Definition

Consider a return time series$r_t=\mu +{\epsilon }_t$, where$\mu$ is the expected return and ${\epsilon }_t$ is a zero-mean white noise. Despite of being serially uncorrelated, the series ${\epsilon }_t$ does not need to be serially independent. For instance, it can present conditional heteroskedasticity. The Generalized Autoregressive Conditional Heteroskedasticity ($\mathrm{GARCH}$) model assumes a specific parametric form for this conditional heteroskedasticity. More specifically, we say that ${\epsilon }_t~\mathrm{GARCH}$ if we can write ${\epsilon }_t={\sigma }_t{z}_t$, where $z_t$ is standard Gaussian and:

$${\sigma }_t^2=\omega +\alpha {\epsilon }_{t-1}^2+\beta {\sigma }_{t-1}^2$$

Estimation

V-Lab estimates all the parameters $\left(\mu ,\omega ,\alpha ,\beta \right)$ simultaneously, by maximizing the log likelihood. The assumption that$z_t$ is Gaussian does not imply the the returns are Gaussian. Even though their conditional distribution is Gaussian, it can be proved that their unconditional distribution presents excess kurtosis (fat tails). In fact, assuming that the conditional distribution is Gaussian is not as restrictive as it seems: even if the true distribution is different, the so- called Quasi-Maximum Likelihood (QML) estimator is still consistent, under fairly mild regularity conditions.

Besides leptokurtic returns, the $\mathrm{GARCH}$ model captures other stylized facts in financial time series, like volatility clustering. The volatility is more likely to be high at time $t$ if it was also high at time $t-1$. Another way of seeing this is noting that a shock at time$t-1$ also impacts the variance at time$t$. However, if$\alpha +\beta <1$, the volatility itself is mean reverting, and it fluctuates around$\sigma$, the square root of the unconditional variance:

$${\sigma }^2≔\text{Var}\left(r_t\right)=\frac{\omega }{1-\alpha -\beta }$$

Usual restrictions on the parameters are $\omega ,\alpha ,\beta >0$ though it is possible to have$\omega =0$ and$\alpha +\beta =1$. The conditional variance is then an integrated process (shocks to the variance are persistent), hence the model is called $\mathrm{IGARCH}$ ( Integrated $\mathrm{GARCH}$). This is the model RiskMetrics uses to compute Value-at-Risk ($\mathrm{VaR}$).

Prediction

Let $r_T$ be the last observation in the sample, and let $\hat{\omega }$,$\hat{\alpha }$ and$\hat{\beta }$ be the QML estimators of the parameters $\omega$, $\alpha$ and$\beta$, respectively. The $\mathrm{GARCH}$ model implies that the forecast of the conditional variance at time$T+h$ is:

$${\hat{\sigma }}_{T+h}^2=\hat{\omega }+\left(\hat{\alpha }+\hat{\beta }\right){\hat{\sigma }}_{T+h-1}^2$$

and so, by applying the above formula iteratively, we can forecast the conditional variance for any horizon $h$. Then, the forecast of the compound volatility at time $T+h$is

$${\hat{\sigma }}_{T+1:T+h}=\sqrt{\sum _{i=1}^h{\hat{\sigma }}_{T+i}^2}$$

Notice that, for large $h$, this forecast of the compound volatility converges to:

$$\sqrt{h}\sqrt{\frac{\hat{\omega }}{1-\hat{\alpha }-\hat{\beta }}}$$

scaling over the forecast horizon with the well known square-root law, times the estimate of the unconditional volatility implied by the$\mathrm{GARCH}$ model.

GARCH(p,q)

The specific model just described can be generalized to account for more lags in the conditional variance. A$\mathrm{GARCH}\left(p,q\right)$ model assumes that:

$${\sigma }_t^2=\omega +\sum _{i=1}^q{\alpha }_i{\epsilon }_{t-i}^2+\sum _{j=1}^p{\beta }_j{\sigma }_{t-j}^2$$

The best model ($p$ and $q$) can be chosen, for instance, by Bayesian Information Criterion (BIC), also known as Schwarz Information Criterion (SIC), or by Akaike Information Criterion (AIC). The former tends to be more parsimonious than the latter. V-Lab uses$p=1$ and$q=1$ though, because this is usually the option that best fits financial time series.

Engle, R. F., 1982. Autoregressive Conditional Heteroskedasticity with Estimates of The Variance of The United Kingdom Inflation. Econometrica 50: 987- 1007. https://www.jstor.org/stable/1912773

Bollerslev, T., 1986. Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics 31: 5- 59. https://www.sciencedirect.com/science/article/pii/0304407686900631