This blog piece intends to introduce a new addin (i.e. SKEWEDUGARCH) that extends the current capability of EViews’ available features for the estimation of univariate GARCH models.
Table of Contents
Introduction
Volatility is an important concept in itself, but it has a special place in finance as it is usually associated with risk. Although investors believe in higher risk higher reward, it is not an easy task to exploit this tradeoff. Price of an asset can change dramatically over a short period of time and in either direction, which makes it exceedingly difficult to predict. Volatility is responsible from such sharp movements, so it is important to develop a gauge to measure and identify its dynamics.One of the critical observations regarding the returns of financial assets was that the volatilities were not fixed over time and tended to cluster around large changes. GARCH models are specifically designed to capture this behavior and describe the movement of volatility more accurately. Details of GARCH estimation in EViews can be found here.
Conditional distribution of error terms of returns (i.e. mean equation) plays an important role in the estimation of GARCHtype models. Currently, EViews offers three different assumptions regarding the specification of this distribution.
Skewed Student’st Distribution
Consistent with the stylized facts of financial markets, distribution of returns has fat tails (i.e. high kurtosis) and are not symmetrical (i.e. positively skewed). Although Student’st and GED specifications can account for the excess kurtosis, they are symmetrical densities by design. Lambert and Laurent (2001) suggest the use of a skewed Student’st density within the GARCH framework. The log likelihood contributions of a standardized skewed Student’st are as follows:\begin{align*} l_t &= \frac{1}{2} \log \rbrace{ \frac{\pi(\nu  2) \Gamma \rbrace{\frac{\nu}{2}}^2}{\Gamma \rbrace{\frac{\nu + 1}{2}} } } + \log \rbrace{\frac{2}{\xi + \frac{1}{\xi}}} + \log(s)\\ &\frac{1}{2}\log(\sigma^2_t)  \frac{\nu + 1}{2} \log \rbrace{1 + \frac{\rbrace{s\rbrace{y_t  X_t^\top \theta} + m}^2}{\sigma_t^2\rbrace{\nu  2}}\xi^{2I_t}} \end{align*} Here, $\xi$ is the asymmetry parameter and $\nu$ is the degreesoffreedom of the distribution. Other parameters, $m,s$ and $I_t$ are given by: \begin{align*} m &= \frac{\Gamma \rbrace{\frac{\nu  1}{2}} \sqrt{\nu  2}}{\sqrt{\pi}\Gamma\rbrace{\frac{\nu}{2}}}\rbrace{\xi  \frac{1}{\xi}}\\ s &= \sqrt{\rbrace{\xi^2 + \frac{1}{\xi^2}  1}  m^2}\\ I_t &= \begin{cases} \phantom{}1 \quad \text{if} \quad \rbrace{\frac{y_t  X_t^\top \theta}{\sigma_t}} \geq  \frac{m}{s}\\ 1 \quad \phantom{\text{if}}\text{otherwise} \end{cases} \end{align*} For a symmetrical distribution, $ξ=1$, but since the addin estimates the logarithmic transformation of the parameter, you should consider $\log(\xi)=0$ for testing the null hypothesis of symmetry.
Below is the comparison of theoretical distribution of Student’st and its (positively) skewed version. Skewness increases the chance of observing extreme values, which has important implications in finance.


Application to USDTRY currency
FX markets are convenient places for studying the dynamics of volatility and Turkish Lira has recently come to the fore among emerging markets due to sudden capital outflows as well as currency shocks (USDTRY.WF1).A simple visual inspection of squared returns shows us the magnitude of the shock that hit the markets on August 10th, 2018 (SKEWEDUGARCH_EXAMPLE.PRG). The impact was so severe that it dwarfed all other volatilities experienced during the analysis period of 20052020.














At this point, one may also wonder if there is any long memory effect in the volatility of returns. In order to do so, we first estimate an ARFIMA model for the squared return series and a simple FIGARCH model for the variance part of regular return series: \begin{align*} &\textbf{Fractional Mean Model}: \quad \rbrace{1  L}^d(r_t^2  \mu) = e_t, \quad \text{where} \quad e_t \sim N(0,\bar{\sigma})\\ &\textbf{Fractional Variance Model}: \quad \sigma_t^2 = \omega + \rbrace{1  \beta_1  \rbrace{1  \alpha_1}\rbrace{1  L}^d}e_{t1}^2 + \beta_1\sigma_{t1}^2, \quad \text{where} \quad \epsilon_t \sim \text{Student}(0,1,\nu) \end{align*}




Since the estimation of fractional difference parameter can be sensitive to the choice of truncation limits, it may not worth the effort unless the statistical properties of results from FIGARCH models are significantly better than that of rival GARCH models. Here, our previous TGARCH(1,1) model with Student’st errors is still the frontrunner in that respect.
What if the positive shocks (i.e. depreciation) happen less frequently but more severe than negative shocks (i.e. appreciation) implied by a symmetric distribution? In order to test this hypothesis, one needs to look for asymmetry towards larger positive extreme values. We can estimate our final model via addin assuming a skewed Student’st distribution and see if we can further improve the fit.


One of the main uses of GARCH models in financial institutions is the estimation of ValueatRisk (VaR), a concept that tracks and calculates the potential loss that might happen during a trading activity of any sort. Commonly used symmetric error distributions for the purpose might lead to underestimation of right tail risk (i.e. in short trading positions). The chart below compares the daily VaR estimations from commonly used distributions and depicts effects of fat tails and skewness for a long position in TL (or a short position in USDTRY).


Files
References
 Lambert P and Laurent S (2001), "Modelling Financial Time Series Using GARCHType Models and a Skewed Student Density", Mimeo, Universite de Liege.
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