Extending the autoregressive AR(1) model
The probabilistic reduction (PR) perspective, as it relates to respecification, provides a systematic way to extend AR(1) in several directions. It must be emphasized, however, the these extensions constitute alternative models. Let us consider a sample of such statistical models.
The extension of the AR(1) to the AR(p) model amounts to replacing the Markov (reduction) assumption with that of Markov of order p, yields:
Vt = a0 + £ akyt-k + ut, f О T,
with the model assumptions 1c-5c modified accordingly.
AR(1) MODEL WITH A TRENDING MEAN
Extending the AR(1) model in order to include a trend, amounts to replacing the reduction assumption of mean stationarity E(yt) = p, for all f О T, with a particular form of mean-heterogeneity, say E(yf) = pf, for all f О T. The implied changes in the bivariate distribution (28.11), via the orthogonal decomposition (28.9), give rise to:
where the statistical parameters ф := (50, 5V a1r a2) Є R2 x (-1, 1) x R+ and
Ф := (p, a1, a0) Є R2 x R+, are interrelated via:
This is an important extension because it provides the backbone of Dickey-Fuller unit root testing (see Dickey and Fuller, 1979, 1981) based on H0 : a 1 = 1. A closer look at the above implicit parameterizations, however, suggests that when a 1 = 1 ^ (50 = p, 51 = 0, a2 = 0); this raises important methodological issues concerning various aspects of these tests (see Spanos and McGuirk, 1999 for further details).
The extension of the AR(1) model to include higher-order trend terms and seasonal effects can be achieved by postulating mean-heterogeneity of the form
E(yt) = XaD + X, t Є T, where (Dit, i = 1, 2,…, m)
can be either sinusoidal functions or/and dummy variables purporting to model the seasonal effects. We conclude this subsection by noting that following an analogous procedure one can specify AR(1) models with a trending variance; see Spanos (1990).
Non-normal autoregressive models
By retaining the reduction assumptions of Markovness and stationarity and replacing normality with an alternative joint distribution one can specify numerous nonnormal autoregressive models; see Spanos (1999, ch. 8). The normal autoregressive model can also be extended in the direction of nonlinear models; see Granger and Terasvirta (1993) for several important nonlinear time series models.