Transforming the Dependent Variable

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If the time series being analyzed is stored as prices in the dataset, it will be necessary to transform it before analysis. GARCH models are most commonly estimated using annualized log returns. The FANPAC keyword function computeLogReturns calculates

\begin{displaymath}r_{t} = \kappa \; log \frac{p_{t}}{p_{t-1}} \end{displaymath}

where the logarithm is the base 10 (the GAUSS function log() rather than ln()), and $\kappa$ is set to the number of periods of the time series in one year, e.g., 250 if the data has a period of one day (because there are about 250 trading days in a year).  For example   
       computeLogReturns amzn 250;

The success of the estimation in GARCH models is very dependent on proper scaling of the time series. The optimization will have serious difficulty when numbers are very large or very small and especially when there are both. For best results scale the time series to approximately a zero mean and unit variance.

Some investigators may desire percent returns rather than log returns. Percent returns are generated by the keyword command computePercentReturns. For example   

       computePercentReturns amzn 100;
This keyword command computes

\begin{displaymath}r_{t} = \lambda \; \frac{p_{t}-p_{t-1}}{p_{t-1}} \end{displaymath}

For interpretation as a percent $\lambda$ has been set to 100.


R. Schoenberg
1999-03-29