Skip to main content
SpringerLink
Log in
Book cover

Restricted Kalman Filtering pp 11–25Cite as

Restricted Kalman Filtering: Theoretical Issues

  • Chapter
  • First Online:

  • 2263 Accesses

Part of the book series: SpringerBriefs in Statistics ((BRIEFSSTATIST,volume 12))

Abstract

This entire chapter will be devoted to a discussion of several topics concerning the theory of imposing linear restrictions enunciated under a quite general form in (2.5) from Assumption 2.1. In Sect. 3.1, I will present and compare three different derivations of the restricted Kalman updating and smoothing equations under an augmented modeling approach. In Sect. 3.2, the statistical efficiency due to the imposition of restrictions is proved, and this shall be done using a geometrical framework. Moving forward, I try in Sect. 3.3 to establish the equivalence between restricted Kalman filtering and something that could be termed a recursive restricted least squares estimator. Finally, in Sect. 3.4, I investigate how initial diffuse state vectors affect the use of the Kalman smoother under linear restrictions.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   29.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   39.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  • ANDERSON, B. D. O. and MOORE, J. B. (1979). Optimal Filtering. Prentice Hall.

    Google Scholar 

  • ANSLEY. C. F. and KOHN, R. (1985). “Estimation, filtering and smoothing in state space models with incompletely specified initial conditions”. Annals of Statistics, Vol. 13, No. 4, pp. 1286–1316.

    Google Scholar 

  • BROCKWELL, P. J. and DAVIS, R. A. (1991). Time Series: Theory and Methods. 2nd edition. Springer.

    Google Scholar 

  • BROCKWELL, P. J. and DAVIS, R. A. (2003). Introduction to Time Series and Forecasting. 2nd edition. Springer Texts in Statistics.

    Google Scholar 

  • BROWN, R., DURBIN, J. and EVANS, J. (1975). “Techniques for testing the constancy of regression relationships Over Time”. Journal of the Royal Statistical Society B, 37, pp. 149–172.

    MathSciNet  MATH  Google Scholar 

  • CAMPA, J.M. and GOLDBERG, L.S. (1995). Exchange Rate Pass-through into Import Prices. Federal Reserve Bank of New York.

    Google Scholar 

  • CARHART, M. M. (1997). “On persistence in mutual fund performance”. Journal of Finance, 52, pp. 57–82.

    Article  Google Scholar 

  • CERQUEIRA, L. F., PIZZINGA, A., and FERNANDES, C. (2009). “Methodological procedure for estimating Brazilian quarterly GDP series”. International Advances in Economic Research, 15, pp. 102–114.

    Article  Google Scholar 

  • CHUNG, K. L. (2001). A Course in Probability Theory. 3rd edition. Academic Press.

    Google Scholar 

  • DAVIDSON, R. and MACKINNON, J. G. (1993). Estimation and Inference in Econometrics. Oxford University Press.

    Google Scholar 

  • DE ROON, F. A., NIJMAN, T. E. and TER HORST, J. R. (2004). “Evaluating style analysis”. Journal of Empirical Finance, Volume 11, Issue 1, pp. 29–53.

    Article  Google Scholar 

  • DE JONG, P. and ZEHNWIRTH, B. (1983). “Claims reserving state-space models and the Kalman filter”. Journal of the Institute of Actuaries, 110, pp. 157–181.

    Article  Google Scholar 

  • DE JONG, P. (1988). “The likelihood for a state space model”. Biometrika, 75, 1, pp. 165–169.

    Article  MathSciNet  MATH  Google Scholar 

  • DE JONG, P. (1989). “Smoothing and Interpolation With the State-Space Models”. Journal of the American Statistical Association, 84, pp. 1085–1088.

    Article  MathSciNet  MATH  Google Scholar 

  • DE JONG, P. (1991). “The diffuse Kalman filter”. Annals of Statistics, Vol. 19, No. 2, pp. 1073–1083.

