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The Kalman Foundations of Adaptive Least Squares: Applications to Unemployment and Inflation

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  • J. Huston McCulloch

Abstract

Adaptive Least Squares (ALS), i.e. recursive regression with asymptotically constant gain, as proposed by Ljung (1992), Sargent (1993, 1999), and Evans and Honkapohja (2001), is an increasingly widely-used method of estimating time-varying relationships and of proxying agents’ time-evolving expectations. This paper provides theoretical foundations for ALS as a special case of the generalized Kalman solution of a Time Varying Parameter (TVP) model. This approach is in the spirit of that proposed by Ljung (1992) and Sargent (1999), but unlike theirs, nests the rigorous Kalman solution of the elementary Local Level Model, and employs a very simple, yet rigorous, initialization. Unlike other approaches, the proposed method allows the asymptotic gain to be estimated by maximum likelihood (ML). The ALS algorithm is illustrated with univariate time series models of U.S. unemployment and inflation. Because the null hypothesis that the coefficients are in fact constant lies on the boundary of the permissible parameter space, the usual regularity conditions for the chi-square limiting distribution of likelihood-based test statistics are not met. Consequently, critical values of the Likelihood Ratio test statistics are established by Monte Carlo means and used to test the constancy of the parameters in the estimated models.

Suggested Citation

  • J. Huston McCulloch, 2005. "The Kalman Foundations of Adaptive Least Squares: Applications to Unemployment and Inflation," Computing in Economics and Finance 2005 239, Society for Computational Economics.
  • Handle: RePEc:sce:scecf5:239
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    File URL: http://www.econ.ohio-state.edu/jhm/papers/KalmanAL.pdf
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    1. Richard Clarida & Jordi Galí & Mark Gertler, 2000. "Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory," The Quarterly Journal of Economics, Oxford University Press, vol. 115(1), pages 147-180.
    2. Bullard, James & Mitra, Kaushik, 2002. "Learning about monetary policy rules," Journal of Monetary Economics, Elsevier, vol. 49(6), pages 1105-1129, September.
    3. Orphanides, Athanasios & Williams, John C., 2005. "The decline of activist stabilization policy: Natural rate misperceptions, learning, and expectations," Journal of Economic Dynamics and Control, Elsevier, vol. 29(11), pages 1927-1950, November.
    4. Sargent, Thomas J, 1971. "A Note on the 'Accelerationist' Controversy," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 3(3), pages 721-725, August.
    5. George W. Evans & Seppo Honkapohja, 2003. "Adaptive learning and monetary policy design," Proceedings, Federal Reserve Bank of Cleveland, pages 1045-1084.
    6. Fabio Milani, 2005. "Adaptive Learning and Inflation Persistence," Working Papers 050607, University of California-Irvine, Department of Economics.
    7. Durbin, James & Koopman, Siem Jan, 2012. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, edition 2, number 9780199641178, June.
    8. Thomas Sargent & Noah Williams & Tao Zha, 2009. "The Conquest of South American Inflation," Journal of Political Economy, University of Chicago Press, vol. 117(2), pages 211-256, April.
    9. Preston, Bruce, 2006. "Adaptive learning, forecast-based instrument rules and monetary policy," Journal of Monetary Economics, Elsevier, vol. 53(3), pages 507-535, April.
    10. McCulloch, J. Huston, 1985. "Interest-risk sensitive deposit insurance premia : Stable ACH estimates," Journal of Banking & Finance, Elsevier, vol. 9(1), pages 137-156, March.
    11. 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.
    12. Perron, Pierre, 1989. "The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis," Econometrica, Econometric Society, vol. 57(6), pages 1361-1401, November.
    13. 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.
    14. McGough, Bruce, 2003. "Statistical Learning With Time-Varying Parameters," Macroeconomic Dynamics, Cambridge University Press, vol. 7(01), pages 119-139, February.
    15. Cooley, Thomas F & Prescott, Edward C, 1973. "An Adaptive Regression Model," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 14(2), pages 364-371, June.
    16. Klein, Benjamin, 1978. "The Measurement of Long- and Short-Term Price Uncertainty: A Moving Regression Time Series Analysis," Economic Inquiry, Western Economic Association International, vol. 16(3), pages 438-452, July.
    17. McCulloch, J Huston, 1997. "Measuring Tail Thickness to Estimate the Stable Index Alpha: A Critique," Journal of Business & Economic Statistics, American Statistical Association, vol. 15(1), pages 74-81, January.
    18. repec:cup:macdyn:v:7:y:2003:i:1:p:119-39 is not listed on IDEAS
    19. Tanaka, Katsuto, 1983. "Non-Normality of the Lagrange Multiplier Statistic for Testing the Constancy of Regression Coefficients," Econometrica, Econometric Society, vol. 51(5), pages 1577-1582, September.
    20. 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.
    21. Prasad V. Bidarkota & J. Huston McCulloch, 1998. "Optimal univariate inflation forecasting with symmetric stable shocks," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 13(6), pages 659-670.
    22. Roy, Anindya & Fuller, Wayne A, 2001. "Estimation for Autoregressive Time Series with a Root Near 1," Journal of Business & Economic Statistics, American Statistical Association, vol. 19(4), pages 482-493, October.
    23. 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.
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    Keywords

    Kalman Filter; Adaptive Learning; Adaptive Least Squares; Time Varying Parameter Model; Natural Unemployment Rate; Inflation Forecasting;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • E37 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Forecasting and Simulation: Models and Applications
    • E31 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Price Level; Inflation; Deflation

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