## Friday, February 9, 2024

### Generalized Autoregressive Score (GAS) approach to univariate GARCH Models

Authors and guest post by Eren Ocakverdi

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.

### 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).
 Figure 1: Squared returns of USDTRY

First, we start by estimating three GARCH models with different distributional assumptions on innovations and then, using the add-in, GAS counterparts are estimated for comparison purposes. Calling the add-in from the equation object will prompt the GUI (see Figure 2).

Apart from slight differences due to optimization setting, results are nearly identical for Gaussian assumption (see Figure 3). This is expected since the coefficients of both GARCH and GAS equations are essentially the same from the mathematical point of view. Note that, Beta coefficient of GAS equation is (and should be) equal to the sum of autoregressive coefficients of GARCH variance equation.
 Figure 3: Comparison of estimation results for Gaussian distribution (GARCH vs GAS)

Estimated conditional variance and score values are indistinguishable from one another in the case of Gaussian distribution (see Figure 4).
 Figure 4: Comparison of estimated conditional score/variance for Gaussian distribution (GARCH vs GAS)

Estimation results for other distributions (i.e. fat-tailed) are not the same nor directly comparable in that sense (see Figures 5 and 6).
 Figure 5: Comparison of estimation results for Students’-t distribution (GARCH vs GAS)

 Figure 6: Comparison of estimation results for Skewed Student’s-t distribution (GARCH vs GAS)

Note here that for Skewed GARCH results, you'll need to use the add-in SKEWEDUGARCH to obtain results

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.
 Figure 7: Comparison of estimated conditional score/variance for Student’s-t distribution (GARCH vs GAS)

 Figure 8: Comparison of estimated conditional score/variance for Skewed Student’s-t distribution (GARCH vs GAS)

GAS models mitigate the impact of the occurrences of large changes in returns as they are usually one-off events and have relatively milder effects on future volatility vis-à-vis smaller shocks. This is mainly due to the resilience and adaptability of markets to new information.

### References

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