Autoregressive Models: Univariate

The objective of this section is to provide a brief overview of the most commonly used time series model, the AR(1), from both, the traditional and the probabilistic reduction (PR) perspectives.

3.1 AR(1): the traditional perspective

The traditional economic theory perspective for an AR(1) time series model was largely determined by the highly influential paper by Frisch (1933) as a linear (constant coefficient) stochastic difference equation:

yt = a 0 + a 1 yt-1 + e t, |a 1| < 1, et ~ NI(0, о2), t = 1, 2,…, (28.3)

with the probabilistic assumptions specified via the error process {et, t £ ¥}:

Подпись: 1a zero mean: jTj 'со' ii о 2a constant variance: jTj ii Q 3a no autocorrelation: E(etet-т) = 0, T Ф 0, 4a Normality: et ~ N(.,.), Подпись: > t £ ¥.(28.4)

These assumptions define the error {et, t £ ¥} to be a Normal, white noise pro­cess. The formulation (28.3) is then viewed as a data generating mechanism (DGM) from right to left, the input being the error process (and the initial condition y0) and the output the observable process {yt, t £ ¥}. The probabilistic structure of the latter process is generated from that of the error process via (28.3) by recur­sive substitution:

Подпись: (t -1 1 (t-1 1 yt = a 0 + a 1yt_1 + et = a y + a 0 Xa1 + X a1^t-; 11=0 ) 1;=0 ) (28.5)

yielding the first two moments:

Подпись:t > 0.

image644 Подпись: cov( yv y+T) image646 Подпись: T > 0.

Using the restriction | aj < 1 we can simplify these to:

(28.6)

image648

Viewed in terms of its first two moments, the stochastic process {yt, t Є T} is both normal and Markov but second-order time heterogeneous. Traditionally, how­ever, the time heterogeneity is sidestepped by focusing on the "steady-state":

(28.7)

image649 image650

Hence, the (indirect) probabilistic assumptions underlying the observable pro­cess {yt, t Є T}, generated via the AR(1) (28.3), are:

As argued in the next subsection, the probabilistic reduction perspective reverses this viewpoint and contemplates (28.3) in terms of probabilistic assumptions regarding the process {yt, t Є T} and not the error process {e t, t Є T}. It must be emphasized that the probabilistic perspective provides an alternative viewpoint for statistical models which has certain advantages over the traditional theory viewpoint when the statistical aspects of modeling are of interest. In contrast, the traditional theory viewpoint has certain advantages when other aspects of model­ing, such as the system properties, are of interest. Hence, the two viewpoints are considered as complimentary.

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