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 need to be serially independent. For example, Ding, Granger, and Engle (1993) find that ${\left|{\epsilon }_t\right|}^d$ often displays strong and persistent autocorrelation for various values of $d$, or rather returns have a long memory property. One way to explain this empirical pattern is through conditional heteroskedasticity. The asymmetric Power $\mathrm{ARCH}$ ($\mathrm{APARCH}$) model assumes a specific parametric form for powers of this conditional heteroskedasticity. More specifically, we say that ${\epsilon }_t\sim \mathrm{APARCH}$ if we can write ${\epsilon }_t={\sigma }_t{z}_t$, where $z_t$ is a standard Gaussian and:

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

Estimation

V-Lab estimates all the parameters $\left(\delta ,\omega ,\alpha ,\gamma ,\beta \right)$ 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 $\mathrm{APARCH}$ 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 $t$. As mentioned, the $\mathrm{APARCH}$ model also delivers the long-memory property of returns discussed in Ding, Granger, and Engle (1993). The $\mathrm{APARCH}$ model, as the $\mathrm{GJR-GARCH}$ model, additionally captures asymmetry in return volatility. That is, volatility tends to increase more when returns are negative, as compared to positive returns of the same magnitude.

Usual restrictions on $\mathrm{APARCH}$ parameters are that $\delta ,\omega ,\alpha ,\beta >0$ and $\mathrm{-1}<\gamma <1$. Many of the volatility models used in V-Lab are in factrestricted versions of the $\mathrm{APARCH}$, which is one of the reasons these models are popular.

Prediction

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

$${\hat{\sigma }}_{T+h}^{\hat{\delta }}=\hat{\omega }+{\hat{\sigma }}_{T+h-1}^{\hat{\delta }}\left\{\hat{\alpha }{𝔼}_T\left[{\left(\left|z_{T+h-1}\right|-\hat{\gamma }{z}_{T+h-1}\right)}^{\hat{\delta }}\right]+\hat{\beta }\right\}$$

where under our assumptions about $z_t$,

$${𝔼}_T\left[{\left(\left|z_{T+h-1}\right|-\hat{\gamma }{z}_{T+h-1}\right)}^{\hat{\delta }}\right]=\frac{1}{\sqrt{2\pi }}\left[{\left(1+\hat{\gamma }\right)}^{\hat{\delta }}+{\left(1-\hat{\gamma }\right)}^{\hat{\delta }}\right]2^{\frac{\hat{\delta }-1}{2}}\Gamma \left(\frac{\hat{\delta }+1}{2}\right)$$

And so, by applying the above formula iteratively, we can forecast conditional volatility (to the power $\hat{\delta }$) for any horizon $h$.

APARCH vs. GARCH

The power of the $\mathrm{APARCH}$ model comes from the fact that it nests many of the other volatility models used by V-Lab. For example, we can obtain the following volatility models as restrictions of parameters of the APARCH model:

• $\mathrm{ARCH}\left(1\right)$ model - set $\delta =2,\gamma =0,\beta =0$
• $\mathrm{GARCH}\left(1,1\right)$ model - set $\delta =2,\gamma =0$
• $\mathrm{GJR-GARCH}\left(1,1\right)$ model - set $\delta =2$

This feature of $\mathrm{APARCH}$ models is especially useful when conducting hypothesis tests concerning the specification of volatility.

APARCH(p,q)

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

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

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.

Ding, Z., Granger C.W., Engle R.F., 1991. A Long Memory Property of Stock Market Returns and a New Model. Journal of Empirical Finance 1 (1993) 83-106. https://doi.org/10.1016/S0304-4076(97)00072-9