Skewed Student-t Distribution Research Articles

A Two-Parameter Inverted Exponentiated Skewed Student-T Distribution: Theory and Application

In this study, a new two-parameter generalization of the skew-t distribution called the Inverted Exponentiated Skew t distribution, has been proposed. Some properties of the model like order statistics, entropy, asymptotic behavior, moments, characteristic function, and quantile function were derived. Parameter estimates using maximum likelihood estimation were consistent and the behavior of the estimates with increase in sample size was studied using Montecarlo simulation. The flexibility of the Inverted Exponentiated Skew Student t model was demonstrated by application to four financial datasets and the results revealed that the Inverted Exponentiated Skew Student t model yielded a better fit

Forecasting VaRs via hybrid EVT with normal and non-normal filters: A comparative analysis from the Chinese stock market

This paper studies the performance of hybrid methods combining normal and non-normal GARCH-type filters with extreme value theory (EVT) in predicting Value-at-Risk (VaR) for four major stock indices in the Chinese stock market. Based on the out-of-sample VaR forecasts results over the 24 models considered, we find that the hybrid EVT approaches illustrate substantial advantage over the corresponding GARCH-type models with parametric distributions, especially in the lower tail and extremal VaR evaluations (1% and 0.5% VaRs). Overall, the hybrid EVT methods with heavy-tailed filters (std-EVT, sstd-EVT) perform the best, followed by the hybrid EVT approaches with light-tailed filters (norm-EVT, snorm-EVT), and then the GARCH-type models with skewed student-t distribution. The conditional normal, skewed normal and student-t models are rejected and excluded from the Model Confidence Set (MCS) in predicting extremal VaRs for most cases, particularly for the lower tail. Additionally, we find that the results for alternative models in predicting VaRs vary sharply among alternative innovation specifications, but present trivial differences across the volatility models. Furthermore, our results based on the MCS tests and comparative backtests also indicate that the scoring function effectively discriminates predictive performance across methods for the lower tail, in contrast to showing less disparity for the upper tail. This emphasizes the importance of model selection in VaR forecasting performance for the lower tail. The above findings are robust to alternative threshold selections and the rolling-step chosen to update parameters. Similar analyses concerning the out-of-sample performance of the 24 models in predicting the Expected Shortfall (ES) show consistent results overall.

Application of Copula Methods in Financial Risk Management: Case of the Zimbabwe Stock Exchange and the Victoria Falls Stock Exchange.

In order to minimize risks and create a safe investing environment, financial risk management is becoming more and more crucial for individuals, financial organizations, and even entire nations. Accurately assessing financial risks and using that information to inform wise investment choices can give an investor a competitive edge as well as significant returns. In actuality, real-world financial variables limit the ability to estimate financial risks. On the other hand, a wealth of data indicates that financial variables typically have asymmetric dependency, skewness, and fat tails. In three ways, the conventional approaches to financial risk management based on normally distributed hypotheses are put to the test by these stylized characteristics of financial variables. First, the univariate normal distribution or other elliptical distributions are unable to adequately fit the distribution of the univariate variable. Second, despite their straightforward tractability, multivariate variables’ extra kurtosis and skewness are not captured by their normal distribution. As a result, the dependence risks associated with multivariate financial variables may be underestimated. Finally, when the joint distribution of various variables is non-elliptical, linear correlation which is typically employed to characterize the dependence of various variables in traditional portfolio risk management is likewise insufficient. This research uses a promising technique based on copulas in conjunction with GARCH and realized volatility models to examine the risks associated with multivariate financial variables in order to address these issues. The multivariate distributions are constructed using copulas in conjunction with GARCH and Realized Volatility models which are then utilized to evaluate portfolio risks in financial market. The findings demonstrate that copula-based models outperform conventional models in fitting financial data. Second, a variety of marginal models have a notable impact on the value at risk of the portfolio, including the GARCH and realized volatility models. Lastly, both the dependence structure and the marginal distribution exhibit notable skewness. Consequently, compared to the normal or Student-t distribution, the skewed Student-t distribution fits some datasets better.

