In this paper: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2ahUKEwjSlIHYnMj1AhWqNOwKHZfHDhkQFnoECAkQAQ&url=https%3A%2F%2Fwww.mdpi.com%2F2076-3387%2F9%2F2%2F40%2Fpdf&usg=AOvVaw12ONJTGtEwL7xIYfPqbhBh they use the following equation for the Conditional Value at Risk using GARCH: $$ ES_q(X_{t})=\mu_{t}+\sigma_{t}ES_q(Z) $$ where $\mu$ is derived from the average prediction models and $\sigma$ is estimated from the volatility prediction models. $ES_q^t(Z)$ is the Conditional Value at Risk of the standardized residuals. I don't understand how to calculate that. Suppose that:
- $r_t$ are the logarithmic returns $r_t = \ln S_{t+1} - \ln S_t$;
- $\varepsilon_t = r_t - \bar{r}$ are the residuals, where $\bar{r}$ is the average return;
- $\sigma_0^2 = \frac{\omega}{1-\alpha -\beta}$ and $\sigma_t^2 = \omega+\alpha \varepsilon_{t-1}^2+\beta \sigma_{t-1}^2$ is the conditional variance at time zero and at time $t$ respectively;
- $\sqrt{\varepsilon_t^2}$ is the realized volatility and $\sqrt{\sigma_t^2}$ is the "fitted" volatility;
Then I don't understand if:
- $ES_q(Z) = ES_q(\sqrt{\varepsilon^2}-\sqrt{\sigma^2})$ or $ES_q(Z) = ES_q(\varepsilon)$ or $ES_q(Z) = ES_q(r)$?
- $ES_q(Z)$ is calculated for the entire time series or only for a sliding window?
- $\mu_t=\bar{r}$ or $\mu_t=$ average loss of the time series?
How can I calculate Conditional Value at Risk using a GARCH model given my conditions?