Parameter estimation for stable distributions with application to commodity futures log returns

📝 Abstract

This paper explores the theory behind the rich and robust family of {\alpha}-stable distributions to estimate parameters from financial asset log-returns data. We discuss four-parameter estimation methods including the quantiles, logarithmic moments method, maximum likelihood (ML), and the empirical characteristics function (ECF) method. The contribution of the paper is two-fold: first, we discuss the above parametric approaches and investigate their performance through error analysis. Moreover, we argue that the ECF performs better than the ML over a wide range of shape parameter values, {\alpha}{\alpha} including values closest to 0 and 2 and that the ECF has a better convergence rate than the ML. Secondly, we compare the t location-scale distribution to the general stable distribution and show that the former fails to capture skewness which might exist in the data. This is observed through applying the ECF to commodity futures log-returns data to obtain the skewness parameter.

💡 Analysis

This paper explores the theory behind the rich and robust family of {\alpha}-stable distributions to estimate parameters from financial asset log-returns data. We discuss four-parameter estimation methods including the quantiles, logarithmic moments method, maximum likelihood (ML), and the empirical characteristics function (ECF) method. The contribution of the paper is two-fold: first, we discuss the above parametric approaches and investigate their performance through error analysis. Moreover, we argue that the ECF performs better than the ML over a wide range of shape parameter values, {\alpha}{\alpha} including values closest to 0 and 2 and that the ECF has a better convergence rate than the ML. Secondly, we compare the t location-scale distribution to the general stable distribution and show that the former fails to capture skewness which might exist in the data. This is observed through applying the ECF to commodity futures log-returns data to obtain the skewness parameter.

📄 Content

Page 1 of 28 ECONOMETRICS | RESEARCH ARTICLE Parameter estimation for stable distributions with application to commodity futures
log-returns M. Kateregga, S. Mataramvura and D. Taylor Cogent Economics & Finance (2017), 5: 1318813 Kateregga et al., Cogent Economics & Finance (2017), 5: 1318813 https://doi.org/10.1080/23322039.2017.1318813 ECONOMETRICS | RESEARCH ARTICLE Parameter estimation for stable distributions with application to commodity futures log-returns M. Kateregga1*, S. Mataramvura1 and D. Taylor1 Abstract: This paper explores the theory behind the rich and robust family of
훼-stable distributions to estimate parameters from financial asset log-returns data. We discuss four-parameter estimation methods including the quantiles, logarithmic moments method, maximum likelihood (ML), and the empirical characteristics function (ECF) method. The contribution of the paper is two-fold: first, we discuss the above parametric approaches and investigate their performance through error analysis. Moreover, we argue that the ECF performs better than the ML over a wide range of shape parameter values, 훼 including values closest to 0 and 2 and that the ECF has a better convergence rate than the ML. Secondly, we compare the t location-scale distribution to the general stable distribution and show that the former fails to capture skewness which might exist in the data. This is observed through applying the ECF to commodity futures log-returns data to obtain the skewness parameter. Subjects: Mathematical Finance; Probability; Statistics Keywords: stable distribution; parameter estimation; density estimation AMS subject classifications: 62G05; 62G07; 62G32 *Corresponding author: M. Kateregga, Actuarial Science, University of Cape Town, Rondebosch, Cape Town 7700, South Africa E-mail: michaelk@aims.ac.za Reviewing editor: Xibin Zhang, Monash University, Australia Additional information is available at the end of the article ABOUT THE AUTHOR Mr M. Kateregga is a finishing PhD student at the University of Cape Town in South Africa. His research is in the field of mathematical finance and his PhD thesis is entitled Stable Distributions with Applications in Finance. The current paper is a chapter in his thesis which is due for submission in August, 2017. Mr Kateregga is also a researcher at the African Collaboration for Quantitative Finance and Risk Research (ACQuFRR) which is the research section of the African Institute of Financial Markets and Risk Management (AIFMRM), which delivers postgraduate education and training in financial markets, risk management and quantitative finance. Mr Kateregga also works with the African Institute for Mathematical Sciences (AIMS) in South Africa as a Research Assistant. PUBLIC INTEREST STATEMENT This paper is entitled parameter estimation for stable distribution with applications to commodity future log-returns. The paper is useful to individuals interested in investing their wealth in financial markets. It provides essential information on how historical asset prices can inform future market movements via parameter estimation. This is crucial to portfolio managers, speculators, and hedgers. It’s imperative that the most accurate estimation method is established. Market data distribution deviates from the normal distribution, it exhibits skews, high or low peaks, and fat or skinny tails. The current paper is geared towards establishing the best estimation method among known methods in economic and financial analysis for skewed data. Received: 22 December 2016 Accepted: 02 April 2017 © 2017 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. Page 2 of 28 M. Kateregga Page 3 of 28 Kateregga et al., Cogent Economics & Finance (2017), 5: 1318813 https://doi.org/10.1080/23322039.2017.1318813

1. Introduction The motivation for this paper derives from the fact that parameter estimation from historical data is an important analysis to financial market participants. It provides useful information for portfolio managers, speculators, and hedgers. It is therefore, imperative that the most accurate estimation method is established. It is a known fact that in general, market data deviates from the Gaussian distribution, its distribution is either skewed, high or low peaked, and/or with fat or skinny tails. The current paper is geared towards establishing a better parameter estimation method among the commonly known ECF, ML, quantile, and logarithm moments methods used in economic and finan- cial analysis for skewed data assumed to flow stable distributions. The application of stable distributions in finance is traced way back in the late 50s when Mandelbrot (1959, 1962, 1963) developed a hypothesis that revolutionalized the way economists viewed and interpreted prices in speculative markets such as grains and securities markets. The hypothesis sug- gested that prices were no

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