GAS-GARCH Student T

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. This characteristic is usually captured by the time variation in a selection of model parameters. For instance, the family of GARCH models updates the conditional variance, which can be viewed as a model parameter, using lagged endogenous variables as well as contemporaneous and lagged exogenous variables. Extending the same logic, the family of Generalized Autoregressive Score (GAS) models proposed by Creal, Koopman, and Lucas (2013) updates the time-varying parameter using the scaled score function of the likelihood function. Formally, ${\epsilon }_t$ follows a GAS model if it is generated by the observation density.$${\epsilon }_t~p\text{(}{\epsilon }_t\text{|}{f}_t,{ℱ}_t\text{;}\theta \text{)}\text{(}1\text{)}$$where $f_t$ is the time-varying parameter, ${ℱ}_t$ is the available information set at time $t$, and $\theta$ is the static parameter. The mechanism for updating $f_t$ follows the familiar autoregressive functional form and uses the score function:$$f_{t+1}=\omega +\alpha s_t+\beta f_t$$$$s_t=S_t\cdot {\mathrm{\∇}}_t$$$${\mathrm{\∇}}_t=\frac{\partial \text{ln}p\text{(}{\epsilon }_t\text{|}{f}_t,{ℱ}_t\text{;}\theta \text{)}}{\partial f_t}$$$$S_t={I^{-1}}_{t\text{|}t-1}$$$$I_{t\text{|}t-1}={𝔼}_{t-1}\text{[}{\mathrm{\∇}}_t{{\mathrm{\∇}}^{\text{'}}}_t\text{]}$$In V-Lab, we consider a variant of the GAS model that resembles the formation of the GARCH model (hence the name GAS-GARCH) by relating the time-varying parameter $f_t$ in the above-mentioned general framework to the conditional variance. More specifically, we say that ${\epsilon }_t~\text{GAS-GARCH-}t$ if we can write ${\epsilon }_t={\sigma }_t{z}_t$ where $z_t$ follows a Student's $t$ distribution with $v$ degrees of freedom, scaled to have unit variance, and $f_t={\sigma }_t^2$ is the time-varying variance. Under the distributional assumption of $z_t$, the above-mentioned general form can lead to this updating equation: $$f_{t+1}=\omega (1-\beta )+\alpha \left(\frac{v+3}{v}\right)\left[{\left(1+\frac{{{\epsilon }_t}^2}{(v-2)f_t}\right)}^{-1}\left(\frac{v+1}{v-2}\right){{\epsilon }_t}^2-f_t\right]+\beta f_t\text{(}2\text{)}$$V-Lab writes the intercept as $\omega (1-\beta )$, so that $\omega$ is the long-run level of the variance, the value reported as ω in the parameter estimates.

Estimation

V-Lab estimates all the parameters $\mathrm{(\omega ,\alpha ,\beta ,v)}$ simultaneously, by maximizing the log likelihood. V-Lab sets $\mu =0$, so the shocks are the returns themselves, with no mean subtracted. The distribution assumption of $z_t$ being Student's $t$ is discussed in the section comparing the GAS-GARCH model and the GARCH model. The GAS-GARCH-$t$ model, like the GARCH model, captures one important stylized fact in financial time series, 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 $t$.

Prediction

Let $r_T$ be the last observation in the sample, and let $\hat{\omega },\hat{\alpha },\hat{\beta }$ and $\hat{v}$ be the Maximum Likelihood (ML) estimates of parameters $\omega ,\alpha ,\beta$ and $v$ respectively. The one-step forecast ${\hat{\sigma }}_{T+1}^2$ is update (2) evaluated at $t=T$ with these estimates (see the section on zero returns for the case of a zero return). The scaled score has an expected value of zero under the model, so the forecast of the conditional variance at time $T+h$ for $h\ge 2$ is:$${\hat{\sigma }}_{T+h}^2=\hat{\omega }(1-\hat{\beta })+\hat{\beta }{\hat{\sigma }}_{T+h-1}^2$$The forecast therefore reverts toward $\hat{\omega }$, and the gap shrinks by the factor $\hat{\beta }$ each day. Then, the forecast of the compound volatility at time $T+h$ is$${\hat{\sigma }}_{T+1\text{:}T+h}=\sqrt{\sum _{i=1}^h{\hat{\sigma }}_{T+i}^2}$$

Zero returns

Thinly traded assets often have days with a return of exactly zero, because the price did not trade or did not change. A Student's $t$ density fitted to those zeros can raise its likelihood without limit by making its peak at zero taller and narrower. It does that by pushing $v$ toward its lower bound of 2, where the variance of the distribution explodes. Following Sucarrat and Grønneberg (2022), V-Lab treats an exact-zero return as a separate event. The return $r_t$ is zero with probability $\pi$, and otherwise it moves as described above. The variance $f_t$ is then the variance of the return given that the price moves.

