When we talk about the linear regression, we have $R^2$ to measure the goodness of fit of the linear model.
Here is the problem, do we have a similar statistical measure to assess the goodness of fit of a GARCH model for the raw data?
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2 Answers
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A GARCH model assumes a perfect fit for the conditional variance equation. This feature is due the definition/construction of the GARCH model (note that there is no error term in the conditional variance equation in a GARCH model).
A class of conditional variance models that allows for imperfect fit are stochastic volatility models.
In any case, measuring the goodness of fit of conditional variance models is problematic -- because the conditional variance is unobserved. Thus conventional techniques such as running a regression of the dependent variable (the conditional variance) on the regressors (e.g. lagged conditional variances and lagged squared error terms) and taking the $R^2$ as a measure of fit do not work.
You can assess how "good" a GARCH model is by looking at model residuals and checking how well they match the model assumptions. Normally you assume the residuals to be i.i.d. You also assume they follow a certain distribution, e.g. a normal distribution. You use this assumption in constructing the likelihood function which is then used for fitting the model via the maximum likelihood estimation (MLE). Thus you could look at the residuals and see how close they are to being i.i.d. and following the assumed distribution.
Aksakal suggested looking at AIC or BIC which gives you model likelihood adjusted for the number of parameters (so as to penalize for overfitting). Looking at model likelihood itself can also be meaningful, but then you have to keep in mind that richer models normally yield higher likelihoods.
answered May 26, 2015 at 15:13
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$\begingroup$ Thanks for yr prompt reply! What measurement will people take to compare the performance of a garch model, now that the fitness is perfect? $\endgroup$– JasonCommented May 26, 2015 at 15:21
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$\begingroup$ Okay, the updated answer has address my question. $\endgroup$– JasonCommented May 26, 2015 at 15:22
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$\begingroup$ If you care about comparing a few competing models, you may use AIC or BIC, as suggested by @Aksakal. You can also look for autocorrelations and remaining ARCH patterns in the standardized residuals from the models. You would prefer a model that has low autocorrelations and low remaining ARCH patterns. $\endgroup$ Commented May 26, 2015 at 15:23
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$\begingroup$ Thank you very much, I learn more about garch models through your kind answer! $\endgroup$– JasonCommented May 26, 2015 at 15:33
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$\begingroup$ In JohnHull’s book, he uses autocorrelation (lack of) as a measure of goodness of fit, but doesn’t explain much. @RichardHardy, do you know anything about that? $\endgroup$– VictorCommented Jul 10, 2020 at 18:55
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GARCH and many other time series models are estimated using maximum likelihood methods (MLE). The common measure of in-sample fit is the information criterion such as Akaike (AIC) or Bayesian (BIC). These are computed using the loglikelihoods.
I sometimes use FVU, which may feel more familiar to you if you're used to $R^2$. Its problem is that it doesn't account for parsimony though, that's why it's my last resort tool when AIC is not applicable
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$\begingroup$ thanks for yr answer! I have heard AIC and BIC in model selecting, but I do not expect that they can be used to measure the fitness. $\endgroup$– JasonCommented May 26, 2015 at 15:16
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$\begingroup$ @KylinYi, I'm not sure why you formed such expectations, but AIC is a goodness-of-fit measure to go for time series. $\endgroup$– AksakalCommented May 26, 2015 at 15:23
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$\begingroup$ thanks for yr suggestions! Both of yr and Richard's answer are very useful, what a pity that I cannot accept two answers. Richard illustrates more details about garch models, thus I hope I will not annoy you when I accept his answer. $\endgroup$– JasonCommented May 26, 2015 at 15:31
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$\begingroup$ @KylinYi, no, we have no egos here :) We're here to help each other. $\endgroup$– AksakalCommented May 26, 2015 at 15:36