Abstract
A dynamic factor model is proposed for the analysis of multivariate nonstationary time series in the time domain. The nonstationarity in the series is represented by a linear time dependent mean function. This mild form of nonstationarity is often relevant in analyzing socio-economic time series met in practice. Through the use of an extended version of Molenaar's stationary dynamic factor analysis method, the effect of nonstationarity on the latent factor series is incorporated in the dynamic nonstationary factor model (DNFM). It is shown that the estimation of the unknown parameters in this model can be easily carried out by reformulating the DNFM as a covariance structure model and adopting the ML algorithm proposed by Jöreskog. Furthermore, an empirical example is given to demonstrate the usefulness of the proposed DNFM and the analysis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akaike, H. (1987). Factor analysis and AIC.Psychometrika, 52, 317–332.
Bartlett, M. S. (1948). Smoothing periodograms from time series with continuous spectra.Nature, 161, 686–687.
Bekker, P. A. (1989). Identification in restricted factor models and the evaluation of rank conditions.Journal of Econometrics, 41, 5–16.
Bentler, P. M. (1980). Multivariate analysis with latent variables: Causal model.Annual Review of Psychology, 31, 419–456.
Bentler, P. M. (1985).Theory and implementation of EQS: A structural equations program. Los Angeles: BMDP Statistical Software.
Box, G. E. P., & Jenkins, G. M. (1970).Time series ananlysis: Forecasting and control. San Francisco: Holden Day.
Box, G. E. P., & Tiao, G. C. (1977). A canonical analysis of multiple time series.Biometrika, 64, 355–365.
Brillinger, D. R. (1975).Time series: Data analysis and theory. New York: Holt, Rinehart and Winston.
Buss, M. (1985). Die Vielseher. Fernseh-Zuschauerforschung in Deutschland. Theorie-Praxis-Ergebnisse [TV viewing behavior in Germany. Theory-Practice-Results]. Frankfurt: Alfred Metzner Verlag.
Colgan, P. W. (Ed.). (1978).Quantitative ethology. New York: Wiley.
Cronbach, L. J., Glaser, G. C., Nanda, H., & Rajarathan, H. (1972).Dependability of behavioral measurements: Theory of generalizability for scores and profiles. New York: Wiley.
Dolan, C., Molenaar, P. C. M., & Boomsma, D. I. (1990). Simultaneous genetic analysis of longitudinal means and covariance structure using the simplex model.Behavior Genetics, 19, 51–62.
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.
Geweke, J. F. (1977). The dynamic factor analysis of economic time series models. In D. J. Aigner & A. S. Goldberger (Eds.),Latent variables in socio-economic models (pp. 365–383). Amsterdam: North-Holland.
Glover, K., & Willems, J. C. (1974). Parametrizations of linear dynamical systems: canonical identifiability.IEEE Transactions on Automatic Control, AC-19, 640–645.
Gottman, J. M., & Ringland, J. T. (1981). The analysis of dominance and bidirectionality in social development.Child Development, 52, 393–412.
Gourieroux, C., Monfort, A., & Trognon, A. (1984). Pseudo maximum likelihood methods: Theory.Econometrica, 17, 287–304.
Granger, C. W. J., & Morris, M. J. (1976). Time series modelling and interpretation.Journal of the Royal Statistical Society, Series A, 139, 246–257.
Gregson, R. A. M. (1983).Time series in psychology. Hillsdale, NJ: Lawrence Erlbaum.
Hannan, E. J. (1970).Multiple time series. New York: Wiley.
Holtzman, W. H. (1967). Statistical models for the study of change in the single case. In C. W. Harris (Ed.),Problems in measuring change (2nd ed., pp. 199–211). Madison: The University of Wisconsin Press.
Jöreskog, K. G. (1976). Some contributions to maximum likelihood factor analysis.Psychometrika, 34, 183–202.
