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Analytical Score Function for Irregularly Sampled Continuous Time Stochastic Processes with Control Variables and Missing Values

Published online by Cambridge University Press:  11 February 2009

Hermann Singer
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Abstract

The unknown structural parameters of a continuous/discrete state space model are estimated by maximum likelihood in the presence of irregular sampling, missing values, and cross-sections of time series (panel data). Exogenous (control) variables are included, and the sampling scheme and missing data pattern can be different for each variable and system. Furthermore, the derived non-linear optimization algorithm with analytical score function can be used for the discrete time case as well.

Type
Research Article
Information
Econometric Theory , Volume 11 , Issue 4 , August 1995 , pp. 721 - 735
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

Anderson, S.J., Jones, R.H., & Swanson, G.D. (1990) Smoothing polynomial splines for bivariate data. SIAM Journal on Scientific and Statistical Computation 11 (4), 749–766.CrossRefGoogle Scholar
Arnold, L. (1974) Stochastic Differential Equations. New York: Wiley.Google Scholar
Bergstrom, A.R. (ed.) (1976) Statistical Inference in Continuous Time Models. Amsterdam: North-Holland.Google Scholar
Campillo, F. & Le Gland, F. (1989) MLE for partially observed diffusions: Direct maximization vs. the EM algorithm. Stochastic Processes and Their Applications 33, 245–274.CrossRefGoogle Scholar
Dacunha-Castelle, D. & Florens-Zmirou, D. (1986) Estimation of the coefficients of a diffusion from discrete observations. Stochastics 19, 263–284.CrossRefGoogle Scholar
de Jong, P. (1989) Smoothing and interpolation with the state-space model. Journal of the American Statistical Association 84, 1085–1088.CrossRefGoogle Scholar
de Jong, P. & Mackinnon, M.J. (1988) Covariances for smoothed estimates in state space models. Biometrika 75, 601–602.CrossRefGoogle Scholar
Dembo, A. & Zeitouni, O. (1986) Parameter estimation of partially observed continuous time stochastic processes via the EM algorithm. Stochastic Processes and Their Applications 23, 91–113.CrossRefGoogle Scholar
Dempster, A.P., Laird, N.M., & Rubin, D.B. (1977) Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society B 39, 1–38.Google Scholar
Dennis, J.E. Jr., & Schnabel, R.B. (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Englewood Cliffs, New Jersey: Prentice-Hall.Google Scholar
Engle, R.F. & Watson, M.W. (1981) A one-factor multivariate time series model of metropolitan wage rates. Journal of the American Statistical Association 76, 774–781.CrossRefGoogle Scholar
Fahrmeir, L. (1976) Approximation von stochastischen Differentialgleichungen auf Digital- und Hybridrechnern. Computing 16, 359–371.CrossRefGoogle Scholar
Florens-Zmirou, D. (1989) Approximate discrete-time schemes for statistics of diffusion processes. Statistics 20, 547–557.CrossRefGoogle Scholar
Genon-Catalot, V. (1990) Maximum contrast estimation for diffusion processes from discrete observations. Statistics 21 (1), 99–116.CrossRefGoogle Scholar
Goodrich, R.L. & Caines, P.E. (1979) Linear system identification from non-stationary cross sectional data. IEEE Transactions on Automatic Control AC-24 (1), 125–127.CrossRefGoogle Scholar
Hamerle, A., Singer, H., & Nagl, W. (1993) Identification and estimation of continuous time dynamic systems with exogenous variables using panel data. Econometric Theory 9, 283–295.CrossRefGoogle Scholar
Harvey, A.C. & Stock, J. (1985) The estimation of higher order continuous time autoregressive models. Econometric Theory 1, 97–112.CrossRefGoogle Scholar
