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I am trying to fit an ARMA/GARCH model to a time series. I found that the best candidate is an ARMA(1,0) + GARCH(1,1) with gaussian white noise model data

It has coefficients with p-values near cero and the residuals are white noise. The problem is that the Jarque Bera Test says the residuals are not normal. The QQ normal plot graph

confirm that. And when I try with several ARMA/GARCH models with t-student white noise, for example:

model data

the QQ t-student plot

graph

fits very well (except for some outliers), but the rest don't (I mean not as good as the first one).

Which one is better?

I have been stuck in this problem for a while.

Thank you very much

Rodrigo

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  • $\begingroup$ I would recommend posting a question such like this over at stats.stackexchange.com $\endgroup$ – Olorun Sep 20 '15 at 9:15
  • $\begingroup$ Questions on GARCH fit better here IMO as it's often used in Quantitative Finance. $\endgroup$ – Bob Jansen Sep 20 '15 at 15:04
  • $\begingroup$ What do you mean by "the rest don't"? The AIC is smaller, so the student-t fit is supposedly a better fit. All p-values are also small, so all coefficients should be relevant. Naively, you would think that the student-t fit would outperform the normal fit, since the latter is a special limit of the former (although AIC does adjust for this). $\endgroup$ – Olaf Nov 24 '15 at 17:48
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To get it out the way: you cannot ask 'what model is better' without a reference to what its use is. Do you want to test for the mean or the AR parameter to trade it? Do you want to calculate VaR? Do you want to forecast volatility over one period? Or over 1000 periods? Or higher moments? Do you want to simulate volatility over one period? Or longer?

For some of these your first model can be perfectly adequate, while for other is will not be.

In general I do not take 'statistical significance' at face value. It does not mean much. The (very small) standard errors you get are not reliable.

Having said that, you can look at the qualitative differences that your models reveal.

  • They agree on their point estimates of mu and ar1

  • The Gaussian model gives very high persistence, alpha1+beta is actually above one. It tries to capture an extremely heavy tail and produces a volatility unit root to achieve that.

  • If you were to simulate vols over longer horizons with the Gaussian model, they would be all over the place.

  • The Student model gives more stable volatility, the sum is .96. Perhaps it reverts too fast. To achieve that it estimated a very heavy tail, if 'shape' means 'degrees of freedom'. So heave that it barely has a variance.

  • Is that mean reverting though? When the error is that fat tailed volatility will look very choppy. A large jump and then exponential decay. Is that how the world looks like?

  • What shocks me is the increase in log-likelihood. If you see the second model as an extension of the first by one parameter, then you have a gigantic improvement. Big enough to make me suspicious.

  • To me it looks like a process with some Garch perhaps, but with structural changes superimposed. Something very bad happens occasionally, which Garch is trying to mimic. Jump diffusion, regime switching, multi fractal, exponential vol, anything that gives rapid structural changes. If you could give us a plot of the time series (before and after differencing) then we could speculate on that.

But as I said, it all depends on what you want your model to do. Simple Gaussian Garch(1,1) is hard to beat out of sample consistently.

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    $\begingroup$ Nice insight. Where did you pick up on this knowledge? Any particular books or articles? $\endgroup$ – Olaf Nov 24 '15 at 20:50
  • $\begingroup$ [Yes, please. ] $\endgroup$ – Rodrigo Guinea Ordóñez Mar 6 '16 at 22:05
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A good rule of thumb is to "test" your models by doing forecasts and to choose the best one. Note however that your choice will be based upon the loss function you selected. If you are concerned about outliers you should (for instance) use Median Squared Errors, if you don't you can use Mean Square Errors. In your particular case the Information Criteria Statistics highly advocates for the use of the Student distribution.

So as general answer it depends of your aim, the choice will differs if you’re estimating for 1) forecasting, 2) parameters calibration, 3) hedging 4) research … I don’t think that a formal answer covering all case can be given. The essential is to understand the implication of (not) choosing a model. (ex: if you are particularly concerned with extreme events, the QQ Plot warns you that the Gaussian hypothesis underestimate these events.)

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  • $\begingroup$ "In your particular case the Information Criteria Statistics highly advocates for the use of the Gaussian distribution. .." Why? The AIC is lower for the student-t, which is better, right? $\endgroup$ – Olaf Nov 24 '15 at 17:50
  • $\begingroup$ you are right, sorry I will correct it. $\endgroup$ – Malick Nov 24 '15 at 17:51

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