Skewed Generalized T Research Articles

Non-Gaussian feature distribution forecasting based on ConvLSTM neural network and its application to robust machine condition prognosis

The prognosis of a machine condition becomes a hot topic nowadays, as the condition monitoring installations provide a massive amount of Health Index (HI) data that could be used for machine remaining lifetime prognosis. However, existing methodologies might not be effective since in many cases the real HI datasets exhibit non-Gaussian characteristics. Thus, to improve efficiency, there is a need to introduce novel approaches which take into account the possible non-Gaussian distribution of HI data. In the proposed methodology, several innovative components are considered to form an efficient solution for the HI time series prediction. As HI is a mixture of trend and random non-Gaussian, non-homogeneous noise, we propose to forecast the HI distribution rather than the direct HI value. The Skewed Generalized t (SGT) distribution is considered as the general non-Gaussian model. Moreover, the combination of convolutional neural network (CNN) and long short-term memory (LSTM) are used to predict SGT distribution parameters. By incorporating the standardization module and the modified activation function alongside a custom time series standardization method, the architecture of the ConvLSTM neural network is created to be appropriate for non-stationary time series data with non-Gaussian behavior. Finally, a pre-training of the model has been proposed, which improves the efficiency of the designed procedure, as usually the amount of HI data is limited and insufficient to reliably train an artificial intelligence (AI) system. The proposed solution is shown to be robust and more resilient to outliers than the classical Gaussian-based approach. Using simulated and real HI data (known as a benchmark), we may conclude that the superiority of our method increases with the increasing non-Gaussianity level of the analyzed time series.

Anticipating extreme losses using score-driven shape filters

Abstract We suggest a new value-at-risk (VaR) framework using EGARCH (exponential generalized autoregressive conditional heteroskedasticity) models with score-driven expected return, scale, and shape filters. We use the EGB2 (exponential generalized beta of the second kind), NIG (normal-inverse Gaussian), and Skew-Gen-t (skewed generalized-t) distributions, for which the score-driven shape parameters drive the skewness, tail shape, and peakedness of the distribution. We use daily data on the Standard & Poor’s 500 (S&P 500) index for the period of February 1990 to October 2021. For all distributions, likelihood-ratio (LR) tests indicate that several EGARCH models with dynamic shape are superior to the EGARCH models with constant shape. We compare the realized volatility with the conditional volatility estimates, and we find two Skew-Gen-t specifications with dynamic shape, which are superior to the Skew-Gen-t specification with constant shape. The shape parameter dynamics are associated with important events that affected the stock market in the United States (US). VaR backtesting is performed for the dot.com boom (January 1997 to October 2020), the 2008 US Financial Crisis (October 2007 to March 2009), and the coronavirus disease (COVID-19) pandemic (January 2020 to October 2021). We show that the use of the dynamic shape parameters improves the VaR measurements.

Modelling the Botswana Pula/Us Dollar exchange rate using the Skewed generalized t (SGT) distributions

The economy of Botswana heavily relies on mineral exports (mainly diamond exports), which are largely dependent on the exchange rate. And, the US Dollar is one of the most important currencies in the basket of currencies to which the Botswana Pula is pegged. Therefore, this paper seeks to empirically establish the baseline characteristics of the Botswana Pula (BWP) and the US Dollar (USD) exchange rate and to identify the most plausible probability distribution from the skewed generalized t (SGT) family that can be used to model the log-returns of the daily BWP/USD exchange rates for the period January 2001 to December 2020. The SGT family is a highly versatile class of models that can capture the skewness and kleptokurticity that are inherent in financial time series. Four probability distributions are considered in this study: skewed t, skewed generalized error, generalized t and skewed generalized t. The maximum likelihood approach is used to estimate the parameters of each model. Model comparison and selection are based on the Akaike information criterion (AIC) and Bayesian information criterion (BIC). The results of the study show that the daily BWP/USD exchange rate series is nonnormal, negatively skewed heavy-tailed. It is also found that, based on the values of both the AIC and BIC, the model that gives the best fit to the data is the skewed t, which is closely followed by the skewed generalized error distribution, while the generalized t gives the worst fit. Keywords: Pula/US Dollar exchange rate, log returns, Generalized t distribution, Skewed generalized error distribution, Skewed generalized t distribution, Skewed t distribution, skewness, kurtosis, maximum likelihood