    Article  MathSciNet  MATH  Google Scholar 

  • DE JONG, P. and CHU-CHUN-LIN, S. (2003). “Smoothing with an unknown initial condition”. Journal of Time Series Analysis, Vol. 24, No. 2, pp. 141–148.

    Article  MathSciNet  MATH  Google Scholar 

  • DOORNICK, J. A. (2001). Ox 3.0: An Object-Oriented Matrix Programming Language. Timberlake Consultants.

    Google Scholar 

  • DORAN, H. (1992). “Constraining Kalman filter and smoothing estimates to satisfy time-varying restrictions”. Review of Economics and Statistics, 74, pp. 568–572.

    Article  Google Scholar 

  • DORAN, H. (1996). “Estimation under exact linear time-varying constraints, with applications to population projections”. Journal of Forecasting, 15, pp. 527–541.

    Article  Google Scholar 

  • DORAN, H. and RAMBALDI, A. (1997). “Applying linear time-varying constraints to econometric models: with an application to demand systems”. Journal of Econometrics, 79, pp. 83–95.

    Article  MathSciNet  MATH  Google Scholar 

  • DURBIN, J. and KOOPMAN, S. J. (2001). Time Series Analysis by State Space Methods. Oxford Statistical Science Series.

    Google Scholar 

  • DURBIN, J. and QUEENNEVILLE, B. (1997). “Benchmarking by state space models”. International Statistical Review, 65, pp. 21–48.

    Article  Google Scholar 

  • ELTON, E. J., GRUBER, M. J., BROWN, S. J. and GOETZMANN, W. (2006). Modern Portfolio Theory and Investment Analysis. 7th edition. John Wiley & Sons.

    Google Scholar 

  • ENDERS, W. (2004). Applied Econometric Time Series. 2nd edition. John Wiley & Sons.

    Google Scholar 

  • FRAGA, A., GOLDFAJN, I. and MINELLA, A. (2003) “Inflation Targeting in Emerging Market Economies”. NBER Working Paper, 10.019.

    Google Scholar 

  • GEETER, J., BRUSSEL, H. and SCHUTTER, J. (1997). “A smoothly constrained Kalman filter”. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19, 10, pp. 1171–1177.

    Article  Google Scholar 

  • GREENE, W. H. (2003), Econometric Analysis. 5th edition. Prentice Hall.

    Google Scholar 

  • HAMILTON, J. D. (1994). Time Series Analysis. Princeton University Press.

    Google Scholar 

  • HARVEY, A. C. (1981). The Econometric Analysis of Time Series. Philip Allan Publishers.

    Google Scholar 

  • HARVEY, A. C. (1989). Forecasting, Structural Time Series Models and The Kalman Filter. Cambridge University Press.

    Google Scholar 

  • HARVEY, A. C. (1993). Time Series Models. 2nd edition. Harvester Wheatsheaf.

    Google Scholar 

  • JENSEN, M. C. (1968). “The performance of mutual funds in the period 1945–1964”. Journal of Finance, Vol. 23, No. 2, pp. 389–416.

    Article  Google Scholar 

  • JOHNSTON, J. and DiNARDO, J. (1997). Econometric Methods. 4th edition. McGraw-Hill.

    Google Scholar 

  • JULIER, S. and UHLMANN, J. (2004). “Unscented filtering and nonlinear estimation”. Proceedings of the IEEE, 92, pp. 401–422.

    Article  Google Scholar 

  • JULIER, S. and LAVIOLA, J. (2007). “Kalman filtering with nonlinear equality con- straints”. IEEE Transactions on Signal Processing, 55, 6, pp. 2774–2784.

    Article  MathSciNet  Google Scholar 

  • KO, S. and BITMEAD, R. R. (2007). “State estimation for linear systems with state equality constraints”. Automatica, 43, 8, pp. 1363–1368.

