Advertisement

Psychometrika

, Volume 65, Issue 2, pp 199–215 | Cite as

Continuous time state space modeling of panel data by means of sem

  • Johan H. L. OudEmail author
  • Robert A. R. G. Jansen
Article

Abstract

Maximum likelihood parameter estimation of the continuous time linear stochastic state space model is considered on the basis of largeN discrete time data using a structural equation modeling (SEM) program. Random subject effects are allowed to be part of the model. The exact discrete model (EDM) is employed which links the discrete time model parameters to the underlying continuous time model parameters by means of nonlinear restrictions. The EDM is generalized to cover not only time-invariant parameters but also the cases of stepwise time-varying (piecewise time-invariant) parameters and parameters varying continuously over time according to a general polynomial scheme. The identification of the continuous time parameters is discussed and an educational example is presented.

Key words

continuous time state space modeling exact discrete model linear stochastic differential equations longitudinal structural equation modeling panel analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arminger, G. (1986). Linear stochastic differential equations for panel data with unobserved variables. In N.B. Tuma (Ed.),Sociological Methodology (pp. 187–212). Washington: Jossey-Bass.Google Scholar
  2. Arminger, G., Wittenberg, J., & Schepers, A. (1996).MECOSA 3: Mean and covariance structure analysis. Friedrichsdorf, Germany: Additive.Google Scholar
  3. Arnold, L. (1974).Stochastic differential equations. New York: Wiley.Google Scholar
  4. Baltagi, B.H. (1995).Econometric analysis of panel data. Chichester: Wiley.Google Scholar
  5. Bergstrom, A.R. (1984). Continuous time stochastic models and issues of aggregation over time. In Z. Griliches & M.D. Intriligator (Eds.),Handbook of econometrics (Vol. 2, pp. 1145–1212). Amsterdam: North-Holland.Google Scholar
  6. Bergstrom, A.R. (1988). The history of continuous-time econometric models.Econometric Theory, 4, 365–383.Google Scholar
  7. Coleman, J.S. (1968). The mathematical study of change. In H.M. Blalock & A. Blalock (Eds.),Methodology in social research (pp. 428–478). New York: McGraw-Hill.Google Scholar
  8. Hamerle, A., Nagl, W., & Singer, H. (1991). Problems with the estimation of stochastic differential equations using structural equations models.Journal of Mathematical Sociology, 16, 201–220.Google Scholar
  9. 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.Google Scholar
  10. Hamilton, J.D. (1986). A standard error for the estimated state vector of a state-space model.Journal of Econometrics, 33, 387–397.Google Scholar
  11. Hansen, L., & Sargent, T.J. (1983). The dimensionality of the aliasing problem.Econometrica, 51, 377–388.Google Scholar
  12. Haughton, D.M.A., Oud, J.H.L., & Jansen, R.A.R.G. (1997). Information and other criteria in structural equation model selection.Communications in Statistics: Simulation and Computation, 26, 1477–1516.Google Scholar
  13. Jansen, R.A.R.G., & Oud, J.H.L. (1995). Longitudinal LISREL model estimation from incomplete panel data using the EM algorithm and the Kalman smoother.Statistica Neerlandica, 49, 362–377.Google Scholar
  14. Jazwinski, A.H. (1970).Stochastic processes and filtering theory. New York: Academic Press.Google Scholar
  15. Jones, R.H. (1984). Fitting multivariate models to unequally spaced data. In E. Parzen (Ed.),Time series analysis of irregularly observed data. Londen: Chapman and Hall.Google Scholar
  16. Jones, R.H. (1993).Longitudinal data with serial correlation: A state space approach. Londen: Chapman and Hall.Google Scholar
