V-Lab
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Documentation>Volatility Analysis>GAS-GARCH Student T

Definition

Consider a return time series rt=μ+εt, where μ is the expected return and εt is a zero-mean white noise. Despite being serially uncorrelated, the series εt 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, εt follows a GAS model if it is generated by the observation density.εt~p(εt|ft,t;θ)(1)where ft is the time-varying parameter, t is the available information set at time t, and θ is the static parameter. The mechanism for updating ft follows the familiar autoregressive functional form and uses the score function:ft+1=ω+αst+βftst=St∇t∇t=lnp(εt|ft,t;θ)ftSt=I-1t|t-1It|t-1=𝔼t-1[∇t∇'t]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 ft in the above-mentioned general framework to the conditional variance. More specifically, we say that εt~GAS-GARCH-t if we can write εt=σtzt where zt follows a Student's t distribution with v degrees of freedom, and ft=σt2 is the time-varying variance. Under the distributional assumption of zt, the above-mentioned general form can lead to this updating equation: f t + 1 = ω + α v + 3 v 1 + ε t 2 ( v - 2 ) f t -1 v + 1 v - 2 ε t 2 - f t + β f t ( 2 )

Estimation

V-Lab estimates all the parameters (μ,ω,α,β,v) simultaneously, by maximizing the log likelihood. The distribution assumption of zt being Student's t is discussed in the section comparing the GAS-GARCH model and the GARCH model. The GAS-GARCH-t model, likethe 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 rt be the last observation in the sample, and let ω^,α^,β^ and v^ be the Maximum Likelihood (ML) estimates of parameters ω,α,β and v respectively. The GAS-GARCH-t model implies that the forecast of the conditional variance at time T+h is: σ ^ T + h 2 = ω ^ + α ^ v ^ + 3 v ^ 4 v ^ - 4 v ^ 2 - 3 v ^ + 4 σ ^ T + h - 1 2 + β ^ σ ^ T + h - 1 2 As ε^T+h-12=v^v^-2σ^T+h-12. Thus we can forecast the conditional variance for any horizon h by applying the above formula iteratively. Then, the forecast of the compound volatility at time T+h isσ^T+1:T+h=i=1hσ^T+i2

GAS-GARCH vs. GARCH

If the disturbance zt is a standard Gaussian, the updating equation for GAS-GARCH reduces to: f t + 1 = ω + α ε t 2 - f t + β f t ( 3 ) which is equivalent to the standard GARCH(1,1) model as given byft+1=ω*+α*εt2+β*ft,ft=σt2(4)where coefficients ω*=ω,α*=α, and β*=α-β are unkown 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 εt 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 εt does not necessitate a substantial increase in the variace. 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:ft+1=ω+i=1pαist-i+1+j=1qβjft-j+1In the specific form of GAS-GARCH-t where εt=σtzt,ft=σt2, and zt follows a Student's t distribution with v degrees of freedom, we can plug in the functional form of the scaled score functions s t-i+1 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 Cireterion (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.

References

Creal, D., S. J. Koopman, and A. Lucas. 2013. Generalized Autoregressive Score Models with Applications. Journal of Applied Econometrics 28:777–795. https://doi.org/10.1002/jae.1279