# ARIMA requires constant variance, so why can we use GARCH for its residuals?

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?

• Why can not get any responses? Jun 17, 2019 at 13:28
• 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. Jun 18, 2019 at 14:31
• Scratch that: I have given it a try. Jun 18, 2019 at 20:02

• 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? Jun 22, 2019 at 10:02
• @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). Jun 22, 2019 at 17:03