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 being serially uncorrelated, the series ${\epsilon }_t$ does not need to be serially independent. For instance, it can present conditional heteroskedasticity. The asymmetric $\mathrm{GARCH}$ ($\mathrm{AGARCH}$) model assumes a specific parametric form for this conditional heteroskedasticity. More specifically, we say that ${\epsilon }_t~\mathrm{AGARCH}$ if we can write ${\epsilon }_t={\sigma }_t{z}_t$, where $z_t$ is a standard Gaussian and:

$${\sigma }_t^2=\omega +\alpha {\left({\epsilon }_{t-1}-\gamma \right)}^2+\beta {\sigma }_{t-1}^2$$

Estimation

V-Lab estimates all the parameters $\left(\omega ,\alpha ,\gamma ,\beta \right)$ 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 $\mathrm{AGARCH}$ model, as 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:

$${\sigma }_2:=\frac{\omega +\alpha {\gamma }^2}{1-\alpha -\beta }$$

Usual restrictions on $\mathrm{AGARCH}$ parameters are that $\omega ,\alpha ,\beta >0$. The $\mathrm{GARCH}$ model is in fact a restricted version of the $\mathrm{AGARCH}$ with $\gamma =0$.

Prediction

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

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

And so, by applying the above formula iteratively, we can forecast conditional volatility for any horizon $h$. Then the forecast of 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 $h$, the forecast of the compound volatility converges to:

$$\sqrt{h}\sqrt{\frac{\hat{\omega }+\hat{\alpha }{\hat{\gamma }}^2}{1-\hat{\alpha }-\hat{\beta }}}$$

AGARCH vs. GARCH

There is a stylized fact that the $\mathrm{AGARCH}$ model captures that is not contemplated by the $\mathrm{GARCH}$ model, which is the empirically observed fact that negative shocks at time $t-1$ have a stronger impact on the variance at time $t$ than positive shocks. This asymmetry is called the leverage effect because the increase in risk was believed to come from the increased leverage induced by a negative shock, but nowadays we know this channel is just too small. Engle and Ng (1993) characterize this asymmetry via the News Impact Curve. In a $\mathrm{GARCH}$ model, this curve is symmetric and centered around ${\epsilon }_{t-1}=0$. In the $\mathrm{AGARCH}$ model, the News Impact Curve is still symmetric, but is centered around ${\epsilon }_{t-1}=\gamma$. The type of asymmetric response discussed above is then associated with positive values of $\gamma$, which we generally find to be statistically significant.

AGARCH(p,q)

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

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

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.

Bollerslev, T., 2008. Glossary to ARCH (GARCH). CREATES Research Paper 2008-49.

Engle, R.F., 1990. Stock Volatility and the Crash of ’87: Discussion. The Review of Financial Studies, Vol. 3, No. 1, pp. 103-106. https://www.jstor.org/stable/2961959

Engle, R.F. and Ng, V.K., 1993. Measuring and Testing the Impact of News on Volatility. Journal of Finance, Vol. 48, No. 5, 1749-1778. https://doi.org/10.1111/j.1540-6261.1993.tb05127.x