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I have the question connected with ARFIMA-GARCH models. I have a time series for which I want to calibrate best model (p,q)-(P, Q) (via BIC) with $ p,q <= 4, P,Q <=2$. GARCH part can be "not vanilla" (GJR, EGARCH, TGARCH, APARCH, other). So, in general, I need to check each possible model and calculate BIC, choosing the best with the lowest BIC value. But... What if I have "redudant" model in ARFIMA part? For example, ar coeffs with indexes 1, 3 in $ARFIMA(3,3)$ are not significant after the model calibration. Is it possbile to say, that EACH model $ARFIMA(3,3)-*GARCH(P, Q)$ will have these ar1, ar3 coefficients as not significant? If "yes", how can it be proven?

Thank you.

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    $\begingroup$ @Pleb, excellent points so far (2/3). I think they should be posted as an answer instead of a series of comments. $\endgroup$
    – Richard Hardy
    Commented May 7 at 11:49
  • $\begingroup$ @Dmitriy I guess it doesn't completely answer your question? Could you provide some feedback, then I can change my answer accordingly. $\endgroup$
    – Pleb
    Commented May 31 at 4:56
  • $\begingroup$ @Pleb I'm sorry, was absent for a time, you was really cool, thank you for your answer!! $\endgroup$
    – Dmitriy
    Commented Jun 4 at 10:27

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It depends on how you fit your combined models

If you do sequential fitting ie. fit the ARFIMA(3,3) model first and then feed the residuals through a GARCH model, then all ARFIMA(3,3)-GARCH(P,Q) models will have redundant ar1 and ar3 variables. However, if you do joint estimation of ARFIMA(p,q)-GARCH(P.Q) you might end up with a model combination where ar1 and ar3 becomes statistically significant, depending on the fitted GARCH model.

In the end, it all boils down to the purpose of the chosen model: Will it act as an explanatory model or a predictive model?

If the goal of the model is to forecast the time-series, then statistical tests of the model variables aren't your main concern. Instead, you should be validating the model performance via out-of-sample test procedures.

If the goal of the model is to explain which variables contribute to the patterns in your time-series, then there is no real need of removing non-significant variables in your time-series. Presumably, you included the variables in your model because you thought that they might play a role in capturing some of the patterns in your time-series.

That the variables failed to reject the null (aka. became redundant) does not imply that the model will perform poorly, it just means that your sample did not detect an effect of the ar1 and ar3 variables. However, if there is a foundational basis to include the extra ar-terms (either from expert opinions or from past experience), then the variables might become statistically significant in the future and hence, they shouldn't be removed from the explanatory model.

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  • $\begingroup$ thank you much. It's so sad that no "fast" algorithm for P,Q parameters selection in GARCH part (like stepwise algorithm for ARIMA in this article Hyndman, RJ and Khandakar, Y (2008) "Automatic time series forecasting: The forecast package for R"). $\endgroup$
    – Dmitriy
    Commented Jun 4 at 10:32
  • $\begingroup$ Sorry, Pleb, one additional question I want you to explain me: imagine I've calibrated ARMA-GARCH model (in rugarch package it's a vanilla ARMA-sGARCH model with normal distribution). So, can I say, that all ARMA-GARCH model with different distributions or *GARCH part with the same P, Q parameters (for GARCH part) - for example, eGARCH, gjrGARCH and other - will also have the same significance for AR/MA coefficients or not? $\endgroup$
    – Dmitriy
    Commented Jun 5 at 12:12
  • $\begingroup$ This depends on how the rugarch package estimates the composite model. If it estimates sequentially: first the ARMA part and then the GARCH part, then the answer would be yes. However, I would read the documentation to get an understanding on how they actually estimate the composite model. :-) $\endgroup$
    – Pleb
    Commented Jun 5 at 15:04
  • $\begingroup$ I would be very grateful if you did this! $\endgroup$
    – Dmitriy
    Commented Jun 5 at 15:07
  • $\begingroup$ It says under the ARFIMAX section that they use a joint estimation scheme for the mean and variance equation. Hence, there will be variability in the ARMA parameters for each different ARMA-GARCH models you estimate. Hope this helps :-) $\endgroup$
    – Pleb
    Commented Jun 6 at 10:38

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