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I'm working on a R project, trying to calibrate a GARCH (so far, (1,1) ) model to the yields of the STOXX50 index over the last 2 years.

I've tried the garch function of the tseries package, but it gave me a "false convergence" result. I tried then the ruGARCH package, and no false convergence so far, but I would like to know if my model is a good fit for the data.

How can I do that ? How can I interpret the results of all the tests done (Box-Liung, etc..)

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  • $\begingroup$ Did you calculate AIC or BIC? $\endgroup$ – Egodym Feb 14 '16 at 17:43
  • $\begingroup$ yes, the function calculated both and it gave me : Akaike -5.8402 Bayes -5.7908 (these are both "minus 5.8...) $\endgroup$ – B2000 Feb 14 '16 at 18:33
  • $\begingroup$ Fit another model(s) to data and compare the values of AIC and BIC. The lower the values, the better the fit of the model. $\endgroup$ – Egodym Feb 14 '16 at 18:42
  • $\begingroup$ just tried on Garch(1,2) and (2,2), AIC and BIC are basically the same (0.05 variation each time, is it significant ? And if this is not, how to interpret the p values in Ljung-Box test on residuals, or in Nyblom stability test ? $\endgroup$ – B2000 Feb 14 '16 at 18:52
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    $\begingroup$ I recommend reading the "Introduction to rugarch package" to see what you are testing. Interpreation of p-value is always the same. $\endgroup$ – Egodym Feb 14 '16 at 19:18
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To test for model misspeicfication:

  1. First ensure that auto correlation of standardized residuals resulted from the ARMA-GARCH model are not significant. Further, you can use Box-Ljung test. It test joint significance of auto correlation upto lag $K$.

  2. Leverage effect is tested by sign bias test. If $p$ value is less than .05 (assumed significance level) then it indicate presence of leverage effect in the data. In this case, try models that capture leverage effects like TGARCH, EGARCH etc.

  3. The chi-squared goodness of fit test compares the empirical distribution of the standardized residuals with the theoretical ones from the chosen density.

But before fitting GARCH model, check for ARCH effects in your data. If there is no ARCH effect then GARCH model is not required at all.

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First of all I would examine whether the model performs the task it is supposed to perform, i.e. account for the conditional heteroskedasticity in the data. That would amount to testing for remaining ARCH effects in the standardized model residuals by the Li-Mak test. If the model fails the test, there is evidence that it does not do its main task well.

Testing for autocorrelations of the standardized residuals, leverage effects and distributional goodness of fit as suggested by @Neeraj also makes sense.

However, be aware that a model that passes all test may be an overfitted model. That is, it describes the sample data well (actually, too well) but it is not likely to generalize successfully, e.g. it would not fit a new sample from the sample underlying population (or data generating process) well. Therefore, use of information criteria (which penalize overfitting) for selecting a model may be justified.

References

  • Li, W. K., and T. K. Mak. "On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity." Journal of Time Series Analysis 15.6 (1994): 627-636.
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