Update (2) is applied only after a nonzero return. A zero return carries no shock to learn from, so when $r_t=0$, the variance reverts toward its long-run level $\omega$:$$f_{t+1}=\omega (1-\beta )+\beta f_t$$A run of more than 21 consecutive zero returns, such as a trading suspension, means the price is not being quoted. From the 22nd consecutive zero on, the variance holds at its last value until the price moves again:$$f_{t+1}=f_t$$

The log likelihood sums only over the days with a nonzero return, because $\pi$ factors out of it. The robust standard errors leave the zero-return days out as well.

The volatility V-Lab publishes, in the fitted series and in the forecasts, is the volatility of the return itself, which allows for the chance that it is zero. The annualized volatility published on day $t$, its forecast for day $t+1$, is$$\sqrt{252(1-{\pi }_t)f_{t+1}}$$where ${\pi }_t$ is the share of zero returns among the last 250 returns through day $t$. Zeros in a run of more than 21 consecutive zeros, counted through day $t$, are left out of ${\pi }_t$, because they show that the price was not quoted, not how often it moves. If every return in the window belongs to such a run, ${\pi }_t$ keeps its value from the day before. Until a series has 250 returns, the missing days count as nonzero returns. On a series without zero returns, ${\pi }_t=0$ and nothing changes.

Forecasts are scaled the same way. The forecast of the variance of the return at time $T+h$ is $(1-{\pi }_T){\hat{\sigma }}_{T+h}^2$, and the compound volatility is computed from these scaled forecasts. The one-step forecast follows the rules above, so it leaves out the score after a zero return. If the sample ends in a run of more than 21 zero returns, the forecast stays at $(1-{\pi }_T)f_T$ for every horizon.

GAS-GARCH vs. GARCH

If the disturbance $z_t$ is a standard Gaussian, the updating equation for GAS-GARCH reduces to: $$f_{t+1}=\omega (1-\beta )+\alpha \left({{\epsilon }_t}^2-f_t\right)+\beta f_t\text{(}3\text{)}$$which is equivalent to the standard GARCH(1,1) model as given by$$f_{t+1}={\omega }^{*}+{\alpha }^{*}{{\epsilon }_t}^2+{\beta }^{*}{f}_t,f_t={{\sigma }_t}^2\text{(}4\text{)}$$where coefficients ${\omega }^{*}=\omega (1-\beta )\text{,}{\alpha }^{*}=\alpha \text{, and}{\beta }^{*}=\beta -\alpha$ are unknown and require certain conditions for stationarity (See GARCH documentation for these requirements). In case $v^{-1}=0$, the Student's $t$ distribution reduces to the Gaussian distribution and update (2) collapses to (3) as required. The recursion in (2), however, has an important difference from the standard GARCH-$t$ model, which has the Student's $t$ density in (1) with the updating equation (4). The denominator of the second term on the right-hand side of (2) causes a smaller increase in the variance for a large realization of $\left|{\epsilon }_t\right|$ as long as $v$ is finite. The intuition is clear: if the errors are modeled by a fat-tailed distribution, a large absolute realization of ${\epsilon }_t$ does not necessitate a substantial increase in the variance. The GAS-GARCH updating mechanism for the model with Student's $t$ errors therefore is substantially different from its familiar GARCH counterpart.

GAS-GARCH(p,q)-t

The specific model just described can be generalized to account for more lags in the update mechanism of the time-varying parameter. A GAS(p,q) model assumes that:$$f_{t+1}=\omega +\sum _{i=1}^p{\alpha }_i{s}_{t-i+1}+\sum _{j=1}^q{\beta }_j{f}_{t-j+1}$$In the specific form of GAS-GARCH-$t$ where ${\epsilon }_t={\sigma }_t{z}_t\text{,}{f}_t={{\sigma }_t}^2$, and $z_t$ follows a Student's $v$ distribution with $t$ degrees of freedom and is scaled to have unit variance, we can plug in the functional form of the scaled score functions $\left\{s_{t-i+1}\right\}$ and arrive at an updating equation which is similar to (2) yet more complicated. 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.