

The concept of a "State" vector came from
the physical sciences. A vector includes all the information about
the system that carries over into the future. For example, the
space shuttle moving through space. Six laws of nature impact
its trajectory. You do not need to know anything about the shuttle's
past, but only the six components of its vector state. Only the
current information is needed to predict its future course.
This is the simplest case. Suppose that same shuttle is moving
through the atmosphere and there exists measurement error. Our
system is not DETERMINISTIC now, but STOCHASTIC. Therefore, the
current state of affairs is described not only on current things,
but also on a past past "process error". This system can be described
by linear equations that we must first determine from the data
itself  not just the six laws of nature.
Since in reality we do not know the true vectors (determinants)
associated with a forecast series, we used the data itself to
give us information about the process. The traditional BoxJenkins
approach does not do this. It tries to identify the correct form
of the ARIMA processes and then estimate and test to see if patterns
remain in the error component. In State Space, the data associated
with the dependent and independent variables are used to extract
sets of linear combinations of past and future values. This extraction
is performed by a common multivariate technique called CANONICAL
CORRELATION.
In canonical correlation, our interest centers on the linear
relationship between one battery of variables (y1,y2) and another
battery of variables (x1,x2). The objective is to find sets of
linear composites subject to certain conditions. First, we compute
the linear composite of y1 called t1. These linear combinations
are the results of applying weights to the data. X1's linear composite,
u1 is also calculated. These linear composites are calculated
in such a way as to obtain the maximum correlation between t1
and u1. Once this correlation is calculated on the linear composites,
its y1 and x2's turn. Y2 and x2's linear composites (t2 and u2)
are calculated for maximum correlation  BUT subject to being
uncorrelated with t1 and t2. Therefore, the correlations between
successive pairs will decline in size. Canonical correlation seeks
to find successive pairs of linear composites that are maximally
correlated, subject to being uncorrelated with the previously
found pair.
Recapping, the State of a time series consists of all the linear
indep. combinations of past and present data that correlates significantly
with the future of the endogenous variables. Because of the "all",
there is no more information that can be extracted from the past.
Because of linear indep endence, there is no erroneous information.
Both past and present data is blended together. State Space Models
exploit past and present autocorrelations. FOr multivariate cases,
crosscorrelations between the variables are also exploited.
State Space Forecasting has characteristics much like ARIMA
 i.e. stationarity requirements, etc. All Box Jenkings models
can be restated in their State Space form, except State Space
tends to overparameterize the models. This, however, generally
happens by including more moving average terms than would have
been estimated by a ARIMA like procedure. Because State Space
will work with one or more data series simultaneously, joint stationarity
conditions must be considered. As a practical stand, this may
mean differencing all variables in the system by the same degree
of differencing. If joint stationarity is not attained, autocorrelations
among the data may be distorted.