Jazwinski, A.H. (1970) Stochastic Processes and Filtering Theory. New York: Academic Press.Google Scholar
Jennrich, R.I. & Bright, P.B. (1976) Fitting systems of linear differential equations using computer generated exact derivatives. Technometrics 18, 385–392.CrossRefGoogle Scholar
Jones, R.H. (1984) Fitting multivariate models to unequally spaced data. In Parzen, E. (ed.), Time Series Analysis of Irregularly Observed Data, pp. 158–188. New York: Springer.CrossRefGoogle Scholar
Jones, R.H. & Ackerson, L.M. (1990) Serial correlation in unequally spaced longitudinal data. Biometrika 77, 721–731.CrossRefGoogle Scholar
Koopman, S.J. (1993) Disturbance smoother for state space models. Biometrika 80, 117–126.CrossRefGoogle Scholar
Koopman, S.J. & Shephard, N. (1992) Exact score for time series models in state space form. Biometrika 79, 823–826.Google Scholar
Le Breton, A. (1976) On continuous and discrete sampling for parameter estimation in diffusion type processes. Mathematical Programming Study 5, 124–144.CrossRefGoogle Scholar
Liptser, R.S. & Shiryayev, A.N. (1977, 1978) Statistics of Random Processes, vols. I and II. New York, Heidelberg, Berlin: Springer.CrossRefGoogle Scholar
Louis, T.A. (1982) Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Association B 44, 226–233.Google Scholar
McDonald, R.P. & Swaminathan, H. (1973) A simple matrix calculus with applications to multivariate statistics. General Systems XVIII, 37–54.Google Scholar
Nowman, K.B. (1991) Open higher order continuous-time dynamic model with mixed stock and flow data and derivatives of the exogenous variables. Econometric Theory 7, 404–408.CrossRefGoogle Scholar
Parzen, E. (ed.) (1984) Time Series Analysis of Irregularly Observed Data. New York: Springer.CrossRefGoogle Scholar
Phillips, P.C.B. (1976) The estimation of linear stochastic differential equations with exogenous variables. In Bergstrom, A.R. (ed.), Statistical Inference in Continuous Time Models, pp. 135–173. Amsterdam: North-Holland.Google Scholar
Rauch, H.E., Tung, F., & Striebel, C.T. (1965) Maximum likelihood estimates of linear dynamic systems. AIAA Journal 3, 1445–1450.CrossRefGoogle Scholar
Robinson, P.M. (1977) Estimation of a time series model from unequally spaced data. Stochastic Processes and Their Applications 6, 9–24.CrossRefGoogle Scholar
Rümelin, W. (1982) Numerical treatment on stochastic differential equations. SIAM Journal of Numerical Analysis 19, 604–613.CrossRefGoogle Scholar
SAS Institute Inc. (1989) SAS/IML Software: Usage and Reference, version 6. Cary, North Carolina: Author.Google Scholar
SAS Institute Inc. (1990) SAS Procedures Guide, version 6. Cary, North Carolina: Author.Google Scholar
Singer, H. (1990) Parameterscädtzung in zeitkontinuierlichen dynamischen Systemen. Constance, Germany: Hartung-Gorre-Verlag.Google Scholar
Singer, H. (1991) LSDE – A Program Package for the Simulation, Graphical Display, Optimal Filtering and Maximum Likelihood Estimation of Linear Stochastic Differential Equations, User's Guide. Meersburg, Germany: Author.Google Scholar
Singer, H. (1993) Continuous time dynamical systems with sampled data, errors of measurement and unobserved components. Journal of Time Series Analysis 14, 527–545.CrossRefGoogle Scholar
Watson, M.W. & Engle, R.F. (1983) Alternative algorithms for the estimation of dynamic factor, mimic and varying coefficient regression models. Journal of Econometrics 23, 385–400.CrossRefGoogle Scholar
Wilcox, R.M. (1967) Exponential operators and parameter differentiation in quantum physics. Journal of Mathematical Physics 8, 962–982.CrossRefGoogle Scholar
Zadrozny, P. (1988) Gaussian likelihood of continuous-time ARMAX models when data are stocks and flows at different frequencies. Econometric Theory 4, 108–124.CrossRefGoogle Scholar