# 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.

AR(p) model

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:

p

Vt = a0 + £ akyt-k + ut, f О T,

k=1

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

m l

E(yt) = XaD + X, t Є T, where (Dit, i = 1, 2,…, m)

i—1 k—1

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.