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I only have a very basic understanding of time series analysis. As I am learning ARIMA and then ARCH/GARCH models, I have some subtle (at least for me) questions on the common procedure to build such models and what statistical tests to use.

Let's say I have followed the Box-Jenkins methodology to find the best $p$ and $q$ for ARIMA. In most literature, the Ljung-Box test is used in the end to test whether autocorrelation at different lags is zero.

  1. But the assumptions of zero mean and constant variance (homoscedasticity) are never mentioned or checked in all the literature I see. Aren't these two along with the autocorrelation the white noise assumption for residuals? Are those two assumptions not that important or that the usual Ljung-Box tells more than just autocorrelation (I feel like zero autocorrelation at different lags somehow implies constant variance, correct me if I am wrong)

  2. This question I think is somehow related to the first one. From the expression or equation of the ARCH/GARCH models, my understanding is that they are trying to model the inconstant variance (heteroskedasticity) of the residuals. Therefore, in my opinion, the natural way to employ ARCH/GARCH is that after building an ARIMA model, if we found residuals with inconstant variance, we fit an ARCH/GARCH process to the variance of residuals (However, like I asked above, inconstant variance is barely, if not never, checked as an assumption).

  3. Somehow follow-up question of 2. If we have already found that the residuals have inconstant variance (heteroskedasticity), do we fit the ARCH/GARCH process to the variance of residuals or directly to the variance of original time series $y$? Is there a common practice to this? Or we could just fit both to see which one is better based on prediction and other metrics?

Thank you in advance. I know this might be too much to ask. But any input would be greatly appreciated!

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  1. Zero autocorrelation at different lags does not imply constant variance. An ARCH(1) time series is illustrative of this, in that the variances are autocorrelated but the values of the time series themselves are not.

  2. That's what happens at an intuitive level (i.e. fit ARIMA and then (G)ARCH). And heteroskedasticity doesn't need to be "checked"; all you need to do is reject the assumption of homoskedasticity. This can be done by looking at, for example, the ACF of squared residuals. There are numerous statistical tests that can test the null hypothesis of homoskedasticity.

  3. In practice, if you discover that the residuals have a GARCH structure, what you do is you first identify the order of the GARCH structure and then re-fit your ARIMA+GARCH model simultaneously (in one shot). You can do this using the rugarch library in R, for example. The effect of including the GARCH structure includes, but is not limited to, interpretation of the GARCH coefficients and also improving prediction intervals for forecasts.

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  • $\begingroup$ Thank you for your response! Follow-up on 1: But why isn't the constant variance checked in the Box-Jenkins method? Is it because the Box-Jenkins method cannot model heteroskedasticity any way? $\endgroup$ – codeedoc Jul 25 '18 at 20:47
  • $\begingroup$ It does not. One resource I found that covers the Box-Jenkins methodology uses tests to look for uncorrelatedness in residuals. However, that fact that the residuals are uncorrelated doesn't imply that the squared residuals are uncorrelated! $\endgroup$ – Kevin Li Jul 25 '18 at 23:33

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