BIC) of the subset ARMA(p,q) models (2^ (p + q) of them!) can be approximately, exhaustively, and quickly computed by the method of regression by leaps and bounds (Furnival and Wilson, 1974) applied to the subset regression of Yt on its own lags and on lags of the residuals from a high-order autoregression of {Yt}.
It is prudent to examine a few best subset ARMA models (in terms of, for example, BIC) in order to arrive at some helpful tentative models for further study. The pattern of which lags of the observed time series and which of the error process enter into the various best subset models can be summarized succinctly in a display like that shown in Exhibit 6.22. This table is based on a simulation of the ARMA(12,12) model shown in Equation (6.5.4). Each row in the exhibit corresponds to a subset ARMA model where the cells of the variables selected for the model are shaded. The models are sorted according to their BIC, with better models (lower BIC) placed in higher rows and with darker shades. The top row tells us that the subset ARMA(14,14) model with the smallest BIC contains only lags 8 and 12 of the observed time series and lag 12 of the error process. The next best model contains lag 12 of the time series and lag 8 of the errors, while the third best model contains lags 4, 8, and 12 of the time series and lag 12 of the errors. In our simulated time series, the second best model is the true subset model. However, the BIC values for these three models are all very similar, and all three (plus the fourth best model) are worthy of further study. However, lag 12 of the time series and that of the errors are the two variables most frequently found in the various subset models summarized in the exhibit, suggesting that perhaps they are the more important variables, as we know they are!
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