From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t-Distribution Case

Abstract

Multivariate modelling of economics data is crucial for risk and profit analyses in companies. However, for the final conclusions, a whole set of variables is usually transformed into a single variable describing a total profit/balance of company’s cash flows. One of the possible transformations is based on the product of market variables. Thus, in this paper, we study the distribution of products of Pareto or Student’s t random variables that are ubiquitous in various risk factors analysis. We review known formulas for the probability density functions and derive their explicit forms for the products of Pareto and Gaussian or log-normal random variables. We also study how the Pareto or Student’s t random variable influences the asymptotic tail behaviour of the distribution of their product with the Gaussian or log-normal random variables and discuss how the dependency between the marginal random variables of the same type influences the probabilistic properties of the final product. The theoretical results are then applied for an analysis of the distribution of transaction values, being a product of prices and volumes, from a continuous trade on the German intraday electricity market.

Integrating a Pareto-Distributed Scale into the Mixed Logit Model: A Mathematical Concept

“…Inspired by Adamska et al [22], who examined the distribution of the product of the lognormal or normal distribution and the Pareto distribution, we propose the integration of a Pareto-distributed scale that can appropriately mitigate (1)-(4) of the CL issues and has behaviorally reasonable interpretability into the MIXL model. We refer to it as a mixed logit with integrated Pareto-distributed scale (MIXL-iPS).…”