The Pearson diffusions: A class of statistically tractable diffusion processes

Michael Sørensen () and Julie Lyng Forman ()
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Julie Lyng Forman: School of Economics and Management, University of Aarhus, Denmark and CREATES, Postal: 8000 Aarhus C, Denmark

CREATES Research Papers from Department of Economics and Business Economics, Aarhus University

Abstract: The Pearson diffusions is a flexible class of diffusions defined by having linear drift and quadratic squared diffusion coefficient. It is demonstrated that for this class explicit statistical inference is feasible. Explicit optimal martingale estimating func- tions are found, and the corresponding estimators are shown to be consistent and asymptotically normal. The discussion covers GMM, quasi-likelihood, and non- linear weighted least squares estimation too, and it is discussed how explicit likeli- hood or approximate likelihood inference is possible for the Pearson diffusions. A complete model classification is presented for the ergodic Pearson diffusions. The class of stationary distributions equals the full Pearson system of distributions. Well-known instances are the Ornstein-Uhlenbeck processes and the square root (CIR) processes. Also diffusions with heavy-tailed and skew marginals are included. Special attention is given to a skew t-type distribution. Explicit formulae for the conditional moments and the polynomial eigenfunctions are derived. The analyti- cal tractability is inherited by transformed Pearson diffusions, integrated Pearson diffusions, sums of Pearson diffusions, and stochastic volatility models with Pearson volatility process. For the non-Markov models explicit optimal prediction based estimating functions are found and shown to yield consistent and asymptotically normal estimators.

Keywords: eigenfunction; ergodic diffusion; integrated diffusion; martingale estimating function; likelihood inference; mixing; optimal estimating function; Pearson system; prediction based estimating function; quasi likelihood; spectral methods; stochastic differential equation; stochastic volatility (search for similar items in EconPapers)
JEL-codes: C22 C51 (search for similar items in EconPapers)
Pages: 32
Date: 2007-09-27
New Economics Papers: this item is included in nep-ecm, nep-ets, nep-ict and nep-ore
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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Related works:
Journal Article: The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes (2008)
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