The data comprises a univariate time series:
In this case, α = 19. The time series is graphed in Exhibit 2:
An example of a multivariate time series would be one comprising daily 1-month, 3-month and 12-month USD Libor rates. This would be a 3-dimensional time series. Presumably, the mechanism that generated a time series will
continue into the future. We are interested in future values, which we
treat as random. To model these, we specify a model called a
stochastic process based upon the time series.
The word stochastic means random. A
stochastic process—or
process—is a
set of random variables
Time series analysis is the fitting of stochastic
processes to time series. This usually entails statistical analyses, but
it is not a straightforward application of statistics. A stochastic
process
Consider again our time series of natural gas prices graphed in Exhibit 2. Are prices from one day to the next interdependent? Certainly! The price one day is likely to be close to that of the previous day. If we considered a longer time series of prices, we might notice other dependencies. For example, natural gas prices often exhibit seasonalities, with prices being higher during the Summer than they are during the rest of the year. Consider a time series of stock prices. Stock prices rise and fall, but on average, they rise with time. Since earlier stock prices are on average lower than later stock prices, those stock prices cannot be identically distributed. Their probability distributions have different means!
Since terms
tX of a process need not be independent, values
realized by earlier terms may affect those realized by later terms, so we
must distinguish between conditional and unconditional distributions for
terms. The conditional distribution of tX as of
time t – k is its distribution conditional on all
information available at time
t – k, including values
To understand the important distinction between the unconditional distribution and a conditional distribution for a term of a stochastic process, consider a somewhat contrived univariate process Y. All terms tY are equal:
and have a U(0,1) unconditional distribution. Two realizations of the process are illustrated in Exhibit 3:
Every term tY has a U(0,1) unconditional distribution, but as soon as we know the value realized by any one term, we know all other terms will realize that same value.
If terms of a univariate process were IID, there would be no
correlations—either conditional or unconditional—between the terms. If
terms of a multivariate process were IID, only correlations
between components of each term
A standard approach of time series analysis is to take a time series that exhibits complicated behavior and try to convert it to a simpler form. Standard techniques for univariate time series are
Multivariate time series can be similarly modified by applying the same techniques to their individual (univariate) components. Optimally, such simplification would yield time series that were so simple that they could reasonably be modeled as IID. In practice—and especially in financial applications—this is rarely possible. A more reasonable goal is to obtain a simplified time series that can reasonably be modeled with some stationary process. Stationarity is a condition similar to IID, but not as strong. Two different forms of stationarity are defined.
Note the similarity of strict stationarity to the IID condition, which
requires, among other things, that the unconditional joint distribution of
any term
Covariance stationarity is the condition that is more frequently assumed in applications. It does require that all first and second moments exist whereas strong stationarity does not. In this one respect, covariance stationarity is a stronger condition.The literature on time series analysis documents numerous standard models for stationary processes. The simplest of these are white noise processes. From white noise processes can be constructed moving average, autoregressive and autoregressive-moving-average processes, which are generally used to model conditionally homoskedastic autocorrelated processes. Other processes are used to model conditionally heteroskedastic processes. Techniques for fitting these processes to actual time series tend to be specific to the particular models.
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, in which case the time
series is 
, in which case it is
or random vectors
may seem similar to a sample
,
and a time series
of a sample, but there is a profound difference. A sample—the province of
statistics—comprises random variables that are assumed independent and
identically distributed (IID). While it is possible that the terms of a
stochastic process might be IID—in which case, time series analysis
reduces to statistics—this is not a particularly interesting case. The
purpose of time series analysis is to study the more interesting case in
which terms corresponding to different points in time have interdependencies.
realized by the stochastic process up to that time. Usually, we don't need to
know all preceding values. Only a handful of the most recent values may be
relevant, but this depends upon the particular process. 
could be non-zero. Absent an IID condition, two
other types of unconditional correlations arise. There are correlations
between corresponding components of two terms lagged a period k
apart,
.
These are called 
.
These are called 
differencing: replacing a time series having terms
tx with one having differenced terms
;
is identical to the unconditional joint distribution of any other segment 
.
Strict stationarity is appealing because it affords a form of homogeneity
across terms without requiring that they be independent.