    Article  MathSciNet  MATH  Google Scholar 

  • KOOP, G., LEON-GONZALES, R. and STRATCHAN, R. (2010). “Dynamic probabilities of restrictions in state space models: an application to the Phillips curve”. Journal of Business & Economic Statistics, Vol. 28, No. 3, pp. 370–379.

    Article  MathSciNet  MATH  Google Scholar 

  • KOOPMAN, S. J. (1997). “Exact initial Kalman filtering and smoothing for nonstationary time series models”. Journal of the American Statistical Association, 92, pp. 1630–1638.

    Article  MathSciNet  MATH  Google Scholar 

  • KOOPMAN, S. J., SHEPHARD, N. and DOORNIK, J. A. (2002). “SsfPack 3.0 beta02: statistical algorithms for models in state space”. Unpublished paper. Department of Econometrics, Free University, Amsterdam.

    Google Scholar 

  • KOOPMAN, S. J. and DURBIN, J. (2003). “Filtering and smoothing of state vector for diffuse state-space models”. Journal of Time Series Analysis, Vol. 24, No. 1, pp. 85–98.

    Article  MathSciNet  MATH  Google Scholar 

  • KUBRUSLY, C. S. (2001). Elements of Operator Theory. Birkhäuser.

    Google Scholar 

  • LEYBOURNE, S. (1993). “Estimation and testing of time-varying coefficient regression models in the presence of linear restrictions”. Journal of Forecasting, 12, pp. 49–62.

    Article  Google Scholar 

  • MASSICOTTE, D., MORAWSKI, R. Z. and BARWICZ, A. (1995). “Incorporation of a positivity constraint into a Kalman-filter-based algorithm for correction of spectrometric data”. IEEE Transactions on Instrumentation and Measurement, 44, 1, pp. 2–7.

    Article  Google Scholar 

  • MENON, J. (1996). “Exchange rate pass-through”. Journal of Economic Surveys, 9(2), pp. 197–231.

    Article  Google Scholar 

  • PAGAN, A. (1980). “Some identification and estimation results for regression models with stochastically varying coefficients”. Journal of Econometrics, 13, pp. 341–363.

    Article  MathSciNet  MATH  Google Scholar 

  • PANDHER, G. S. (2002). “Forecasting multivariate time series with linear restrictions using constrained structural state-space models”. Journal of Forecasting, 21, pp. 281–300.

    Article  Google Scholar 

  • PANDHER, G. S. (2007). “Modelling & controlling monetary and economic identities with constrained state space models”. International Statistical Review, Vol. 75, No. 2, pp. 150–169.

    Article  Google Scholar 

  • PARSLEY, D. (1995). “Anticipated future shocks and exchange rate pass-through in the presence of reputation”. International Review of Economics 4(2).

    Google Scholar 

  • PIZZINGA, A. (2009). “Further investigation into restricted Kalman filtering”. Statistics & Probability Letters, 79, pp. 264–269.

    Article  MathSciNet  MATH  Google Scholar 

  • PIZZINGA, A. (2010). “Constrained Kalman filtering: additional results”. International Statistical Review, Vol. 78, No. 2, pp. 189–208.

    Article  MathSciNet  Google Scholar 

  • PIZZINGA, A. (2012)“Diffuse restricted Kalman filtering”. Communications in Statistics: Theory and Methods (to appear).

    Google Scholar 

  • PIZZINGA, A., RUGGERI, E. and GUEDES, Q. (2005). “Relatório Técnico Estatístico: Projeto Hooke (in Portuguese)”. Technical report. DCT.T/Furnas Centrais Elétricas S.A.

    Google Scholar 

  • PIZZINGA, A. and FERNANDES, C. (2006). “State space models for dynamic style analysis of portfolios”. Brazilian Review of Econometrics, Vol. 26, 1, pp. 31–66.