  17. Jöreskog, K.G. (1974). Analyzing psychological data by structural analysis of covariance matrices. In D.H. Krantz, R.C. Atkinson, R.D. Luce & P. Suppes (Eds.),Contemporary developments in mathematical psychology: Vol. II (pp. 1–56). San Francisco: Freeman.Google Scholar
  18. Jöreskog, K.G., & Sörbom, D. (1985). Simultaneous analysis of longitudinal data from several cohorts. In W.M. Mason & S.E. Fienberg (Eds.),Cohort analysis in social research: Beyond the identification problem (pp. 323–341). New York: Springer.Google Scholar
  19. Jöreskog, K.G., & Sörbom, D. (1989).LISREL 7: User's reference guide. Mooresville IN: Scientific Software.Google Scholar
  20. Jöreskog, K.G., & Sörbom, D. (1993).New features in LISREL. Chicago: Scientific Software International.Google Scholar
  21. Mommers, M.J.C., & Oud, J.H.L. (1992). Monitoring reading and spelling achievement. In L. Verhoeven & J.H.A.L. de Jong (Eds.),The construct of language proficiency (pp. 49–60). Amsterdam: John Benjamins.Google Scholar
  22. Neale, M.C. (1997).Mx: Statistical Modeling (4th ed.). Richmond, VA: Department of Psychiatry.Google Scholar
  23. Oud, J.H.L. (1978).Systeem-methodologie in sociaal-wetenschappelijk onderzoek [Systems methodology in social science research]. Doctoral dissertation. Nijmegen, The Netherlands: Alfa.Google Scholar
  24. Oud, J.H.L., & Jansen, R.A.R.G. (1996). Nonstationary longitudinal LISREL model estimation from incomplete panel data using EM and the Kalman smoother. In U. Engel & J. Reinecke, (Eds.),Analysis of change: Advanced techniques in panel data analysis (pp. 135–159). New York: de Gruyter.Google Scholar
  25. Oud, J.H.L., van den Bercken, J.H.L., & Essers, R.J. (1990). Longitudinal factor score estimation using the Kalman filter.Applied Psychological Measurement, 14, 395–418.Google Scholar
  26. Oud, J.H.L., van Leeuwe, J.F.J., & Jansen, R.A.R.G. (1993). Kalman filtering in discrete and continuous time based on longitudinal LISREL models. In J. H. L. Oud & A. W. van Blokland-Vogelesang (Eds.),Advances in longitudinal and multivariate analysis in the behavioral sciences (pp. 3–26). Nijmegen: ITS.Google Scholar
  27. Phillips, P.C.B. (1973). The problem of identification in finite parameter continuous time models. In A.R. Bergstrom (Ed.),Statistical inference in continuous time models (pp. 135–173). Amsterdam: North-Holland.Google Scholar
  28. Raftery, A.E. (1993). Bayesian model selection in structural equation models. In K.A. Bollen & J.S. Long (Eds.),Testing structural equation models. Newbury Park CA: Sage.Google Scholar
  29. Ruymgaart, P.A., & Soong, T.T. (1985).Mathematics of Kalman-Bucy filtering. Berlin: Springer.Google Scholar
  30. Schwarz, G. (1978). Estimating the dimension of a model.Annals of Statistics, 6, 461–464.Google Scholar
  31. 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: Author.Google Scholar
  32. Singer, H. (1993). Continuous-time dynamical systems with sampled data, errors of measurement and unobserved components.Journal of Time Series Analysis, 14, 527–545.Google Scholar
  33. Singer, H. (1995). Analytic score function for irregularly sampled continuous time stochastic processes with control variables and missing values.Econometric Theory, 11, 721–735.Google Scholar
  34. Singer, H. (1998). Continuous panel models with time dependent parameters.Journal of Mathematical Sociology, 23, 77–98.Google Scholar
  35. Tuma, N.B., & Hannan, M. (1984).Social dynamics. New York: Academic Press.Google Scholar

Copyright information

© The Psychometric Society 2000

Authors and Affiliations

  1. 1.Institute of Special educationUniversity of NijmegenNijmegenThe Netherlands

Personalised recommendations