Forecasting risk of European carbon emissions trading market with DySco-SKST model

Influenced by energy prices, macroeconomic factors and policy factors, the price of the European carbon emissions trading market has changed significantly. Effective measurement of price risk of the European carbon emissions trading market is of practical importance for market participants. Two measures Value at Risk (VaR) and Expected shortfall (ES) are used to assess price risk. Based on the dynamic score (DySco) model and skewed Student-t (SKST) distribution, the DySco-SKST model is constructed and used to predict the price risk. The unconditional/conditional coverage tests, dynamic quantile test, Actual over Expected ratio, mean and maximum Absolute Deviation, quantile loss and FZ loss are used to evaluate the VaR and ES prediction performance. The daily spot closing prices of EUAs from January 3, 2013 to November 23, 2018 are used. The empirical results show that the VaR and ES prediction performance of the DySco-SKST model is better than that of the DySco-N and DySco-ST models. The VaR and ES prediction performance is affected by the parameter re-estimation scheme, but not by the parameter re-estimation frequency. The DySco-SKST model is better at predicting VaR and ES under the rolling window than under the expanding window.

Risks in Major Cryptocurrency Markets: Modeling the Dual Long Memory Property and Structural Breaks

This study estimates the effects of the dual long memory property and structural breaks on the persistence level of six major cryptocurrency markets. We apply the Bai and Perron structural break test, Inclán and Tiao’s iterated cumulative sum of squares (ICSS) algorithm, and the fractionally integrated generalized autoregressive conditional heteroscedasticity (FIGARCH) model, with different distributions. The results show that long memory and structural breaks characterize the conditional volatility of cryptocurrency markets, confirming our hypothesis that ignoring structural breaks leads to an underestimation of the persistence of volatility modeling. The ARFIMA-FIGARCH model, with structural breaks and a skewed Student-t distribution, fits the cryptocurrency market’s price dynamics well.

Modelling the Impact of the COVID-19 Pandemic on Some Nigerian Sectorial Stocks: Evidence from GARCH Models with Structural Breaks

This study provides evidence of the impact of COVID-19 on five (5) Nigerian Stock Exchange (NSE) sectorial stocks (NSE Insurance, NSE Banking, NSE Oil and Gas, NSE Food and Beverages, and NSE Consumer Goods). To achieve the goal of this paper, daily stock prices were obtained from a secondary source ranging from 2 January 2020 to 25 March 2021. Because of the importance of incorporating structural breaks in modelling stock returns, the Zivot–Andrews unit root test revealed 20 January 2021, 26 March 2020, 27 July 2020, 23 March 2020 and 23 March 2020 as potential break points for NSE Insurance, NSE Food, Beverages and Tobacco, NSE Oil and Gas, NSE Banking, and NSE Consumer Goods, respectively. This study investigates the volatility in daily stock returns for the five (5) Nigerian Stock Exchange (NSE) sectorial stocks using nine versions of GARCH models (sGARCH, girGARCH, eGARCH, iGARCH, aPARCH, TGARCH, NGARCH, NAGARCH, and AVGARCH); in addition, the half-life and persistence values were obtained. The study used the Student t- and skewed Student t-distributions. The results from the GARCH models revealed a negative impact of COVID-19 on the NSE Insurance, NSE Food, Beverages and Tobacco, NSE Banking, and NSE Consumer Goods stock returns; however, the NSE Oil and Gas returns showed a positive correlation with the COVID-19 pandemic. This study recommends that the shareholders, investors, and policy players in the Nigerian Stock Exchange markets should be adequately prepared in the form of diversification of investment in stocks that can withstand future possible crises in the market.

Bootstrapping Time-Varying Uncertainty Intervals for Extreme Daily Return Periods

This study aims to overcome the problem of dimensionality, accurate estimation, and forecasting Value-at-Risk (VaR) and Expected Shortfall (ES) uncertainty intervals in high frequency data. A Bayesian bootstrapping and backtest density forecasts, which are based on a weighted threshold and quantile of a continuously ranked probability score, are developed. Developed backtesting procedures revealed that an estimated Seasonal autoregressive integrated moving average-generalized autoregressive score-generalized extreme value distribution (SARIMA–GAS–GEVD) with a skewed student-t distribution had the best prediction performance in forecasting and bootstrapping VaR and ES. Extension of this non-stationary distribution in literature is quite complicated since it requires specifications not only on how the usual Bayesian parameters change over time but also those with bulk distribution components. This implies that the combination of a stochastic econometric model with extreme value theory (EVT) procedures provides a robust basis necessary for the statistical backtesting and bootstrapping density predictions for VaR and ES.

