Tuesday, February 2, 2021

Univariate GARCH Models with Skewed Student’s-t Errors

Authors and guest post by Eren Ocakverdi

This blog piece intends to introduce a new add-in (i.e. SKEWEDUGARCH) that extends the current capability of EViews’ available features for the estimation of univariate GARCH models.

Table of Contents

  1. Introduction
  2. Skewed Student’s-t Distribution
  3. Application to USDTRY currency
  4. Files
  5. References

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 trade-off. 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 GARCH-type models. Currently, EViews offers three different assumptions regarding the specification of this distribution.



Skewed Student’s-t 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’s-t 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’s-t density within the GARCH framework. The log likelihood contributions of a standardized skewed Student’s-t 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 degrees-of-freedom 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 add-in 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’s-t and its (positively) skewed version. Skewness increases the chance of observing extreme values, which has important implications in finance.


Figure 1: Skewed t-Distribution

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 2005-2020.


Figure 2: Squared Returns

In order to estimate the conditional variance of returns, we start by fitting two alternative models (i.e. GARCH(1,1) and TGARCH(1,1)) with two different distributional assumptions (i.e. Normal and Student’s-t). Mean equation is same for all models: \begin{align*} r_t &= \bar{r} + e_t\\ e_t &= \epsilon_t \sigma_t \end{align*} \begin{align*} \textbf{Model 1}: \quad \sigma_t^2 &= \omega + \alpha_1 e_{t-1}^2 + \beta_1\sigma_{t-1}^2, \quad \text{where} \quad \epsilon_t \sim N(0,1)\\ \textbf{Model 2}: \quad \sigma_t^2 &= \omega + \alpha_1 e_{t-1}^2 + \beta_1\sigma_{t-1}^2 + \gamma_1 e_{t-1}^2(e_t < 0), \quad \text{where} \quad \epsilon_t \sim N(0,1)\\ \textbf{Model 3}: \quad \sigma_t^2 &= \omega + \alpha_1 e_{t-1}^2 + \beta_1\sigma_{t-1}^2, \quad \text{where} \quad \epsilon_t \sim \text{Student}(0,1,\nu)\\ \textbf{Model 2}: \quad \sigma_t^2 &= \omega + \alpha_1 e_{t-1}^2 + \beta_1\sigma_{t-1}^2 + \gamma_1 e_{t-1}^2(e_t < 0), \quad \text{where} \quad \epsilon_t \sim \text{Student}(0,1,\nu) \end{align*}


Figure 3a: Model 1 Results
Figure 3b: Model 2 Results



Figure 4a: Model 3 Results
Figure 4b: Model 4 Results

From a purely statistical point of view ($p$-values and information criteria that is), fat tails and/or leverage effects better represent the Turkish Lira’s volatility dynamics. Distribution fit to standardized residuals and the analysis of news impact can be provided as supporting evidence in that respect.



Figure 5a: Leverage
Figure 5b: News Impact Curve

Extreme events seem to occur more often than suggested by the normal distribution and the volatility response to these shocks are more severe in the case of depreciation than that of appreciation.

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_{t-1}^2 + \beta_1\sigma_{t-1}^2, \quad \text{where} \quad \epsilon_t \sim \text{Student}(0,1,\nu) \end{align*}


Figure 6a: Fractional Mean Model
Figure 6b: Fractional Variance Model

Fractional difference parameter is significantly different from 0 and 1 in both models, but it is also significantly smaller than 0.5 in the ARFIMA model suggesting that the squared return series has long memory properties. However, modelling the variance of the return series explicitly we have successfully explained the behaviour of volatility and mitigated the impact of (and need for) long memory.

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’s-t 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 add-in assuming a skewed Student’s-t distribution and see if we can further improve the fit.


Figure 7: Skewed GARCH Estimates

Estimated parameter values change slightly vis-à-vis our original TGARCH model, but the asymmetry parameter seems to be positive and significant, supporting the evidence of skewness. Information criteria favors this version of the model over all other specifications above.

One of the main uses of GARCH models in financial institutions is the estimation of Value-at-Risk (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).


Figure 8: Value-at-Risk

At its peak around the summer of 2018, currency shock led 99% VaR threshold of a TL-denominated asset or portfolio to jump to a daily loss of 14.5%. It would have been considered as an astronomical event a year ago, since it was only around 1% back then. Increasing the likelihood of extreme events and incorporating the asymmetric tail behaviour of the shocks, would further add 5.1 and 3.5 bps, respectively and would carry this limit to 23.1%!




Files




References

  1. Lambert P and Laurent S (2001), "Modelling Financial Time Series Using GARCH-Type Models and a Skewed Student Density", Mimeo, Universite de Liege.

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