A comparison of the GB2 and skewed generalized log-t distributions with an application in finance

Several families of statistical distributions have been used to model financial data. The four-parameter generalized beta of the second kind (GB2) and five-parameter skewed generalized t (SGT) have been fit to return and log-return data, respectively. We introduce the skewed generalized log-t (SGLT) distribution and note that the GB2 and SGLT share such distributions as the asymmetric log-Laplace (ALL), log-Laplace (LL), and log-normal (LN). We then compare the relative performance of the GB2 and SGLT in modeling the distribution of daily, weekly, and monthly stock return data. We find that the GB2 and SGLT perform similarly and that the three-parameter log-t (LT) distribution is quite robust.

Distributional Assumptions and the Estimation of Contingent Valuation Models

Contingent valuation methods are well-established techniques for measuring the value of goods and services not transacted in markets and have been applied in many different settings. Some of these applications include estimating the value of outdoor recreation, reducing risk, decreasing pollution, or reducing transportation time. The parameter estimates depend upon the survey design, the model specification, and the method of estimation. Distributional misspecification or heteroskedasticity can lead to inconsistent estimators. This paper introduces a partially adaptive estimation procedure, based on two families of flexible probability density functions [the generalized beta of the second kind (GB2) and the skewed generalized t (SGT)], to adjust for distributional misspecification and accommodate possible heteroskedasticity. Using a linear link function, these methods are applied to the problem of estimating the willingness to pay to protect Australia’s Kakadu Conservation Zone from mining. In this application, the assumption of homoskedasticity is not rejected for the GB2 family, but is rejected for the SGT. A Monte Carlo simulation confirms the importance of the homoskedasticity assumption as well as the impact of the bid design. For this example, many of the more flexible distributions are in fairly close agreement with some of their special cases. However, this application illustrates how flexible nested distributions can be used to accommodate diverse distributional characteristics, including possible heteroskedasticity.

Risk Measures for Investment Values and Returns Based on Skewed-Heavy Tailed Distributions: Analytical Derivations and Comparison

The skewed generalized t (SGT) displays an exceptional ability in modelling the tails of the empirical distributions of returns of financial and other assets. This feature makes it an appealing candidate for the computation of value at risk and expected shortfall measures, used by regulators, investors, portfolio managers and actuaries to measure and manage the risk exposure of their assets. This paper makes a specific contribution by deriving the analytical equations for the computation of value at risk, expected shortfall and downside risk measures for asset values and returns based on the SGT distribution. An assessment using simulations and estimation show that risk measures based on returns overestimate risk exposure.

Skewed Generalized T and Nested Probability Distributions: Specification and Moments

This paper presents a unified framework for the popular Skewed Generalized t (SGT) distribution and its special cases the Skewed Generalized Error Distribution (SGED), the skewed student’s t, the skewed Laplace and skewed normal distributions. The analytical moment equations presented can be useful to researchers in statistics, finance and econometrics working in the areas of estimation, asset pricing and risk management. The unified framework presented helps avoid confusions related to the interpretation of their parameters and results.

Highly flexible distributions to fit multiple frequency financial returns

Financial data are usually studied via low flexible distributions, independently of the frequency of the data, due to their simplicity and analytical tractability. In this paper we analyze two highly flexible five-parameter distributions into fitting financial returns, these are the skewed generalized t (SGT) and the generalized hyperbolic (GH). Applications carried on two exchange rates (Euro–Dollar and Dollar–Yen), and two indexes (S&P 500 and Nikkei 225) over four frequencies: weekly, daily, 30-min and 5-min, confirm the superiority of the SGT and GH in approximating the distribution of a given data at a remarkable precision. Moreover, as we move from higher to lower frequency, the distribution’s overall shape does indeed change radically, and the estimated parameters refute the tendency to normality, which calls into question the aggregational Gaussianity’s stylized fact.

