The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes

34 Pages Posted: 23 Jun 2008

See all articles by Michael Sorensen

Michael Sorensen

University of Copenhagen - Institute for Mathematical Sciences; University of Aarhus

Julie Lyng Forman

affiliation not provided to SSRN

Margit Sommer

School of Economics and Management, University of Aarhus; School of Economics and Management; CREATES Research Papers

Date Written: September 27, 2007

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 functions 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 likelihood 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 formula for the conditional moments and the polynomial eigenfunctions are derived. The analytical 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

JEL Classification: C22, C51

Suggested Citation

Sorensen, Michael and Forman, Julie Lyng and Sommer, Margit, The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes (September 27, 2007). Available at SSRN: https://ssrn.com/abstract=1150110 or http://dx.doi.org/10.2139/ssrn.1150110

Michael Sorensen

University of Copenhagen - Institute for Mathematical Sciences ( email )

Universitetsparken 5
DK-2100 Copenhagen, DK - 2200
Denmark
+45 3532 0899 (Phone)
+45 3532 0772 (Fax)

University of Aarhus ( email )

DK-8000 Aarhus C
Denmark

Julie Lyng Forman

affiliation not provided to SSRN

Margit Sommer (Contact Author)

School of Economics and Management, University of Aarhus ( email )

School of Economics and Management
Building 1322, Bartholins Alle 10
DK-8000 Aarhus C
Denmark

School of Economics and Management ( email )

Building 350
DK-8000 Aarhus C
Denmark

CREATES Research Papers ( email )

School of Economics and Management
Building 1322, Bartholins Alle 10
DK-8000 Aarhus C
Denmark

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