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A note on exact correspondences between adaptive learning algorithms and the Kalman filter

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  • Michele Berardi
  • Jaqueson K. Galimberti

Abstract

Digressing into the origins of the two main algorithms considered in the literature of adaptive learning, namely Least Squares (LS) and Stochastic Gradient (SG), we found a connection between their non-recursive forms and their interpretation within a state-space unifying framework. Based on such connection, we extend the correspondence between the LS and the Kalman filter recursions to a formulation with time-varying gains of the former, and also present a similar correspondence for the case of the SG. Our correspondences hold exactly, in a computational implementation sense, and we discuss how they relate to previous approximate correspondences found in the literature.

Suggested Citation

  • Michele Berardi & Jaqueson K. Galimberti, 2012. "A note on exact correspondences between adaptive learning algorithms and the Kalman filter," Centre for Growth and Business Cycle Research Discussion Paper Series 170, Economics, The University of Manchester.
    • Handle: RePEc:man:cgbcrp:170
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    1. Timothy Cogley & Thomas J. Sargent, 2005. "Drift and Volatilities: Monetary Policies and Outcomes in the Post WWII U.S," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 8(2), pages 262-302, April.
    2. Thomas J. Sargent & Noah Williams, 2005. "Impacts of Priors on Convergence and Escapes from Nash Inflation," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 8(2), pages 360-391, April.
    3. George W. Evans & Seppo Honkapohja & Noah Williams, 2010. "Generalized Stochastic Gradient Learning," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 51(1), pages 237-262, February.
    4. Berardi, Michele & Galimberti, Jaqueson K., 2013. "A note on exact correspondences between adaptive learning algorithms and the Kalman filter," Economics Letters, Elsevier, vol. 118(1), pages 139-142.
    5. Marcet, Albert & Sargent, Thomas J., 1989. "Convergence of least squares learning mechanisms in self-referential linear stochastic models," Journal of Economic Theory, Elsevier, vol. 48(2), pages 337-368, August.
    6. Albert Marcet & Juan P. Nicolini, 2003. "Recurrent Hyperinflations and Learning," American Economic Review, American Economic Association, vol. 93(5), pages 1476-1498, December.
    7. James H. Stock & Mark W. Watson, 2003. "Has the Business Cycle Changed and Why?," NBER Chapters, in: NBER Macroeconomics Annual 2002, Volume 17, pages 159-230, National Bureau of Economic Research, Inc.
    8. Stock, James H & Watson, Mark W, 1996. "Evidence on Structural Instability in Macroeconomic Time Series Relations," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(1), pages 11-30, January.
    9. Bray, Margaret, 1982. "Learning, estimation, and the stability of rational expectations," Journal of Economic Theory, Elsevier, vol. 26(2), pages 318-339, April.
    10. Branch, William A. & Evans, George W., 2006. "A simple recursive forecasting model," Economics Letters, Elsevier, vol. 91(2), pages 158-166, May.
    11. Christopher A. Sims & Tao Zha, 2006. "Were There Regime Switches in U.S. Monetary Policy?," American Economic Review, American Economic Association, vol. 96(1), pages 54-81, March.
    12. Margaritis, Dimitris, 1990. "A time-varying model of rational learning," Economics Letters, Elsevier, vol. 33(4), pages 309-314, August.
    13. Evans, George W. & Honkapohja, S., 1998. "Stochastic gradient learning in the cobweb model," Economics Letters, Elsevier, vol. 61(3), pages 333-337, December.
    14. Thomas Sargent & Noah Williams & Tao Zha, 2006. "Shocks and Government Beliefs: The Rise and Fall of American Inflation," American Economic Review, American Economic Association, vol. 96(4), pages 1193-1224, September.
    15. McGough, Bruce, 2003. "Statistical Learning With Time-Varying Parameters," Macroeconomic Dynamics, Cambridge University Press, vol. 7(1), pages 119-139, February.
    16. Bullard, James, 1992. "Time-varying parameters and nonconvergence to rational expectations under least squares learning," Economics Letters, Elsevier, vol. 40(2), pages 159-166, October.
    17. Barucci, Emilio & Landi, Leonardo, 1997. "Least mean squares learning in self-referential linear stochastic models," Economics Letters, Elsevier, vol. 57(3), pages 313-317, December.
    18. Bray, Margaret M & Savin, Nathan E, 1986. "Rational Expectations Equilibria, Learning, and Model Specification," Econometrica, Econometric Society, vol. 54(5), pages 1129-1160, September.
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    Cited by:

    1. Christina Strobach & Carin van der Cruijsen, 2015. "The formation of European inflation expectations: One learning rule does not fit all," DNB Working Papers 472, Netherlands Central Bank, Research Department.
    2. André Marine Charlotte & Dai Meixing, 2018. "Learning, robust monetary policy and the merit of precaution," The B.E. Journal of Macroeconomics, De Gruyter, vol. 18(2), pages 1-20, June.
    3. Berardi, Michele & Galimberti, Jaqueson K., 2017. "Empirical calibration of adaptive learning," Journal of Economic Behavior & Organization, Elsevier, vol. 144(C), pages 219-237.
    4. Jaqueson K. Galimberti, 2020. "Information weighting under least squares learning," CAMA Working Papers 2020-46, Centre for Applied Macroeconomic Analysis, Crawford School of Public Policy, The Australian National University.
    5. Berardi, Michele & Galimberti, Jaqueson K., 2019. "Smoothing-Based Initialization For Learning-To-Forecast Algorithms," Macroeconomic Dynamics, Cambridge University Press, vol. 23(3), pages 1008-1023, April.
    6. Schaefer, Daniel & Singleton, Carl, 2018. "Unemployment and econometric learning," Research in Economics, Elsevier, vol. 72(2), pages 277-296.
    7. Galimberti, Jaqueson K., 2019. "An approximation of the distribution of learning estimates in macroeconomic models," Journal of Economic Dynamics and Control, Elsevier, vol. 102(C), pages 29-43.
    8. Michele Berardi, 2020. "A probabilistic interpretation of the constant gain learning algorithm," Bulletin of Economic Research, Wiley Blackwell, vol. 72(4), pages 393-403, October.
    9. Berardi, Michele & Galimberti, Jaqueson K., 2013. "A note on exact correspondences between adaptive learning algorithms and the Kalman filter," Economics Letters, Elsevier, vol. 118(1), pages 139-142.
    10. Marine Charlotte André & Meixing Dai, 2015. "Central bank accountability under adaptive learning," Working Papers of BETA 2015-32, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
    11. Berardi, Michele & Galimberti, Jaqueson K., 2017. "On the initialization of adaptive learning in macroeconomic models," Journal of Economic Dynamics and Control, Elsevier, vol. 78(C), pages 26-53.
    12. Berardi, Michele, 2019. "A probabilistic interpretation of the constant gain algorithm," MPRA Paper 94023, University Library of Munich, Germany.
    13. Michele Berardi & Jaqueson K. Galimberti, 2012. "On the plausibility of adaptive learning in macroeconomics: A puzzling conflict in the choice of the representative algorithm," Centre for Growth and Business Cycle Research Discussion Paper Series 177, Economics, The University of Manchester.
    14. Michele Berardi & Jaqueson K. Galimberti, 2012. "On the initialization of adaptive learning algorithms: A review of methods and a new smoothing-based routine," Centre for Growth and Business Cycle Research Discussion Paper Series 175, Economics, The University of Manchester.

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    More about this item

    JEL classification:

    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness
    • D84 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Expectations; Speculations

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