The frequency of regime switching in financial market volatility

Abstract The mechanism of risk responses to market shocks is considered as stagnant in recent financial literature, whether during normal or stress periods. Since the returns are heteroskedastic, a little consideration was given to volatility structural breaks and diverse states. In this study, we conduct extensive simulations to prove that the switching regime GARCH model, under the highly flexible skewed generalized t (SGT) distribution, is remarkably efficient in detecting different volatility states. Next, we examine the switching regime in the S&P 500 volatility for weekly, daily, 10-minute and 1-minute returns. Results show that the volatility switches regimes frequently, and differences between the distributions of the high and low volatility states become more accentuated as the frequency increases. Moreover, the SGT is highly preferable to the usually employed skewed t distribution.

Conditional higher order moments in metal asset returns

This study examines the role of higher order moments in the returns of four important metals, aluminium, copper, gold and silver, using the asymmetric GARCH (AGARCH) model with a conditional skewed generalized-t (SGT) distribution. Implications of higher order moments in metal returns are evaluated by comparing the performances of conditional value-at-risk measures obtained from the AGARCH models with SGT distributions to those obtained from the AGARCH models with normal and student-t distributions. With the exception of gold, the AGARCH model with the SGT distribution appears to have the best fit for all metals examined.

High-order moments and extreme value approach for value-at-risk

Abstract We modify a two-step approach by McNeil and Frey (2000) for forecasting Value-at-Risk (VaR). Our approach combines the asymmetric GARCH (GJR) model that allows the high-order moments (i.e., skewness and kurtosis) of the skewed generalized t (SGT) distribution to rely on the past information set to estimate volatility, and the modified Hill estimator (Huisman et al., 2001) for estimating the innovation distribution tail of the GJR model. Using back-testing of the daily return series of 10 stock markets, the empirical results show that our proposed approach could give better one-day VaR forecasts than McNeil and Frey (2000) and the GJR/GARCH models with alternative distributions. In addition, our proposed approach also provides the accuracy of expected shortfall estimates. The evidence demonstrates that our proposed two-step approach that incorporates the modified Hill estimator into the GJR model based on the SGT density with autoregressive conditional skewness and kurtosis provides consistently accurate VaR forecasts in the short and longer sample periods.

Estimating Flexible, Fat-Tailed Asset Return Distributions

We introduce new robust numerical methods, based on the minimum relative U−entropy (MRUE) principle, to estimate univariate probability density functions for power-law (fat-tailed) random variables. The semi-parametric models that we estimate via convex programming are flexible enough to conform well to potentially plentiful data for not-too-extreme values, while allowing for power-law tails (which need not be symmetric). We observe that a number of well-known power-law models, including the exponential, Pareto, Student-t, and skewed generalized-t (SGT) distributions, are special cases of the family of power-law probability densities that we consider. We benchmark our method against state-of-the-art asset return models on S&P500 index returns, individual stock returns, and power price returns and find that our models outperform the benchmarks out-of-sample. We attribute this out-performance to simultaneously conforming to data where it is plentiful, while building in reasonably conservative tails.

The role of the SGT Density with Conditional Volatility, Skewness and Kurtosis in the Estimation of VaR: A Case of the Stock Exchange of Thailand

One of primary tools used to assess the financial risk is Value-at-Risk (VaR). It turns to be a standard measure of downward risk among financial intermediaries and regulators recently as it summarized the risk into just a single and easy-to-understand number. Despite the simplicity of VaR's concept, an accurate calculation of VaR is still challenging. This paper aims to propose an alternative approach which is believed to provide more accurate VaR rather than the traditional ones. Instead of the conventional Gaussian distribution, the more flexible skewed generalized t (SGT) density function is assumed for return series. Its volatility is characterized by eight types of GARCH process. Meanwhile, conditional skewness and kurtosis is modeled to exhibit time-varying feature by their past information set and autoregressive term. Daily returns on the SET index will be used to explore the performance of estimated VaR. The finding shows that this new approach can provide more accurate and robust estimates of the actual VaR threshold, especially with TS-GARCH model, than any other approaches that have been applied earlier.

