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

Author

Listed:
  • 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)

  • Joanna Janczura

    (Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland)

  • Agnieszka Wyłomańska

    (Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland)

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.

Suggested Citation

  • Julia Adamska & Łukasz Bielak & Joanna Janczura & Agnieszka Wyłomańska, 2022. "From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t -Distribution Case," Mathematics, MDPI, vol. 10(18), pages 1-29, September.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:18:p:3371-:d:916795
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    References listed on IDEAS

    as
    1. Nadarajah, Saralees & Kotz, Samuel, 2006. "On The Product And Ratio Of Gamma And Weibull Random Variables," Econometric Theory, Cambridge University Press, vol. 22(2), pages 338-344, April.
    2. Yongning Wang & Ruey S. Tsay, 2013. "On Diagnostic Checking of Vector ARMA-GARCH Models with Gaussian and Student-t Innovations," Econometrics, MDPI, vol. 1(1), pages 1-31, April.
    3. Chunli Huang & Xu Zhao & Weihu Cheng & Qingqing Ji & Qiao Duan & Yufei Han, 2022. "Statistical Inference of Dynamic Conditional Generalized Pareto Distribution with Weather and Air Quality Factors," Mathematics, MDPI, vol. 10(9), pages 1-25, April.
    4. Kan, Raymond, 2008. "From moments of sum to moments of product," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 542-554, March.
    5. Deepesh Bhati & Enrique Calderín-Ojeda & Mareeswaran Meenakshi, 2019. "A New Heavy Tailed Class of Distributions Which Includes the Pareto," Risks, MDPI, vol. 7(4), pages 1-17, September.
    6. Saralees Nadarajah, 2008. "On the product of generalized Pareto random variables," Applied Economics Letters, Taylor & Francis Journals, vol. 15(4), pages 253-259.
    7. Cline, D. B. H. & Samorodnitsky, G., 1994. "Subexponentiality of the product of independent random variables," Stochastic Processes and their Applications, Elsevier, vol. 49(1), pages 75-98, January.
    8. Rui Li & Saralees Nadarajah, 2020. "A review of Student’s t distribution and its generalizations," Empirical Economics, Springer, vol. 58(3), pages 1461-1490, March.
    9. Mridula Garg & Ajay Sharma & Pratibha Manohar, 2016. "The distribution of the product of two independent generalized trapezoidal random variables," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(21), pages 6369-6384, November.
    10. T. Pham-Gia & N. Turkkan, 2002. "The product and quotient of general beta distributions," Statistical Papers, Springer, vol. 43(4), pages 537-550, October.
    11. Yusuke Uchiyama & Kei Nakagawa, 2020. "TPLVM: Portfolio Construction by Student's $t$-process Latent Variable Model," Papers 2002.06243, arXiv.org.
    12. Oh Kang Kwon & Stephen Satchell, 2020. "The Distribution of Cross Sectional Momentum Returns When Underlying Asset Returns Are Student’s t Distributed," JRFM, MDPI, vol. 13(2), pages 1-19, February.
    13. Chen Li & Xiaohu Li, 2018. "On the Optimal Risk Sharing in Reinsurance with Random Recovery Rate," Risks, MDPI, vol. 6(4), pages 1-16, October.
    14. Sooie-Hoe Loke & Enrique Thomann, 2018. "Numerical Ruin Probability in the Dual Risk Model with Risk-Free Investments," Risks, MDPI, vol. 6(4), pages 1-13, October.
    15. Hamedani, G. G. & Walter, G. G., 1985. "On the product of symmetric random variables," Statistics & Probability Letters, Elsevier, vol. 3(5), pages 251-253, September.
    16. Yusuke Uchiyama & Kei Nakagawa, 2020. "TPLVM: Portfolio Construction by Student’s t -Process Latent Variable Model," Mathematics, MDPI, vol. 8(3), pages 1-10, March.
    17. Bielak, Łukasz & Grzesiek, Aleksandra & Janczura, Joanna & Wyłomańska, Agnieszka, 2021. "Market risk factors analysis for an international mining company. Multi-dimensional, heavy-tailed-based modelling," Resources Policy, Elsevier, vol. 74(C).
    18. White Halbert & Granger Clive W.J., 2011. "Consideration of Trends in Time Series," Journal of Time Series Econometrics, De Gruyter, vol. 3(1), pages 1-40, February.
    19. Nadarajah, Saralees, 2005. "On the product XY for some elliptically symmetric distributions," Statistics & Probability Letters, Elsevier, vol. 75(1), pages 67-75, November.
    20. Mahmoudi, Eisa, 2011. "The beta generalized Pareto distribution with application to lifetime data," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(11), pages 2414-2430.
    21. Saralees Nadarajah, 2012. "Exact Distribution of the Product of N Student’s t RVs," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 997-1009, December.
    22. Wecker, William E., 1978. "A note on the time series which is the product of two stationary time series," Stochastic Processes and their Applications, Elsevier, vol. 8(2), pages 153-157, December.
    23. Tang, Jen & Gupta, A. K., 1984. "On the distribution of the product of independent beta random variables," Statistics & Probability Letters, Elsevier, vol. 2(3), pages 165-168, May.
    24. Glen, Andrew G. & Leemis, Lawrence M. & Drew, John H., 2004. "Computing the distribution of the product of two continuous random variables," Computational Statistics & Data Analysis, Elsevier, vol. 44(3), pages 451-464, January.
    25. Saralees Nadarajah, 2009. "The product t density distribution arising from the product of two Student’s t PDFs," Statistical Papers, Springer, vol. 50(3), pages 605-615, June.
    26. Rafal Weron, 2006. "Modeling and Forecasting Electricity Loads and Prices: A Statistical Approach," HSC Books, Hugo Steinhaus Center, Wroclaw University of Science and Technology, number hsbook0601, December.
    27. Saralees Nadarajah & Samuel Kotz*, 2005. "On the Product and Ratio of Gamma and Beta Random Variables," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 89(4), pages 435-449, November.
    28. Sel Ly & Kim-Hung Pho & Sal Ly & Wing-Keung Wong, 2019. "Determining Distribution for the Product of Random Variables by Using Copulas," Risks, MDPI, vol. 7(1), pages 1-20, February.
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