# A PROBABILISTIC FRAMEWORK FOR TIME SERIES

A time series is defined as a finite sequence of observed data {y1, y2,…, yT} where the index t = 1, 2,…, T, denotes time. The probabilistic concept which corresponds to this series is that of a real stochastic process: {Yt, t Є T}, defined on the probability space (S, %, P(-)) : Y(-,-) : {S x T} ^ RY, where S denotes the outcomes set, % is the relevant о-field, P(-) : % ^ [0, 1], T denotes the relevant index set (often T := {1, 2,…, T,…}) and RY denotes a subset of the real line.

The probabilistic foundation of stochastic processes comes in the form of the finite joint distribution of the process {Yt, t Є T} as formulated by Kolmogorov (1933) in the form of Kolmogorov’s extension theorem. According to this result, the probabilistic structure of a stochastic process (under certain mild conditions) can be fully described by a finite dimensional joint distribution of the form:

f(y1, y2,…, Vt; v), for all (y1, y2,…, Vt) є RY. (28.2)

This joint distribution provides the starting point for the probabilistic reduction (PR) approach to modeling. The probabilistic structure of a stochastic process can be conveniently defined in the context of the joint distribution by imposing cer­tain probabilistic assumptions from the following three categories:

Distribution: normal, student’s t, gamma, beta, logistic, exponential, etc.

Dependence: Markov( p), ergodicity, m-dependence, martingale, mixing, etc.

Heterogeneity: identically distributed, stationarity (strict, kth order), etc.

Time series modeling can be viewed as choosing an appropriate statistical model which captures all the systematic information in the observed data series. Systematic statistical information comes to the modeler in the form of chance regularity patterns exhibited by the time series data. For example, the cycles exhibited by the time series data in Figures 28.1-28.2 constitute a chance regularity pattern associated with positive autocorrelation because they are not deterministic cycles that would indicate seasonality. For an extensive discussion on numerous chance regularity patterns and how they can be detected using a variety of graphical techniques including t-plots, scatter-plots, P-P and Q-Q plots, see Spanos (1999, chs 5-6).

The success for empirical modeling depends crucially on both recognizing these regularity patterns and then choosing the appropriate probabilistic con­cepts (in the form of assumptions) in order to model this information. The choice
of these probabilistic assumptions amounts to specifying an appropriate statist­ical model. In the context of the PR approach all possible models (P) are viewed as reductions from the joint distribution (28.2). That is, the chosen model P0 £ P, constitutes an element which arises by imposing certain reduction assumptions from the above three categories on the process {Yt, t £ ¥}. This methodological perspective, introduced by Spanos (1986), differs from the traditional view in so far as it does not view these models as just stochastic equations (linear, differ­ence, differential, integral, etc.). In the next section we discuss the two alternative perspectives using the AR(1) model.