Econometrics II

We were occupied with an imprecise definition of a stochastic process:

An -valued stochastic process is a collection of random variables parameterized by an index (or parameter) set , that satisfies some conditions, which imply that it is equivalently a random element (an appropriatele measurable function) with values a the set of functions , and thereby it defines a probability measure on the latter set. Any such function is termed as a sample path of the process.

If is a non-empty finite (and ordered) subset of the parameter set, then the finite dimensional distribution (fidi) of the process corresponding to is simply the joint distribution of the random vector consisting of the random variables of the process that are indexed by the elements of according to its ordering.

The (Daniell-) Kolmogorov Extension Theorem implies that such a process, or equivalently the probability measure that it defines on the the set of functions , is equivalently "described" by the set of all fidis, given some consistency conditions w.r.t. to permutations and integration.

A "natural class" of stochastic processes is that of the Gaussian ones. A Gaussian process is a stochastic process for which every fidi is a Normal distribution. A simple example of a Gaussian process can be found here. 

When the parameter set is totally ordered, hence it could represent time, then the process is called time series (the term time series can be also used in order to characterize the sample paths of such a process-we will by convention use it for the process itself). 

We will be occupied with time series examples with parameter sets that are usually subsets of the real line. When those subsets are discrete, then the time series is said to evolve in discrete time, while when they are continuous then it is said to evolve in continuous time.

When  or  then the time series is also called a double stochastic sequence and a stochastic sequence respectively, and usually denoted by , and  respectively.

Typical examples of such stochastic sequences, and the more accessible to us at this early stage of the course are the iid ones, or more generally the ones comprised by independent random variables.

Given the fidi characterization of a process, several potential properties can be described via fdis.

In this respect, for such a sequence and a fidi of it, the latter is said to be invariant w.r.t. time translations iff the fidi remains the same when the time indices that define it are arbitrarily translated.

This implies the following property. A time series is called (strictly-) stationary iff every fidi remains invariant w.r.t. time translations-try to extend the definition when the parameter space is the real line. We have also defined weak (or second order, or covariance) stationarity, via a set of (potentially joint) moment conditions.

Notes for the above can be found here.