The risk-return relationship in South Africa: tail optimization of the GARCH-M approach

<abstract> <p>The risk-return relationship is of fundamental significance in the field of economics and finance. It is used to structure investment strategies, allocate resources, as well as assist in the construction of policy and regulatory frameworks. The accurate forecast of the risk-return relationship ensures sound financial decisions, whereas an inaccurate one can underestimate risk and thus lead to losses. The GARCH-M approach is one of the foremost models used in South African literature to investigate the risk-return relationship. This study made a novel and significant contribution, on a local and international level, as it was the first study to investigate GARCH-M type models with different innovation distributions. This study analyzed the JSE ALSI returns of the South African market for the sample period from 05 October 2004 to 05 October 2021. Results revealed that the EGARCH (1, 1)-M with the Skewed Student-t distribution (Skew-t) is optimal relative to the standard GARCH, APARCH and GJR. However, the EGARCH-M Skew-t failed to capture the financial data's asymmetric, volatile and random nature. To improve forecast accuracy, this study applied different nonnormal innovation distributions: the Pearson Type Ⅳ distribution (PIVD), Generalized Extreme Value distribution (GEVD), Generalized Pareto distribution (GPD) and Stable. Model diagnostics revealed that the nonnormal innovation distributions adequately captured asymmetry. The Value at Risk and backtesting procedure found that the PIVD, followed by Stable, outperformed the Extreme Value Theory distributions (GEVD and GPD). Thus, investors, risk managers and policymakers would opt to use the EGARCH-M in combination with the PIVD when modelling the risk-return relationship. The main contribution of this study was to confirm that applying GARCH type models with the conventional and normal type distributions, to a volatile emerging market, is considered ineffective. Therefore, this study recommended the exploration of other innovation distributions, that were not included in the scope of this study, for future research purposes.</p> </abstract>

Volatility spillovers between oil and equity markets and portfolio risk implications in the US and vulnerable EU countries

This paper examines the frequency dynamics of volatility spillovers between Brent crude oil and stock markets in the US (S&P500 index), Europe (STOXX600 index), Asia (Dow Jones Asia index) and stock markets of five vulnerable European Union (EU) countries known as the GIPSI (Greece, Ireland, Portugal, Spain, and Italy). We use the methodologies developed by Diebold and Yilmaz (2012) and Baruník and Křehlík (2018) and the in-sample VaR performance to achieve our objectives. The results show that the spillover effect between the oil and the considered stock markets is time-varying, crisis-sensitive, and frequency-dependent. In addition, this effect had intensified during the onset of the 2008–2009 Global Financial Crisis, the plunge of oil prices that started in the mid-2014, and the occurrence of COVID-19. Furthermore, the results also demonstrate that oil is a net receiver of risk in the system, irrespective of the time horizon. Among the GIPSI markets, the larger and highly indebted EU economies of Italy and Spain are net contributors of risk, while the smaller EU economies of Greece, Ireland, and Portugal as well as Asia are net receivers of risk, also regardless of the time horizons. The US and European stock markets are net contributors of the spillovers, whereas the Asia equity index is a net receiver, irrespective of the frequencies. Finally, the AR(1)-FIGARCH model with the skewed Student-t distribution provides more accurate in-sample estimates in all markets. The hedging is more expensive in the long term than in the short term. The hedging is also more expensive during the ESDC period than the other turbulent periods considered in the study. The hedging effectiveness is also higher in the long term than in the short term, is crisis-sensitive, and depends on the time-investment horizon factor.

Forecasting of the Nigeria Stock Returns Volatility Using GARCH Models with Structural Breaks

This study examines the stock returns series using Symmetric and Asymmetric GARCH models with structural breaks in the presence of some varying distribution assumptions. Volatility models of Symmetric GARCH (1,1), Asymmetric Power GARCH (1,1) and GJR-GARCH(1,1) models were considered in estimating and measuring shock persistence, leverage effects and mean reversion rate with structural breaks considering dummy variable for these structural changes and varying distributions . The skewed student-t distribution is considered best distribution for the models; moreover findings showed the high persistence of shock in returns series for the estimated models. However, when structural breaks were incorporated in the estimated models by including dummy variable in the conditional variance equations of all the models, there was significant reduction of shock persistence parameter and mean reversion rate. The study found the GJR-GARCH (1,1) with skewed student-t distribution best fit the series. The volatility was forecasted for 12 months period using GJR-GARCH (1,1) model and the values are compared with the actual values and the results indicates a continuous increase in unconditional variance.

