Hostname: page-component-6bf8c574d5-9nwgx Total loading time: 0 Render date: 2025-02-23T02:30:05.165Z Has data issue: false hasContentIssue false
  • English
  • Français

LEARNING MARKOV PROCESSES WITH LATENT VARIABLES

Published online by Cambridge University Press:  13 February 2025

Ayden Higgins  and
Koen Jochmans
Ayden Higgins
Affiliation:
University of Exeter Business School
Koen Jochmans*
Affiliation:
Toulouse School of Economics
*
Address correspondence to Koen Jochmans, Toulouse School of Economics, Université Toulouse Capitole, Toulouse, France, e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the problem of identifying the parameters of a time-homogeneous bivariate Markov chain when only one of the two variables is observable. We show that, subject to conditions that we spell out, the transition kernel and the distribution of the initial condition are uniquely recoverable (up to an arbitrary relabelling of the state space of the latent variable) from the joint distribution of four (or more) consecutive time-series observations. The result is, therefore, applicable to (short) panel data as well as to (stationary) time series data.

Type
ARTICLES
Information
Econometric Theory , First View , pp. 1 - 13
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We are grateful to Anna Mikusheva, Peter Phillips, and two anonymous referees for constructive feedback. Financial support from the European Research Council, grant ERC-2016-STG-715787, and from the French Government and the ANR under the Investissements d’ Avenir program, grant ANR-17-EURE-0010 is gratefully acknowledged.

References

REFERENCES

Ailliot, P., & Pène, F. (2015). Consistency of the maximum likelihood estimate for non-homogeneous Markov-switching models. ESAIM: Probability and Statistics , 19, 268–292.CrossRefGoogle Scholar
Alexandrovich, G., Holzmann, H., & Leister, A. (2016). Nonparametric identification and maximum likelihood estimation of hidden Markov models. Biometrika , 103, 423–434.CrossRefGoogle ScholarPubMed
Allman, E. S., Matias, C., & Rhodes, J. A. (2009). Identifiability of parameters in latent structure models with many observed variables. Annals of Statistics , 37, 3099–3132.CrossRefGoogle Scholar
Anderson, T. W. (1954). On estimation of parameters in latent structure analysis. Psychometrika , 19, 1–10.CrossRefGoogle Scholar
Arcidiacono, P., & Miller, R. A. (2011). Conditional choice probability estimation of dynamic discrete choice models with unobserved heterogeneity. Econometrica , 79, 1823–1867.Google Scholar
Bajari, P., Benkard, L., & Levin, J. (2007). Estimating dynamic models of imperfect competition. Econometrica , 75, 1331–1370.CrossRefGoogle Scholar
Bonhomme, S., Jochmans, K., & Robin, J.-M. (2016a). Estimating multivariate latent-structure models. Annals of Statistics , 44, 540–563.CrossRefGoogle Scholar
Bonhomme, S., Jochmans, K., & Robin, J.-M. (2016b). Non-parametric estimation of finite mixtures from repeated measurements. Journal of the Royal Statistical Society Series B , 78, 211–229.CrossRefGoogle Scholar
Cappé, O., Moulines, E., & Rydén, T. (2005). Inference in hidden Markov models , Springer Series in Statistics. Springer.CrossRefGoogle Scholar
Connault, B. (2016). Hidden Rust models . Mimeo.Google Scholar
De Lathauwer, L., De Moor, B., & Vandewalle, J. (2004). Computation of the canonical decomposition by means of a simultaneous generalized Schur decomposition. SIAM Journal of Matrix Analysis and Applications , 26, 295–327.CrossRefGoogle Scholar
Gassiat, E., Cleynen, A., & Robin, S. (2016). Finite state space non parametric hidden Markov models are in general identifiable. Statistics and Computing , 26, 61–71.CrossRefGoogle Scholar
Gupta, R., Kumar, R., & Vassilvitskii, S. (2016). On mixtures of Markov chains. Thirtieth conference on neural information processing systems , Barcelona.Google Scholar
Hall, P., & Zhou, X.-H. (2003). Nonparametric estimation of component distributions in a multivariate mixture. Annals of Statistics , 31, 201–224.CrossRefGoogle Scholar
Henry, M., Kitamura, Y., & Salanié, B. (2014). Partial identification of finite mixtures in econometric models. Quantitative Economics , 5, 123–144.CrossRefGoogle Scholar
Higgins, A., & Jochmans, K. (2021). Joint approximate asymmetric diagonalization of non-orthogonal matrices . Mimeo.Google Scholar
Higgins, A., & Jochmans, K. (2023). Identification of mixtures of dynamic discrete choices. Journal of Econometrics , 237, 105462.CrossRefGoogle Scholar
Hotz, J., & Miller, R. (1993). Conditional choice probabilities and the estimation of dynamic models. Review of Economic Studies , 60, 497–529.CrossRefGoogle Scholar
Hu, Y. (2008). Identification and estimation of nonlinear models with misclassification error using instrumental variables: A general solution. Journal of Econometrics , 144, 27–61.CrossRefGoogle Scholar
Hu, Y., & Shum, M. (2012). Nonparametric identification of dynamic models with unobserved state variables. Journal of Econometrics , 171, 32–44.CrossRefGoogle Scholar
Jochmans, K., Henry, M., & Salanié, B. (2017). Inference on two-component mixtures under tail restrictions. Econometric Theory , 33, 610–635.CrossRefGoogle Scholar
Kasahara, H., & Shimotsu, K. (2009). Nonparametric identification of finite mixture models of dynamic discrete choices. Econometrica , 77, 135–175.Google Scholar
Kleibergen, F., & Paap, R. (2006). Generalized reduced rank tests using the singular value decomposition. Journal of Econometrics , 133, 97–126.CrossRefGoogle Scholar
Miller, R. A. (1984). Job matching and occupational choice. Journal of Political Economy , 71, 1565–1578.Google Scholar
Pouzo, D., Psaradakis, Z., & Sola, M. (2022). Maximum likelihood estimation in Markov regime-switching models with covariate-dependent transition probabilities. Econometrica , 90, 1681–1710.CrossRefGoogle Scholar
Teicher, H. (1963). Identifiability of finite mixtures. Annals of Mathematical Statistics , 34, 1265–1269.CrossRefGoogle Scholar
Vandermeulen, R. A., & Scott, C. D. (2020). An operator theoretic approach to nonparametric mixture models. Annals of Statistics , 47, 2704–2733.Google Scholar