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

Julia Adamska, Łukasz Bielak, Joanna Janczura and Agnieszka Wyłomańska ()
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Julia Adamska: Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
Łukasz Bielak: KGHM, M. Skłodowskiej-Curie 48, 59-301 Lubin, Poland
Agnieszka Wyłomańska: Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland

Mathematics, 2022, vol. 10, issue 18, 1-29

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.

Keywords: multivariate random variables; product; Pareto distribution; Student’s t distribution; heavy tails; transaction value; electricity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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