This blog piece intends to introduce a new add-in (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 observation-driven time series models referred to as generalized autoregressive score (GAS) models. The observation-driven 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 well-known models such as GARCH.GAS Model Specification
Let’s denote $y_t$ as the dependent variable of interest, $f_t$ as time-varying 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, time-varying $f_t$ 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 fat-tailed 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 2015-2023 (gasmodelu_example.prg).
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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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