    Google Scholar 

  • PIZZINGA, A., FERNANDES, C. and CONTRERAS, S. (2008). “Restricted Kalman filtering revisited”. Journal of Econometrics, 144, 2, pp. 428–429.

    Article  MathSciNet  Google Scholar 

  • PIZZINGA, A., VEREDA, L., ATHERINO, R. and FERNANDES, C. (2008). “Semi-strong dynamic style analysis with time-varying selectivity measurement: applications to Brazilian exchange rate funds”. Applied Stochastic Models in Business and Industry, Vol. 24, 1, pp. 3–12.

    Article  MathSciNet  MATH  Google Scholar 

  • PIZZINGA, A., VEREDA, L. and FERNANDES, C. (2011). “A dynamic style analysis of exchange rate funds: the case of Brazil at the 2002 election”. Advances and Applications in Statistical Sciences. Vol. 6, Issue 2, pp. 111–135.

    MathSciNet  MATH  Google Scholar 

  • SHARPE, W. F. (1988). “Determining a fund’s effective asset mix”. Investment Management Review, pp. 59–69.

    Google Scholar 

  • SHARPE, W. F. (1992). “Asset allocation: management style and performance measurement”. Journal of Porfolio Management, Winter, pp. 7–19.

    Google Scholar 

  • SHUMWAY, R. H. and STOFFER, D. S. (2006). Time Series Analysis and Its Applications (With R Examples). Springer.

    Google Scholar 

  • SIMON, D. (2009). “Kalman Filtering with State Constraints: a Survey of Linear and Nonlinear Algorithms”. IET Control Theory & Applications. (in press)

    Google Scholar 

  • SIMON, D. and CHIA, T. (2002). “Kalman filtering with state equality constraints”. IEEE Transactions on Aerospace and Electronic Systems, 38, 1, pp. 128–136.

    Article  Google Scholar 

  • SIMON, D. and SIMON, D. L. (2004). “Aircraft turbofan engine health estimation using constrained Kalman filtering”. Journal of Engineering for Gas Turbines and Power, 126, pp. 1–6.

    Article  Google Scholar 

  • SOUZA, R. M., MACIEL, L. and PIZZINGA, A. (2011). “Using a restricted Kalman filtering approach for the estimation of a dynamic exchange-rate pass-through”. In Gomez, J. M. (ed.) Kalman Filtering, Chap. 9, pp. 255–268. Nova Publishers.

    Google Scholar 

  • SWINKELS, L. and VAN DER SLUIS, P. J. (2006). “Return-based style analysis with time-varying exposures”. European Journal of Finance, Vol. 12, pp. 529–552.

    Article  Google Scholar 

  • TANIZAKI, H. (1996). Nonlinear Filters. 2nd edition. Springer.

    Google Scholar 

  • TAYLOR, J. (2000) “Low inflation, pass-through and the pricing power of firms”. European Economic Review, 44, pp. 1389–1408.

    Article  Google Scholar 

  • TEIXEIRA, B. O. S., CHANDRASEKAR, J., TORRES, L. A. B., AGUIRRE, L. A. and BERNSTEIN, D. S. (2009). “State estimation for linear and nonlinear equality-constrained systems”. International Journal of Control, 82, 5, pp. 918–936.

    Article  MathSciNet  MATH  Google Scholar 

  • WEST, M. and HARRISON, J. (1997). Bayesian Forecasting and Dynamic Models. 2nd edition. Springer.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics Institute of Mathematics and Statistics, Fluminense Federal University, Rio de Janeiro, Brazil

    Adrian Pizzinga

Authors
  1. Adrian Pizzinga
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media New York

About this chapter

Cite this chapter

Pizzinga, A. (2012). Restricted Kalman Filtering: Theoretical Issues. In: Restricted Kalman Filtering. SpringerBriefs in Statistics, vol 12. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4738-2_3

Download citation

Publish with us

Policies and ethics

Search

Navigation