This blog piece intends to introduce a new addin (i.e. GASMODELU) that estimates selected univariate GARCH models within the Generalized Autoregressive Score (GAS) framework.
Table of Contents
Introduction
Creal et. al. (2013) proposed a class of observationdriven time series models referred to as generalized autoregressive score (GAS) models. The observationdriven approach allows the use of lagged dependent variables or contemporaneous and lagged exogenous variables when modeling the time variation in the parameters. Since the updating mechanism of the parameters over time is based on the scaled score of the likelihood function, GAS approach encompasses other wellknown models such as GARCH.GAS Model Specification
Let’s denote $y_t$ as the dependent variable of interest, $f_t$ as timevarying parameter vector and $\theta$ as the vector of static parameters. Then the GAS framework can be outlined as follows:\begin{align*} y_t &\sim p\left(y_t  Y_{t  1}, f_t ; \theta\right), \quad Y_t = \{y_1, y_2, \ldots, y_t\}\\ f_t &= \omega + A s_t + B f_t\\ s_t &= S_t \overline{V}_t\\ S_t &= \mathcal{J}_{t  t  1}^{1} = E_{t  1} \left[ \frac{\partial^2 ln\left( p\left(y_t  Y_{t  1}, f_t ; \theta \right)\right)}{\partial f_t f_t^{'}} \right]^{1}\\ \overline{V}_t &= \frac{\partial ln\left( p\left(y_t  Y_{t  1}, f_t ; \theta \right)\right)}{\partial f_t} \end{align*} The approach is based on the observation density, $p\left(y_t  Y_{t  1}, f_t ; \theta \right)$ for a given parameter, $f_t$. When an observation $y_t$ is realized, timevarying $f_t$ to the next period is updated accordingly. Scaling matrix, $S_t$ is usually defined as a function of the variance of score. Doing so yields the GARCH model as a special case: \begin{align*} y_t &= \sigma_t \epsilon_t, \quad \epsilon_t \sim NID\left(0, 1\right)\\ f_{t+1} &= \omega + A s_t + B f_t, \quad \text{where} \quad f_t = \sigma^2_t\\ ln\left( p\left(y_t  Y_{t  1}, f_t ; \theta \right)\right) &= 0.5 \left(ln(2\pi) + ln(f_t) + \frac{y_t^2}{f_t} \right)\\ \overline{V}_t &= \frac{1}{2f_t^2}\left(y_t^2  f_t \right)\\ S_t &=  E_{t  1} \left[\frac{1}{2f_t^2}  \frac{y_t^2}{f_t^3} \right]^{1}\\ &=  \left[ \frac{1}{2f_t^2}  \frac{f_t}{f_t^3} \right]\\ &= 2f_t^2\\ \sigma_{t+1}^2 &= \omega + \alpha y_t^2 + \beta \sigma_t^2, \quad \text{where} \quad \alpha = A \quad \text{and} \quad \beta = B  A \end{align*} Although the final result collapses to a standard GARCH(1,1) model for Gaussian distribution, this is not the case for fattailed distributions. For instance, the GAS updating mechanism for the model with Student’s$t$ errors is very different from its GARCH counterpart: \begin{align*} \sigma_{t+1}^2 = \omega + \alpha (1 + 3 v^{1}) \left( \frac{\left(1 + v^{1}\right)}{\left(1  2 v^{1}\right)\left(1 + v^{1} \frac{y_t^2}{1  2 v^{1}} \right) \sigma_t^2} \right) \end{align*} Please note that, if $y_t$ takes up a very large positive/negative value, it is pulled down accordingly and therefore increase in the variance remains smaller to that of a regular GARCH with Student’s$t$ errors. This result is also consistent with the stylized facts of financial markets, where the observed impact of a sudden jump on future volatility is smaller than the standard models predict.
Application to USDTRY currency
Turkish FX market provides a useful laboratory for exploring the impact of currency shocks on volatility dynamics (usdtry.wf1). Figure 1 below shows the squared values of daily returns, where at least three severe shocks can be visually identified during the sample period of 20152023 (gasmodelu_example.prg).















Figures 7 and 8 depict the comparison of estimated conditional variances and scores for Student’s$t$ and Skewed version of Student’s$t$ distributions.




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References
 Creal, D., Koopman, S. J., and Lucas, A. (2013). Generalized Autoregressive Score Models with Applications, Journal of Applied Econoemtrics, 28:281–314.
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