# Interprete GARCH residuals QQ plot

How to conclude, that time series' volatilty is not constant? I used GARCH, but have trouble with interpretation.

I did the Kolmogorov-Smirnov test for normal distribution with GARCH residuals, which tells me that they are not normally distributed, but again I have trouble with interpretation of what that means for the time series. I m not very skilled in statistics.

• Is your time series a zero-mean one?
– Emil
Sep 7 '18 at 16:07
• The title and the body of the question do not match. Could you change the title to reflect the body? Sep 23 '18 at 17:23

You are conflating a possible characteristic (volatility clustering) of the time series with checking how true or not one of the assumptions (normal residuals) of the model (GARCH) used for the series is.

Let's start with conditional heteroscedasticity. Assuming you have a zero-mean series $y_t$, then below are two possible ways of checking whether conditional heteroscedasticity is present:

1. The first way, who is also the more subjective of the two, is to simply plot the series over time and check whether you see any "clusters" where the series shows spikes that stand out a lot when compared to the mean (which you expect to be zero). The series then could look something like this image, taken from here. I've highlighted in red what I would consider as volatility spikes:
2. Of course, one might say that those circles are not the only ones that seem to stand out, and another might say (as an extreme example) that they don't stand out at all. Therefore, it's always a good idea to actually perform a test and proceed according to its results. In this case, the test I am referring to is the standard portmanteau, Ljung-Box $Q(m)$ test on the squared series $y_t^2$, whose null hypothesis is that the first $m$ lags of the autocorrelation function of $y_t^2$ are zero. A rejection of this would be strong indication that volatility of the series is indeed not constant over time.

After concluding that your (again, zero-mean) series $y_t$ has indeed a time-varying volatility, you decide to model it using, let's say, a typical GARCH(1,1) specification, so this becomes

$$\begin{cases} y_t = \sigma_t \varepsilon_t, \\ \sigma_t^2 = \alpha_0 + \alpha_1 y_{t-1}^2 + \beta\sigma_{t-1}^2 \end{cases}$$

where you assume that $\varepsilon_t \sim\mathcal{N}(0,1)$ and of course iid. After you estimate this model, you get the parameter estimates $\hat{\alpha}_0, \hat{\alpha}_1,$ and $\hat{\beta}$, but also your estimated conditional variances $\hat{\sigma}_t^2$ and standardized residuals $\hat{\varepsilon}_t$ (depending on the software you use, you might get them automatically as part of the output or have to compute them yourself), which according to the formula used above, they are equal to $\hat{\varepsilon}_t = y_t/\hat{\sigma}_t$.

Now this series of $\hat{\varepsilon}_t$ is very useful. By running the Kolmogorov-Smirnov test on $\hat{\varepsilon}_t$ (or by checking the QQ plot), you can check to see whether your prespecified distribution assumption (in this case $\mathcal{N}(0,1)$) was correct, whereas by performing the same Ljung-Box test mentioned above on $\hat{\varepsilon}_t^2$ you can check the validity of the volatility equation for $\sigma_t^2$ you specified.

You should read Chapter 3 of Analysis of Financial Time Series by R. S. Tsay, it's a great book that explains these concepts in a very pedagogical way and includes multiple practical demonstrations as well.

• Thanks. It is zero mean series and your answer is very helpful. Sep 8 '18 at 10:23
• A nice answer, except that once a GARCH model is fit, neither the Ljung-Box test for squared standardized residuals $\hat\varepsilon_t$ nor the Engle LM test will be approapriate. Li-Mak test should be used there. The former two tests are only appropriate for raw data or residuals from a conditional mean model that have not been modelled as a GARCH process. Sep 23 '18 at 17:25