A New Generalization of the Student’s t Distribution with an Application in Quantile Regression

In this work, we present a new generalization of the student’s t distribution. The new distribution is obtained by the quotient of two independent random variables. This quotient consists of a standard Normal distribution divided by the power of a chi square distribution divided by its degrees of freedom. Thus, the new symmetric distribution has heavier tails than the student’s t distribution and extensions of the slash distribution. We develop a procedure to use quantile regression where the response variable or the residuals have high kurtosis. We give the density function expressed by an integral, we obtain some important properties and some useful procedures for making inference, such as moment and maximum likelihood estimators. By way of illustration, we carry out two applications using real data, in the first we provide maximum likelihood estimates for the parameters of the generalized student’s t distribution, student’s t, the extended slash distribution, the modified slash distribution, the slash distribution generalized student’s t test, and the double slash distribution, in the second we perform quantile regression to fit a model where the response variable presents a high kurtosis.

Value at Risk estimation using GAS models with heavy tailed distributions for cryptocurrencies

Risk management and prediction of market losses of cryptocurrencies are of notable value to risk managers, portfolio managers, financial market researchers and academics. One of the most common measures of an asset’s risk is Value-at-Risk (VaR). This paper evaluates and compares the performance of generalized autoregressive score (GAS) combined with heavy-tailed distributions, in estimating the VaR of two well-known cryptocurrencies’ returns, namely Bitcoin returns and Ethereum returns. In this paper, we proposed a VaR model for Bitcoin and Ethereum returns, namely the GAS model combined with the generalized lambda distribution (GLD), referred to as the GAS-GLD model. The relative performance of the GAS-GLD models was compared to the models proposed by Troster et al. (2018), in other words, GAS models combined with asymmetric Laplace distribution (ALD), the asymmetric Student’s t-distribution (AST) and the skew Student’s t-distribution (SSTD). The Kupiec likelihood ratio test was used to assess the adequacy of the proposed models. The principal findings suggest that the GAS models with heavy-tailed innovation distributions are, in fact, appropriate for modelling cryptocurrency returns, with the GAS-GLD being the most adequate for the Bitcoin returns at various VaR levels, and both GAS-SSTD, GAS-ALD and GAS-GLD models being the most appropriate for the Ethereum returns at the VaR levels used in this study.

UTILIZATION OF STUDENT’S T DISTRIBUTION TO HANDLE OUTLIERS IN TECHNICAL EFFICIENCY MEASUREMENT

Stochastic frontier analysis (SFA) is the favorite method for measuring technical efficiency. SFA decomposes the error term into noise and inefficiency components. The noise component is generally assumed to have a normal distribution, while the inefficiency component is assumed to have half normal distribution. However, in the presence of outliers, the normality assumption of noise is not sufficient and can produce implausible technical efficiency scores. This paper aims to explore the use of Student’s t distribution for handling outliers in technical efficiency measurement. The model was applied in paddy rice production in East Java. Output variable was the quantity of production, while the input variables were land, seed, fertilizer, labor and capital. To link the output and inputs, Cobb-Douglas or Translog production functions was chosen using likelihood ratio test, where the parameters were estimated using maximum simulated likelihood. Furthermore, the technical efficiency scores were calculated using Jondrow method. The results showed that Student’s t distribution for noise can reduce the outliers in technical efficiency scores. Student’s t distribution revised the extremely high technical efficiency scores downward and the extremely low technical efficiency scores upward. The performance of model was improved after the outliers were handled, indicated by smaller AIC value.