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Asymptotic Behavior

If you are working with long-range forecast horizons, the following asymptotic behaviors hold for the outputs of garchpred:

• As noted earlier, the conditional standard deviation forecast sigmaForecast, which is the first garchpred output, approaches the unconditional standard deviation of {ε_t}.

  For GARCH(P,Q) models it is

  

  For GJR(P,Q) models, it is

  

  And for EGARCH(P,Q) models, it is

  

• GARCH effects do not affect the MMSE forecast of the conditional mean meanForecast, which is the second garchpred output. The forecast approaches the unconditional mean of {y_t} as in the constant variance case. That is, the presence of GARCH effects introduces dependence in the variance process. It only affects the uncertainty of the mean forecast, leaving the mean forecast itself unchanged.

• The mean square error of the conditional mean meanRMSE^2, which is the square of the fourth garchpred output, approaches the unconditional variance of {y_t}.

• EGARCH(P,Q) models represent the logarithm of the conditional variance as the output of a linear filter, rather than the conditional variance process itself. Because of this, the MMSE forecasts derived from EGARCH(P,Q) models are optimal for the logarithm of the conditional variance. They are, however, generally downward-biased forecasts of the conditional variance process itself. The following output arrays are based on the conditional variance forecasts:

  • SigmaForecast

  • SigmaTotal

  • MeanRMSE

  Thus, these outputs generally underestimate their true expected values for conditional variance forecasts derived from EGARCH(P,Q) models. The important exception is the one-period ahead forecast, which is unbiased in all cases. For unbiased multiperiod forecasts of SigmaForecast, SigmaTotal, and MeanRMSE, you can perform Monte Carlo simulation using garchsim. For an example, see Example Workflow.

Presample Data

Examples

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