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# GARCH-M Model

In finance, the return of a security may depend on its volatility (risk). To model such phenomena, the GARCH^{i}-in-mean (GARCH-M^{i}) model adds a heteroskedasticity term into the mean equation. It has the specification:

Where:

- is the time series value at time t.
- is the mean of GARCH model.
- is the volatility coefficient (risk premium) for the mean.
- is the model's residual at time t.
- is the conditional standard deviation (i.e. volatility) at time t.
- is the order of the ARCH component model.
- are the parameters of the the ARCH component model.
- is the order of the GARCH component model.
- are the parameters of the the GARCH component model.
- are the standardized residuals:

- is the probability distribution function for . Currently, the following distributions are supported:
- Normal distribution

- Student's t-distribution

- Generalized error distribution (GED
^{i})

- Normal distribution

## Remarks

- A positive risk-premium (i.e. ) indicates that data series is positively related to its volatility.
- Furthermore, the GARCH-M model implies that there are serial correlations in the data series itself which were introduced by those in the volatility process.
- The mere existence of risk-premium is, therefore, another reason that some historical stocks returns exhibit serial correlations.

## Files Examples

## References

- Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
- Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740