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According to what I have found so far, in order to implement ARIMA we need to have a stationary (constant mean and variance) transformed data set. In addition, I have also seen that the square of the residuals may be in relation as a result of an ARIMA model, which is why we involve an ARCH or GARCH model. Actually, volatility (variance) may be time-dependent as a result of ARIMA.

How is it possible that although our transformed data have constant variance, in order to implement the ARIMA model, the sum of squared residuals, which means variance as a result of the ARIMA model, may not be constant (time-dependent)?

We say that to employ ARIMA we need to have constant variance data, but afterwards according to the residuals of ARIMA we can see that variance is not constant and time-dependent, so let's include an ARCH GARCH model? Isn't this a contradiction?

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  • $\begingroup$ Why can not get any responses? $\endgroup$
    – mertcan
    Jun 17, 2019 at 13:28
  • $\begingroup$ Here is my perspective. There are rather few users who are both capable of and interested in answering questions related to GARCH models. I think I have been the most active one over the last few years, but right now I am pretty busy at work with hardly any free time to spend here. $\endgroup$
    – Richard Hardy
    Jun 18, 2019 at 14:31
  • $\begingroup$ Scratch that: I have given it a try. $\endgroup$
    – Richard Hardy
    Jun 18, 2019 at 20:02

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This is indeed a contradiction, and this is one of the reasons why simultaneous ARIMA-GARCH estimation is preferred to stepwise estimation (first ARIMA, then GARCH).
(I have mentioned this in a few of my earlier answers. You can look them up using keywords "simulatneous estimation" or "simultaneously", "ARIMA" or "ARMA", "GARCH" in questions tagged by .)
Many of the optimality results for estimators of ARIMA models assume constant conditional error variance, and if this is violated, optimality is no longer guaranteed. Similarly, optimality results for GARCH estimators also assume the conditional mean model is well specified.

Now, a practical problem arises from the fact that we do not know what model (e.g. what lag orders within the ARIMA-GARCH class) is adequate for the data before we explore them. Therefore, we have to try out different candidate models and check their adequacy until we find a satisfactory model. In the process, we of course fit some models just to learn that they are not adequate (e.g. fit an ARIMA model without GARCH errors but discover that the errors are plagued by autoregressive conditional heteroskedasticity), and this is essentially unavoidable. What is important is that our final model is adequate.

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  • $\begingroup$ I am so glad to have a response from you @Richard Hardy. As I see you also agree with me. Even we apply our ARIMA-GARCH model stepwise we can not evade from contradiction. The most important assumption ARIMA involves is the variance of the errors is constant and it is known as homoscedasticity. I mean at the end of ARIMA model we must have constant variance, but when we take squared of residuals we may say there is a heteroscedasticity let's apply GARCH ARCH model. Still contradiction exist? $\endgroup$
    – mertcan
    Jun 19, 2019 at 19:18
  • $\begingroup$ @mertcan, Yes, I think I agree with you. But read my last sentence: if you happen to find an adequate model, then all assumptions are satisfied and estimators have their optimality properties. So the process of model discovery might be plagued by some contradictions, but the end result may be free of them. $\endgroup$
    – Richard Hardy
    Jun 20, 2019 at 15:48
  • $\begingroup$ Again thanks for return @Richard Hardy. I am aware that seeing high ACF or PACF value in ARIMA residuals may be a sign to employ ARIMA-GARCH model. But what if ARIMA-GARCH model and its variants lead to non-normal $$v_t = \frac {u_t} {\sigma_t}$$ whereas $$y_t=ARIMA model terms+u_t$$ and $$u_t=v_t*\sigma_t = N(0,\sigma_t^2)$$ and $$v_t = N(0,1)$$ is supposed to be normal? What other model must we consider? $\endgroup$
    – mertcan
    Jun 22, 2019 at 10:02
  • $\begingroup$ @mertcan, ARIMA-GARCH are often specified with nonnormal standardized errors, e.g. Student-$t$. Also, lack of normality is not all that big of a problem for obtaining well-behaved coefficient estimates. Often you can say that instead of MLE you just do quasi MLE (QMLE). $\endgroup$
    – Richard Hardy
    Jun 22, 2019 at 17:03
  • $\begingroup$ Let's say that when we decide to inspect v_t terms, we see they are not standard normal as it supposed to be or considered. As you know, when we encounter such a case in ARIMA residuals we decide to employ ARIMA GARCH or GARCH models. But what method can we employ if GARCH or ARIMA GARCH model assumption towards v_t is violated @Richard Hardy? How can we improve our model selection? $\endgroup$
    – mertcan
    Jun 22, 2019 at 21:16

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