Volatility of Some Selected Currencies Against the Naira Using Generalized Autoregressive Score Models

The role exchange rate plays in international trade and bilateral agreement between countries cannot be over-emphasize. Fluctuations in exchange rate has direct impact on the economy of any country especially a country like Nigeria which depends largely on import goods. So, there is need to identify appropriate model that can adequately describe the dynamics of the exchange rate volatilities. This article investigated the volatility of exchange rates in Nigeria by selecting the U.S dollars, Pound Sterling and Euro against the Naira using daily data over the period of January 02, 2002 to August 31, 2018. The GAS model with its variants was applied to study the volatility of the exchange rates assuming three different probability distributions for the innovations of the models namely; Normal distribution (N), Student-t distribution (T) and Skewed-Student-t distribution (SKT). The AIC and SBIC estimates obtained were used to access fitness performance. The GAS model and its variants’ forecasting ability were access using several forecast measures. Using the estimates of the AIC and SBIC, GAS-T, EGAS-T and EGAS-STK were selected for US dollars/Naira, Pound sterling/Naira and Euro/Naira exchange rates respectively as the best fitted models. Based on the estimates of MAE and RMSE, GAS-T, EGAS-T and EGAS-SKT were selected for forecasting the volatility of US dollars/Naira, Pound sterling/Naira and Euro/Naira exchange rates respectively.

Analysis of the Duration–Hardness Ratio Plane of Gamma-Ray Bursts Using Skewed Distributions

Abstract The two widely accepted classes of gamma-ray bursts (GRBs), short and long, are with confidence ascribed to mergers of compact objects and collapse of massive stars, respectively. A third, intermediate/soft class, remains putative. Its existence was claimed based on univariate and bivariate analyses of GRB observables modeled with Gaussian distributions. This, however, may not be the appropriate approach, as it has already been shown that the univariate distributions of durations are better described by mixtures of two skewed components rather than three Gaussian ones. This paper investigates whether data in the duration–hardness ratio plane is better modeled by mixtures of skewed bivariate distributions than by normal ones. The archival data set of the Compton Gamma-Ray Observatory/BATSE and Fermi/Gamma-ray Burst Monitor data from the most recent catalog release are examined. The preferred model is chosen based on two information criteria, Akaike and Bayesian. It is found that the best description is given by a two-component mixture of skewed Student-t distributions, which outperforms any other model considered. This implies that the distribution of the studied parameters is intrinsically skewed, introducing spurious Gaussian components, and hence the third class is unlikely to be a real phenomenon. Its existence, based on statistical inference, is therefore rejected as unnecessary to explain the observations.

Forecasting and Backtesting of <i>VaR</i> in International Dry Bulk Shipping Market under Skewed Distributions

It is extremely important to model the empirical distributions of dry bulk shipping returns accurately in estimating risk measures. Based on several commonly used distributions and alternative distributions, this paper establishes nine different risk models to forecast the Value-at-Risk (VaR) of dry bulk shipping markets. Several backtests are explored to compare the accuracy of VaR forecasting. The empirical results indicate the risk models based on commonly used distributions have relatively poor performance, while the alternative distributions, i.e. Skewed Student-T (SST) distribution, Skewed Generalized Error Distribution (SGED), and Hyperbolic distribution (HYP) produce more accurate VaR measurement. The empirical results suggest risk managers further consider more flexible empirical distributions when managing extreme risks in dry bulk shipping markets.

Analysis of Markov switching seemingly unrelated regression model with skewed distributions, and its application to Thai cassava market

This study conducts a Markov switching seemingly unrelated regression with skewed distribution. We analyse MS-SUR model under skewed distributions. In other words, the multivariate skew-normal and skew-t with unknown skewness parameter are adopted to be the likelihood for the MS-SUR model. The simulation results show that parameters of MS-SUR model appear to be quite reasonable. In addition, we apply our model to study Thai cassava market and the results provide an evidence that given skewed student-t distribution for both demand and supply equations in the MS-SUR models present the lowest AIC and BIC. Moreover, the empirical results show that exchange rate and export price are two main factors that affect, respectively, demand for and supply of Thai cassava in market downturn regime.

A Skewed Student-T Value-at-Risk Approach for Long Memory Volatility Processes in Japanese Financial Markets

This paper investigates the relevance of skewed Student-t distributions in capturing long memory volatility properties in the daily return series of Japanese financial data (Nikkei 225 Index and JPY-USD exchange rate). For this purpose, we assess the performance of two long memory Value-at-Risk (VaR) models (FIGARCH and FIAPARCH VaR model) with three different distribution innovations: the normal, Student-t, and skewed Student-t distributions. From our results, we find that the skewed Student-t distribution model produces more accurate VaR estimations than normal and Student-t distribution models. Thus, accounting for skewness and excess kurtosis in the asset return distribution can provide suitable criteria for VaR model selection in the context of long memory volatility and enhance the performance of risk management in Japanese financial markets.

Forecasting the VaR of crude oil market: Do alternative distributions help?

