Abstract
Unknown parameters, including regression coefficients, in state space models can be estimated by maximum likelihood. An alternative approach is to augment the state vector to include regression coefficients. However, the state estimator obtained by the Kalman filter is numerically different from the maximum likelihood estimator. We address the discrepancy by a novel method based on proper distributions returned by the ordinary Kalman filter without dependency on diffuse initialization. We prove that maximizing a low-dimensional objective function that combines the likelihood, the filtering mean and variance can reproduce the high-dimensional maximum likelihood results.
References
Ansley, C. F., and R. Kohn. 1985. “Estimation, Filtering and Smoothing in State Space Models with Incompletely Specified Initial Conditions.” Annals of Statistics 13: 1286–316. https://doi.org/10.1214/aos/1176349739.Search in Google Scholar
Commandeur, J., S. J. Koopman, and M. Ooms. 2011. “Statistical Software for State Space Methods.” Journal of Statistical Software 41: 1–18. https://doi.org/10.18637/jss.v041.i01.Search in Google Scholar
Cornwell, C., and P. Rupert. 1988. “Efficient Estimation with Panel Data: An Empirical Comparison of Instrumental Variable Estimators.” Journal of Applied Econometrics 3: 149–55. https://doi.org/10.1002/jae.3950030206.Search in Google Scholar
De Jong, P. 1988. “The Likelihood for a State-Space Model.” Biometrika 75: 165–9. https://doi.org/10.1093/biomet/75.1.165.Search in Google Scholar
De Jong, P. 1991. “The Diffuse Kalman Filter.” Annals of Statistics 19 (2): 1073–83. https://doi.org/10.1214/aos/1176348139.Search in Google Scholar
Durbin, J., and S. J. Koopman. 2012. Time Series Analysis by State Space Methods, 2nd ed. Oxford: Oxford University Press.10.1093/acprof:oso/9780199641178.001.0001Search in Google Scholar
Francke, M. K., S. J. Koopman, and A. F. de Vos. 2010. “Likelihood Functions for State Space Models with Diffuse Initial Conditions.” Journal of Time Series Analysis 31: 407–14. https://doi.org/10.1111/j.1467-9892.2010.00673.x.Search in Google Scholar
Gelman, A., J. Carlin, H. Stern, D. Dunson, A. Vehtari, and D. Rubin. 2014. Bayesian Data Analysis, 3rd ed. Boca Raton: CRC Press.10.1201/b16018Search in Google Scholar
Greene, W. H. 2008. Econometric Analysis, 6th ed. New Jersey: Prentice Hall.Search in Google Scholar
Harvey, A. C. 1990. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge: Cambridge University Press.10.1017/CBO9781107049994Search in Google Scholar
Hooker, M. A. 1994. “Analytic First and Second Derivatives for the Recursive Prediction Error Algorithm’s Log Likelihood Function.” IEEE Transactions on Automatic Control 39 (3): 662–4. https://doi.org/10.1109/9.280783.Search in Google Scholar
Koopman, S. J. 1997. “Exact Initial Kalman Filtering and Smoothing for Nonstationary Time Series Models.” Journal of the American Statistical Association 92: 1630–8. https://doi.org/10.1080/01621459.1997.10473685.Search in Google Scholar
Koopman, S. J., and J. Durbin. 2003. “Filtering and Smoothing of State Vector for Diffuse State-Space Models.” Journal of Time Series Analysis 24 (1): 85–98. https://doi.org/10.1111/1467-9892.00294.Search in Google Scholar
Koopman, S. J., and N. G. Shephard. 1992. “Exact Score for Time Series Models in State Space Form.” Biometrika 79 (4): 823–6. https://doi.org/10.2307/2337237.Search in Google Scholar
Nagakura, D. 2021. “Computing Exact Score Vectors for Linear Gaussian State Space Models.” Communications in Statistics – Simulation and Computation 50 (8): 2313–26. https://doi.org/10.1080/03610918.2019.1601216.Search in Google Scholar
Shephard, N. G., and A. C. Harvey. 1990. “On the Probability of Estimating a Deterministic Component in the Local Level Model.” Journal of Time Series Analysis 11: 339–47. https://doi.org/10.1111/j.1467-9892.1990.tb00062.x.Search in Google Scholar
Wooldridge, J. 2010. Econometric Analysis of Cross Section and Panel Data, 2nd ed. Cambridge: The MIT Press.Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Research Articles
- The Story of a Model: The First-Order Diagonal Bilinear Autoregression
- Maximum Likelihood Estimation of Regression Effects in State Space Models
- Software
- QR.break: An R Package for Structural Breaks in Quantile Regression
- Practitioner's Corner
- Fast Algorithms for Quantile Regression with Selection
Articles in the same Issue
- Frontmatter
- Research Articles
- The Story of a Model: The First-Order Diagonal Bilinear Autoregression
- Maximum Likelihood Estimation of Regression Effects in State Space Models
- Software
- QR.break: An R Package for Structural Breaks in Quantile Regression
- Practitioner's Corner
- Fast Algorithms for Quantile Regression with Selection