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STATE SPACE FORECASTING

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 Box-Jenkins 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, cross-correlations 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 over-parameterize 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.


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