Accurate modeling of the empirical distribution of crude oil market returns is extremely important in estimating risk measures. In addition to several commonly used distributions, alternative distributions are explored in this study, some of which account for the asymmetry and heavy tails simultaneously found in distributions, and contain more tail parameters to separately depict the right and left tails when forecasting the Value-at-Risk (VaR) of crude oil markets during highly volatile periods. Seven backtests are also conducted to compare the VaR forecasting accuracy among different distributions. The empirical results indicate that a highly volatile environment challenges the commonly used distributions, and the four risk models based on commonly used distributions are rejected about 27% to 38% of the time. The alternative distributions, i.e., skewed general error distributions (SGED), generalized hyperbolic skewed Student-t distributions (GHST), and generalized asymmetric Student-t (GAST) distributions, generally produce more accurate VaR measurement, and GAST gives the best measurement accuracy. The empirical results imply that risk managers or policymakers should further consider more flexible distributions, such as SGED, GHST, or GAST in particular, when quantifying or managing the risk in turbulent market times.

The HARX-GJR-GARCH skewed-t multipower realized volatility modelling for S&P 500

The heterogeneous autoregressive (HAR) models are used in modeling high frequency multipower realized volatility of the S&P 500 index. Extended from the standard realized volatility, the multipower realized volatility representations have the advantage of handling the possible abrupt jumps by smoothing the consecutive volatility. In order to accommodate clustering volatility and asymmetric of multipower realized volatility, the HAR model is extended by the threshold autoregressive conditional heteroscedastic (GJR-GARCH) component. In addition, the innovations of the multipower realized volatility are characterized by the skewed student-t distributions. The extended model provides the best performing in-sample and out-of-sample forecast evaluations.

Volatility Modelling and Parametric Value-At-Risk Forecast Accuracy: Evidence from Metal Products

In this paper, we investigate the one-day-ahead VaR and ES accuracy of four metal daily return series including Aluminium, Copper, Nickel and Zinc. Since, all sample presents volatility clustering, volatility asymmetry, and volatility persistence, we have assessed five GARCH-type models including three fractionary integrated models assuming three alternative distributions (normal, Student-t and skewed Student-t distributions). Estimates results reveal the performance of AR (1) - FIAPARCH model under a skewed Student-t distribution. We have computed one-day ahead VaR and (ES) for both short and long trading positions. Backtesting results show very clearly that the skewed Student-t FIAPARCH model provides the best results for both short and long VaR estimations. These results present several potential implications for metal markets risk quantifications and hedging strategies.

Wartość zagrożona w warunkach ufinansowienia rynków surowców rolnych

Purpose – The main objective of this study was to analyze the usefulness of selected conditional VaR estimation models to measure the risk of an investment in the agricultural commodities futures markets in conditions of the financialisation of these markets. An additional aim was to evaluate and compare the level of an investment risk in the futures contract on particular agricultural commodities on different exchanges. Design/Methodology/approach – To estimate the VaR of the futures contracts for agricultural commodities the following conditional models were used: GARCH with a skewed Student-t distribution, GARCH-EVT and GARCH-FHS. To verify the effectiveness of tested models the following tests were implemented: Kupiec test, Christoffersen test, Christoffersen and Pelletier test. Findings – GARCH with a skewed Student-t distribution and GARCH-EVT allowed for correct estimation of the value at risk of the futures contracts for agricultural commodities during the financialisation of the agricultural commodities markets. There were differences in the level of an investment risk in the futures contracts on agricultural raw material on various exchanges. Originality/value – According to the author's knowledge the issue of measuring the value at risk of futures contracts for agricultural commodities rarely been undertaken in the literature, especially in the case of contracts traded on various exchanges.

Porównanie bayesowskich modeli Copula-AR(1)-GARCH(1,1) z asymetrycznością rozkładów warunkowych

The main aim of the paper is to formally assess the relative explanatory power of competing bivariate Copula-AR-GARCH models with symmetric and skewed Student t distributions on the example of data from the The Warsaw Stock Exchange. The subject of comparison were 22 Copula-AR(1)-GARCH (1,1) models, which differed in assumptions on the copula and the occurrence of skewness in marginal distributions. In the context of the models under consideration, Monte Carlo Important Sampling methods were used to estimate the characteristics of a posteriori distribution and the marginal density of the observation matrix. For analysing empirical data, a posteriori models turned out to be ones more likely to have symmetrical conditional t-Student distributions. For the logarithmic daily growth rates of the two sub-indicies of the stock index WIG, the highest a posteriori probability was obtained by the Clayton-Gumbel copula model. The use of the skewed Student's t-distribution did not improve the explanatory power of the Copula-GARCH models.