# Pre-Treatment of Time Series and Assessment of Models

• Hans-Peter Deutsch
Chapter
Part of the Finance and Capital Markets Series book series (FCMS)

## Abstract

The pre-treatment for the transformation of a given data set into a stationary time series has been mentioned several times in the preceding sections and will receive detailed treatment in this section. The basis for pre-treating a time series is its decomposition into a trend component gt, a seasonal component,St, and a random component1 Z t :

$$\matrix{ {{X_t}\, = \,{g_t}\, + \,{s_t}\, + \,{Z_t}} \cr }$$
(34.1)

Z f then represents the stationary time series with E[Z t ] = 0. The trend g t is a deterministic function of the time variable f, which represents a long-term development, for example a polynomial or an exponential function.2 A weaker trend can sometimes be more readily recognized after a compression of the time axis. The season s t represents a periodic component with a period p:

$$\matrix{ {{s_t}\, = \,{s_{t\, + \,p}}} \cr }$$
(34.2)

It follows that the sum $$\sum\nolimits_{i\, = \,1}^p {{s_{t\, + \,1}}}$$ of p successive values is a constant. This constant can be incorporated into the trend g t so that, without loss of generality, the sum can be assumed to be equal to zero:

$$\sum\limits_{i\, = \,1}^p {{s_{t\, + \,i}}} \, = \,0.$$

## Keywords

Time Series Move Average Internal Model Conditional Variance Time Series Model
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