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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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Hans-Peter Deutsch 2009

Authors and Affiliations

  • Hans-Peter Deutsch
    • 1
  1. 1.FrankfurtGermany

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