GMM Estimation of Continuous-Time Bilinear Processes

  • Abdelouahab Bibi Larbi Ben M’hidi University
  • Fateh Merahi Abbes Laghrour University
Keywords: Continuous-time bilinear processes, Evolutionary transfer functions, Spectral representation, GMM estimation.


This paper examines the moments properties in frequency domain of the class of first order continuous-timebilinear processes (COBL(1,1) for short) with time-varying (resp. time-invariant) coefficients. So, we used theassociated evolutionary (or time-varying) transfer functions to study the structure of second-order of the process and its powers. In particular, for time-invariant case, an expression of the moments of any order are showed and the continuous-time AR (CAR) representation of COBL(1,1) is given as well as some moments properties of special cases. Based on these results we are able to estimate the unknown parameters involved in model via the so-called generalized method of moments (GMM) illustrated by a Monte Carlo study and applied to modelling two foreign exchange rates of Algerian Dinar against U.S-Dollar (USD/DZD) and against the single European currency Euro (EUR/DZD).


A. R. Bergstrom, Continuous Time Econometric Modelling, Advanced Texts In Econometrics, Oxford University Press, Oxford, 1990.

A. Bibi, Evolutionary Transfer Functions of Bilinear Processes with Time-Varying Coefficients, Computers and Mathematics with Applications, vol. 52, pp. 331–338, 2006.

A. Bibi, and F. Merahi, A note on L2-structure of continuous-time bilinear processes with time-varying coefficients, International Journal of Statistics and Probability, vol. 4, no. 3, pp. 150–160, 2015.

A. Bibi, and F. Merahi, Moment method estimation of first-order continuous-time bilinear processes, Communications in Statistics-Simulation and Computation, vol. 48, no. 4, pp. 1070–1087, 2019.

T. Bollerslev, and H. Zhoub, Estimating stochastic volatility diffusion using conditional moments of integrated volatility, Journal of Econometrics, vol. 109, pp. 33–65, 2002.

M. Carrasco, and J-P. Florens. Simulation-Based Method of Moments and Efficiency, Journal of Business & Economic Statistics, vol. 20, no. 4, pp. 482–492, 2002.

K. C. Chan, G. A. Karolyi, F. A. Longstaff, and A. B. Sanders, An Empirical Comparison of Alternative Models of the Short-Term Interest Rate, Journal of finance, vol. 47, no. 3, pp. 1209–1227, 1992.

S. Haug, C. Klüppelberg, A. Lindner, and M. Zapp, Method of moment estimation in the COGARCH(1,1) model, Econometrics Journal, vol. 10, no. 2, pp. 320–341, 2007.

L. P. Hansen, Large Sample Properties of Generalized Method of Moments Estimators, Econometrica, vol. 50, no. 4, pp. 1029–1054, 1982.

J. Hlouskova, and L. Sögner, GMM Estimation of Affine Term Structure Models, Econometrics and Statistics, vol. 13, pp. 2–15, 2020.

E. Iglói, and G. Terdik, Bilinear stochastic systems with fractional Brownian motion input, The Annals of Applied Probability, vol. 9, no. 1, pp. 46–77, 1999.

J. Kallsen, and J. Muhle-Karbe, Method of moment estimation in time-changed Lévy models, Statistics & Decisions vol. 28, no. 2,pp. 169–194, 2011.

W. G. Kelley, and A. C. Peterson The theory of differential equations, Springer-Verlag, New York, 2010.

M. Kessler, Estimation of an ergodic diffusion from discrete observations, Scand. J. Statistics, vol. 24, no. 2, pp. 211–229, 1997.

Y. A. Kutoyants, Statistical inference for ergodic diffusion processes, Springer-Verlag, London, 2004.

A. Le Breton, and M. Musiela, A study of one-dimensional bilinear differential model for stochastic processes, probability and mathematical statistics, vol. 4, no. 1, pp. 91–107, 1984.

P. Major, Multiple Wiener-Itˆ o integrals, Lecture Notes in Mathematics 849, Springer-Verlag, New York, 1981.

B. ∅Ksendal, Stochastic Differential Equations: An Introduction with Applications (Sixth Edition), Springer-Verlag, Berlin, 2003.

B. L. S. Prakasa Rao, Statistical Inference for Fractional Diffusion Processes, Wiley, 2010.

H. Zhou, Finite sample properties of EMM, GMM, QMLE and MLE for a square-root interest rate diffusion model, Journal of Computational Finance, vol. 5, no. 2, pp. 89–122, 2001.

How to Cite
Bibi, A., & Merahi, F. (2020). GMM Estimation of Continuous-Time Bilinear Processes. Statistics, Optimization & Information Computing, 9(4), 990-1009.
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