Jöreskog, K. G. (1979). Statistical estimation of structural models in longitudinal-developmental investigations. In J. R. Nesselroade & P. B. Baltes (Eds.),Longitudinal research in the study of behavior and development (pp. 303–351). New York: Academic Press.
Jöreskog, K. J., & Sörbom, D. (1988).LISREL 7. Chicago: SPSS.
Kashyap, R. L., & Rao, A. R. (1976). Dynamics stochastic models from empirical data. New York: Academic Press.
MacCallum, R., & Ashby, F. G. (1986). Relationships between linear systems theory and covariance structure modeling.Journal of Mathematical Psychology, 30, 1–27.
McDonald, D. G. (1986). Generational aspects of television coviewing.Journal of Broadcasting & Electronic Media, 30, 75–85.
McFarland, D. J. (1971). Feedback mechanisms in animal behavior. London: Academic Press.
Molenaar, P. C. M. (1985). A dynamic factor model for the analysis of multivariate time series.Psychometrika, 50, 181–202.
Molenaar, P. C. M. (1987). Dynamic assessment and adaptive optimization of the psychotherapeutic process.Behavioral Assessment, 9, 389–416.
Molenaar, P. C. M. (1989). Aspects of dynamic factor analysis. inProceedings of the Symposium on the Analysis of Statistical Information (pp. 183–199). Tokyo: The Institute of Statistical Mathematics.
Molenaar, P. C. M., & De Gooijer, J. G. (1988). On the identification of the latent covariance structure in dynamic nonstationary factory models. In M. G. H. Jansen & W. H. Van Schuur (Eds.), The many faces of multivariate analysis (p. 196–209. Groningen: RION.
Molenaar, P. C. M., & Roelofs, J. W. (1987). The analysis of multiple habituation profiles of single trial evoked potentials.Biological Psychology, 24, 1–21.
Otter, P. W. (1986). Dynamic structural systems under indirect observation: Identifiability and estimation aspects from a system theoretic perspective.Psychometrika, 51, 415–428.
Peña, D., & Box, G. E. P. (1987). Identifying a simplifying structure in time series.Journal of the American Statistical Association, 82, 836–843.
Reinsel, G. C. (1983). Some results on multivariate autoregressive index models.Biometrika, 70, 145–156.
Schmitz, B. (1989).Einführung in die Zeitreihenanalyse: Modelle, Software, Anwendungen [Introduction to time series analysis: Models, software, and applications]. Bern: Huber.
Schmitz, B. (1990). Univariate and multivariate time-series models: The analysis of intraindividual variability and intraindividual relationships. In A. Eye (Ed.),Statistical methods in longitudinal research, Vol. II: Time series and categorical longitudinal data (pp. 351–386). New York: Academic Press.
Schwarz, G. (1978). Estimating the dimension of a model.Annals of Statistics, 6, 461–464.
Soong, R. (1988). The statistical reliability of people meter ratings.Journal of Advertisting Research, February/March, 50–56.
Sörbom, D. (1976). A statistical model for the measurement of change in true scores. In D. N. M. de Gruijter & L. J. T. van der Kamp (Eds.)Advances in psychological and educational measurement (pp. 159–169). New York: Wiley.
Thatcher, R. W., Krause, P. J., & Hrybyk, M. (1986). Cortico-cortical association and EEG coherence: A two-compartmental model.Electroencephalography and Clinical Neurophysiology, 64, 123–143.
Tiao, G. C., & Tsay, R. S. (1989). Model specification in multivariate time series.Journal of the Royal Statistical Society, Series B, 51, 157–213 (with discussion).
Voeten, M. J. M. (1985).Sequential analysis of teacher-student interaction. Unpublished doctoral dissertation, Catholic University of Nijmegen.
Winfree, A. T. (1980).The geometry of biological time. New York: Springer Verlag.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Molenaar, P.C.M., De Gooijer, J.G. & Schmitz, B. Dynamic factor analysis of nonstationary multivariate time series. Psychometrika 57, 333–349 (1992). https://doi.org/10.1007/BF02295422
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02295422