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Inference for systems of stochastic differential equations from discretely sampled data: a numerical maximum likelihood approach

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  • Thomas Lux

Abstract

Maximum likelihood estimation of discretely observed diffusion processes is mostly hampered by the lack of a closed form solution of the transient density. It has recently been argued that a most generic remedy to this problem is the numerical solution of the pertinent Fokker–Planck (FP) or forward Kolmogorov equation. Here we expand existing work on univariate diffusions to higher dimensions. We find that in the bivariate and trivariate cases, a numerical solution of the FP equation via alternating direction finite difference schemes yields results surprisingly close to exact maximum likelihood in a number of test cases. After providing evidence for the efficiency of such a numerical approach, we illustrate its application for the estimation of a joint system of short-run and medium-run investor sentiment and asset price dynamics using German stock market data. Copyright Springer-Verlag Berlin Heidelberg 2013

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  • Thomas Lux, 2013. "Inference for systems of stochastic differential equations from discretely sampled data: a numerical maximum likelihood approach," Annals of Finance, Springer, vol. 9(2), pages 217-248, May.
    • Handle: RePEc:kap:annfin:v:9:y:2013:i:2:p:217-248
    DOI: 10.1007/s10436-012-0219-9
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    References listed on IDEAS

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    1. Lux, Thomas, 2009. "Rational forecasts or social opinion dynamics? Identification of interaction effects in a business climate survey," Journal of Economic Behavior & Organization, Elsevier, vol. 72(2), pages 638-655, November.
    2. A. S. Hurn & J. I. Jeisman & K. A. Lindsay, 0. "Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations," Journal of Financial Econometrics, Oxford University Press, vol. 5(3), pages 390-455.
    3. Creedy, John & Lye, Jenny & Martin, Vance L, 1996. "A Non-linear Model of the Real US-UK Exchange Rate," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 11(6), pages 669-686, Nov.-Dec..
    4. Lux, Thomas, 2012. "Estimation of an agent-based model of investor sentiment formation in financial markets," Journal of Economic Dynamics and Control, Elsevier, vol. 36(8), pages 1284-1302.
    5. Thomas Lux, 2009. "Rational Forecasts or Social Opinion Dynamics? Identification of Interaction Effects in a Business Climate Survey," Post-Print hal-00720175, HAL.
    6. Rheinlaender Thorsten & Steinkamp Marcus, 2004. "A Stochastic Version of Zeeman's Market Model," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 8(4), pages 1-25, December.
    7. Thomas Lux, 2011. "Sentiment dynamics and stock returns: the case of the German stock market," Empirical Economics, Springer, vol. 41(3), pages 663-679, December.
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    Cited by:

    1. Shi, Yong & Tang, Ye-ran & Long, Wen, 2019. "Sentiment contagion analysis of interacting investors: Evidence from China’s stock forum," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 246-259.

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    More about this item

    Keywords

    Stochastic differential equations; Numerical maximum likelihood; Fokker–Planck equation; Finite difference schemes; Asset pricing; C58; G12; C13;
    All these keywords.

    JEL classification:

    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General

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