Forecasting the volatility of S&P depositary receipts using GARCH-type models under intraday range-based and return-based proxy measures

We employ four various GARCH-type models, incorporating the skewed generalized t (SGT) errors into those returns innovations exhibiting fat-tails, leptokurtosis and skewness to forecast both volatility and value-at-risk (VaR) for Standard & Poor's Depositary Receipts (SPDRs) from 2002 to 2008. Empirical results indicate that the asymmetric EGARCH model is the most preferable according to purely statistical loss functions. However, the mean mixed error criterion suggests that the EGARCH model facilitates option buyers for improving their trading position performance, while option sellers tend to favor the IGARCH/EGARCH model at shorter/longer trading horizon. For VaR calculations, although these GARCH-type models are likely to over-predict SPDRs' volatility, they are, nevertheless, capable of providing adequate VaR forecasts. Thus, a GARCH genre of model with SGT errors remains a useful technique for measuring and managing potential losses on SPDRs under a turbulent market scenario.

Skewness and leptokurtosis in GARCH-typed VaR estimation of petroleum and metal asset returns

Abstract This paper utilizes the most flexible skewed generalized t (SGT) distribution for describing petroleum and metal volatilities that are characterized by leptokurtosis and skewness in order to provide better approximations of the reality. The empirical results indicate that the forecasted Value-at-Risk (VaR) obtained using the SGT distribution provides the most accurate out-of-sample forecasts for both the petroleum and metal markets. With regard to the unconditional and conditional coverage tests, the SGT distribution produces the most appropriate VaR estimates in terms of the total number of rejections; this is followed by the nonparametric distribution, generalized error distribution (GED), and finally the normal distribution. Similarly, in the dynamic quantile test, the VaR estimates generated by the SGT and nonparametric distributions perform better than that generated by other distributions. Finally, in the superior predictive test, the SGT distribution has significantly lower capital requirements than the nonparametric distribution for most commodities.

Risk Measurement Performance of Alternative Distribution Functions

AbstractThis paper evaluates the performance of three extreme value distributions, i.e., generalized Pareto distribution (GPD), generalized extreme value distribution (GEV), and Box‐Cox‐GEV, and four skewed fat‐tailed distributions, i.e., skewed generalized error distribution (SGED), skewed generalized t (SGT), exponential generalized beta of the second kind (EGB2), and inverse hyperbolic sign (IHS) in estimating conditional and unconditional value at risk (VaR) thresholds. The results provide strong evidence that the SGT, EGB2, and IHS distributions perform as well as the more specialized extreme value distributions in modeling the tail behavior of portfolio returns. All three distributions produce similar VaR thresholds and perform better than the SGED and the normal distribution in approximating the extreme tails of the return distribution. The conditional coverage and the out‐of‐sample performance tests show that the actual VaR thresholds are time varying to a degree not captured by unconditional VaR measures. In light of the fact that VaR type measures are employed in many different types of financial and insurance applications including the determination of capital requirements, capital reserves, the setting of insurance deductibles, the setting of reinsurance cedance levels, as well as the estimation of expected claims and expected losses, these results are important to financial managers, actuaries, and insurance practitioners.

A conditional-SGT-VaR approach with alternative GARCH models

This paper proposes a conditional technique for the estimation of VaR and expected shortfall measures based on the skewed generalized t (SGT) distribution. The estimation of the conditional mean and conditional variance of returns is based on ten popular variations of the GARCH model. The results indicate that the TS-GARCH and EGARCH models have the best overall performance. The remaining GARCH specifications, except in a few cases, produce acceptable results. An unconditional SGT-VaR performs well on an in-sample evaluation and fails the tests on an out-of-sample evaluation. The latter indicates the need to incorporate time-varying mean and volatility estimates in the computation of VaR and expected shortfall measures.

The empirical distribution of stock returns: evidence from an emerging European market

There is now substantial evidence that daily equity returns are not normally distributed, but instead display significant leptokurtosis and, in many cases, skewness. Considerable effort has been made in order to capture these empirical characteristics using a range of statistical distributions. However, the evidence to date is confined entirely to the returns of developed stock markets, and in particular to the USA. In this paper, the daily returns of a large emerging European stock market, Turkey, are modelled. Two very flexible families of distributions that have recently been introduced are employed: the exponential generalized beta (EGB) and the skewed generalized t (SGT). These distributions permit very diverse levels of skewness and kurtosis and, between them, nest most of the distribution previously considered in the literature.