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The two fundamental steps in time series analysis are (1) identifi
cation of the appropriate model and (2) estimation of parameters. The
following discussions outline these two steps.
Univariate Time Series
An observed time series (x^,...,xt) may be considered a realization
of some theoretical stochastic process (Granger and Newbold, 1977). In
a general sense, the observed time series is selected from a finite set
of jointly distributed random variables, such that there exists some
probability distribution function P(x^,...,xt) that assigns probabili
ties to the possible combinations of normally distributed x^, il,...,t.
Unfortunately, except for very small t, the probability functions of the
outcomes (xj,...,xt) are not completely known. However, it is possible
to generate a model that captures most of the underlying stochastic
properties and, thus, the random behavior of the series.
Each time series possesses a unique characteristicthe autocorre
lation function. This function, which is independent of the unit of
measurement, indicates whether the process moves in the same or opposite
direction through time. In other words, the autocorrelation function
provides a measure of how much interdependence (memory) there is between
data points in a given time series. The autocorrelation function is
given as
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