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My question is simple. When shall I stop when trying the value for p and q?

I have got the loglikelihood from ARCH(1) to ARCH(10). It's increasing. And then I tried GARCH(1,1), GARCH(2,1) etc. The loglikelihood is always increasing so I'm confused. Do I need to continue to apply for larger lags until the log likelihood start to decrease? What argument shall I make when I stop trying?

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    $\begingroup$ Are you using straight log likelihood or a penalized version like AIC / BIC? $\endgroup$ Dec 9, 2014 at 18:13
  • $\begingroup$ I'm using straight log likelihood. Shall I include AIC as another critieria? $\endgroup$ Dec 9, 2014 at 18:46
  • $\begingroup$ I'm not an expert in time series, but unless I'm strongly mistaken, moving from (eg) ARCH(1) to ARCH(2) amounts to adding another parameter / predictor to the model. So of course it would make the model look like it fits better (even if it doesn't really). You would need some sort of penalty to account for this fact. $\endgroup$ Dec 9, 2014 at 19:13
  • $\begingroup$ Thank you very much for your help =] You are right. I just checked and I decides to use AIC because AIC is like the likelihood plus the penalty of adding more parameters. $\endgroup$ Dec 9, 2014 at 20:00
  • $\begingroup$ I agree with gung that you should use AIC/BIC instead of likelihood for this kind of problem. Also, you may look at residual diagnostics to see how good a job your model does in terms of capturing the conditional heteroskedasticity. $\endgroup$ Dec 10, 2014 at 1:40

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As the commenters point out, increasing ARCH/GARCH orders amounts to including additional degrees of freedom, so the (log) likelihood is guaranteed to increase. If you simply follow this increase in (log) likelihood, you will overfit.

One possibility would be to balance the gain in (log) likelihood against model complexity by adding a penalty term. Information criteria like the AIC or BIC are helpful here.

Here is an alternative. ARCH and GARCH are fundamentally ways to forecast future volatility. They aim at producing good density forecasts, by modeling the conditional heteroskedasticity. So one way of choosing model orders would be to fit each model, then create density forecasts for a holdout sample and assess which model gives the best density forecast. Possible tools would be the Probability Integral Transform or (proper) .

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