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ARFIMA-GARCH Selection: Joint Estimation and Model Purpose

Article Quant Q&A · Author: Dmitriy
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Summary

This note addresses whether nonsignificant AR terms in a fitted ARFIMA model can be treated as redundant across candidate ARFIMA-GARCH combinations. It distinguishes sequential fitting, where an ARFIMA model is estimated first and its residuals are then modeled with GARCH, from joint estimation of the mean and volatility components. In the sequential setup, the ARFIMA estimates remain fixed; under joint estimation, changing the GARCH specification can affect the AR coefficient estimates and their significance.

The answer also distinguishes forecasting from explanation. For a predictive model, it recommends judging candidates by out-of-sample performance rather than relying chiefly on coefficient significance. For an explanatory model, a nonsignificant estimate means the sample did not detect an effect, not that the term must be removed; substantive reasons may justify retaining it. No proof, fitted examples, or comparison of specific information criteria is provided, so the guidance is conceptual and depends on the estimation procedure and modeling objective.

Key ideas

  • Whether AR terms remain nonsignificant across ARFIMA-GARCH candidates depends on sequential versus joint estimation.
  • Joint estimation can change AR coefficient estimates as the GARCH specification changes.
  • Forecasting models should be compared using out-of-sample predictive performance.
  • A nonsignificant coefficient indicates insufficient detected evidence in the sample, not proof of no effect.
  • Explanatory modeling may retain terms supported by prior knowledge even when their estimates are nonsignificant.

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StatisticsMachine learningBacktesting
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# Reducing possible models count for calibration in ARFIMA-GARCH models


# Reducing possible models count for calibration in ARFIMA-GARCH models












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.

## Answer by Pleb (score 1, accepted)

https://quant.stackexchange.com/a/79301

### 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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.