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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.

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Paper provided by Society for Computational Economics in its series Computing in Economics and Finance 2005 with number 239.

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Date of creation: 11 Nov 2005
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Handle: RePEc:sce:scecf5:239

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

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Find related papers by JEL classification:
C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions
E37 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Forecasting and Simulation
E31 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Price Level; Inflation; Deflation

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