Comparing volatility among time series-GARCH Models

Question

I have 10 time series on which I want to compare their volatility by using ARMA-GARCH models. I have estimated the ARMA-GARCH models by using eviews.

1. According to my supervisor now, I should compute the conditional coefficient of variation by dividing the conditional standard deviation to the conditional mean. As conditional mean takes both positive and negative values, I was wondering if I should take their absolute value. Negative coefficient of variation makes no sense, and then perform a $t$-test for means equality.

2. My thought is, why not to take the time series of the conditional standard deviation, make a descriptive statistics (in order to find the average st. dev.) and then perform a $t$-test of equality in means?

Answer 1

1.

To second your concern, Wikipedia's article on coefficient of variation ($c_v$) suggests that [t]he coefficient of variation should be computed only for data measured on a ratio scale rather than interval scale, as it is meaningless in the latter case. So you probably should not use it.

Now, how would you define the conditional coefficient of variation? Would you have one value per time point, like $c_{v,t}:=\frac{\sigma_t}{\mu_t}$? If you then try to remedy the problem of negative values by taking the absolute value, you would end up with a new measure $c^*_{v,t}:=\frac{\sigma_t}{|\mu_t|}$. It looks alright to me, just think whether it is exactly what you need (how exactly you define volatility).

2.

Your second approach seems better suited for the task. You can estimate the unconditional (or long-run) variances of the series and compare them. Here are two threads that discuss the topic: "Estimating unconditional variance in time series" and "What is the long run variance?". The question is, how do we get the distribution of the estimator for the unconditional variance? We need it for deriving the distribution of the test statistics for testing the equality of the two variances.