# How to calculate volatility using ARCH and GARCH model

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?

• Using daily relative returns would be easier to model. Jun 29, 2021 at 14:35
• @MehmetSuzen can i ask whether this formula: rt=Pt-Pt-1 is correct to calculate the stock return using daily closing price? Jul 1, 2021 at 14:09
• @Jonas_Dim ooo i seee okayyy notedd!!!! Jul 1, 2021 at 14:20

## 1 Answer

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.

• ooo i seee. but is it possible to add the dummy in TGARCH or EGARCH model to identify if the death/cases are good news or bad news ? Can this method solve my second objective below? My first objective is to compare the volatility between 4 selected stock indexes that i have chosen (& to identify which sector is the most and least impacted by the pandemic) & my second objective is to identify the impact of COVID-19 on the 4 selected stock indexes. so i think of using the GARCH, EGARCH and TGARCH. Jul 1, 2021 at 13:57
• Sure, you can add the dummy variables in the conditional variance equations of EGARCH or TGARCH models. This was just an example for a GARCH(1,1) model. Maybe use the GARCH(1,1) as the benchmark model and then look how EGARCH or TGARCH perform.
– Lars
Jul 1, 2021 at 14:07
• aaa i see! i think i have mistaken between ARMA and ARIMA...... "a low order ARMA model is choosen to model the mean dynamics of the time series." like AR(1) right @jonas_Dim Jul 2, 2021 at 3:33
• Yes, or MA(1), ARMA(1,1) etc. Since there is no strong autocorrelation in returns, this is sufficient most of the time.
– Lars
Jul 2, 2021 at 5:46
• I think i just understand yr answer above :') so in order to conduct this thesis, i will first calculate the stock return, then test the stationarity of the mean equation(a low ARMA), then proceed to test the existence of ARCH effect. Next, go to the GARCH model of the daily return to calculate the volatility (This step will let me know the volatility of each index right?). Next, GARCH-X model (insert the dummy variable of death/cases to get the impact result). & if i want to check if it indicate a good/bad news i can use EGARCH/TGARCH. correct me if i'm wrong :)) Jul 2, 2021 at 6:53