How to calculate volatility using ARCH and GARCH model

Question

I'm conducting a thesis about the impact of COVID-19 to stock market.. I use the ARCH and GARCh model to compare the volatility of the stock market during the pandemic. I also use the dummy variable of daily cases and death cases of COVID-19 to check the effect on the stock market. But I'm not sure, how to conduct it because i actually don't really understand this topic time series analysis :( and did the data should be the daily price or the stock return?

Answer 1

With (G)ARCH models you do not model prices but returns. More precisely, you model the volatility of asset returns. Volatility in this context is the conditional variance of the returns given the returns from yesterday, the day before yesterday and so on. Let ${\cal F}_{t-1}=\left\{r_{t-1},r_{t-2},\dots\right\}$ be the information set at trading day $t$, then you try to model $Var(r_t\vert {\cal F}_{t-1})$. (G)ARCH models do so, by assuming that the daily returns can be modeled as: \begin{align} r_t&=\mu_t+\epsilon_t \\ \epsilon_t&=\sigma_tz_t,\quad z_t\overset{iid}{\sim}(0,1) \end{align} Where $\mu_t=E(r_t\vert {\cal F}_{t-1})$ is the conditional expectation of the returns. Usually, a low order ARMA model is choosen to model the mean dynamics of the time series. It is easy to show that $Var(r_t\vert {\cal F}_{t-1})=\sigma_t^2$ and the main question is what functional form does $\sigma_t^2$ have? Since you want to use the data for daily COVID cases and deaths, a GARCH-X model would be a good choice. For example: $$ \sigma_t^2=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1\sigma_{t-1}^2+\gamma_1I_1\epsilon_{t-1}^2+\gamma_2I_2\epsilon_{t-1}^2 $$ Where $I_1$ and $I_2$ are indicator variables with value 1 if deaths/cases are greater than a choosen treshold. The parameters $\gamma_1$ and $\gamma_2$ are the main parameters of interest then, because they show you what the effect on volatility is if the deaths/cases exceed the threshold. Since the choose of such a treshold is very subjective, I think a smooth-transition model would be a better choice. However, since you are unfamiliar with the use of GARCH models and ST-GARCH models are kind of exotic, I recommend to take the GARCH-X model.