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Frequency Dependence in Regression Model Coefficients: An Alternative Approach for Modeling Nonlinear Dynamic Relationships in Time Series

  • Richard Ashley
  • Randal Verbrugge

This article proposes a new class of nonlinear time series models in which one of the coefficients of an existing regression model is frequency dependent—that is, the relationship between the dependent variable and this explanatory variable varies across its frequency components. We show that such frequency dependence implies that the relationship between the dependent variable and this explanatory variable is nonlinear. Past efforts to detect frequency dependence have not been satisfactory; for example, we note that the two-sided bandpass filtering used in such efforts yields inconsistent estimates of frequency dependence where there is feedback in the relationship. Consequently, we provide an explicit procedure for partitioning an explanatory variable into frequency components using one-sided bandpass filters. This procedure allows us to test for and quantify frequency dependence even where feedback may be present. A distinguishing feature of these new models is their potentially tight connection to macroeconomic theory; indeed, they are perhaps best introduced by reference to the frequency dependence in the marginal propensity to consume posited by the Permanent Income Hypothesis (PIH) of consumption theory. An illustrative empirical application is given, in which the Phillips Curve relationship between inflation and unemployment is found to be negligible at low frequencies, corresponding to periods ≥ 18 months, but inverse at higher frequencies, just as predicted by Friedman and Phelps in the 1960s.

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File URL: http://www.tandfonline.com/doi/abs/10.1080/07474930802387753
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Article provided by Taylor & Francis Journals in its journal Econometric Reviews.

Volume (Year): 28 (2009)
Issue (Month): 1-3 ()
Pages: 4-20

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Handle: RePEc:taf:emetrv:v:28:y:2009:i:1-3:p:4-20
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  1. Jushan Bai & Pierre Perron, 1998. "Estimating and Testing Linear Models with Multiple Structural Changes," Econometrica, Econometric Society, vol. 66(1), pages 47-78, January.
  2. Marianne Baxter & Robert G. King, 1995. "Measuring Business Cycles Approximate Band-Pass Filters for Economic Time Series," NBER Working Papers 5022, National Bureau of Economic Research, Inc.
  3. Engle, Robert F, 1978. "Testing Price Equations for Stability across Spectral Frequency Bands," Econometrica, Econometric Society, vol. 46(4), pages 869-81, July.
  4. Tan, Hui Boon & Ashley, Richard, 1999. "Detection And Modeling Of Regression Parameter Variation Across Frequencies," Macroeconomic Dynamics, Cambridge University Press, vol. 3(01), pages 69-83, March.
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  8. Richard A. Ashley & Randall J. Verbrugge., 2006. "Mis-Specification in Phillips Curve Regressions: Quantifying Frequency Dependence in This Relationship While Allowing for Feedback," Working Papers e06-11, Virginia Polytechnic Institute and State University, Department of Economics.
  9. Lawrence J. Christiano & Terry J. Fitzgerald, 1999. "The Band pass filter," Working Paper 9906, Federal Reserve Bank of Cleveland.
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  12. repec:cup:macdyn:v:3:y:1999:i:1:p:69-83 is not listed on IDEAS
  13. Diego Comin & Mark Gertler, 2003. "Medium Term Business Cycles," NBER Working Papers 10003, National Bureau of Economic Research, Inc.
  14. Hodrick, Robert J & Prescott, Edward C, 1997. "Postwar U.S. Business Cycles: An Empirical Investigation," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 29(1), pages 1-16, February.
  15. Dean Corbae & Sam Ouliaris & Peter C.B. Phillips, 1997. "Band Spectral Regression with Trending Data," Cowles Foundation Discussion Papers 1163, Cowles Foundation for Research in Economics, Yale University.
  16. Lawrence J. Christiano & Terry J. Fitzgerald, 2003. "Inflation and monetary policy in the twentieth century," Economic Perspectives, Federal Reserve Bank of Chicago, issue Q I, pages 22-45.
  17. Edmund S. Phelps, 1968. "Money-Wage Dynamics and Labor-Market Equilibrium," Journal of Political Economy, University of Chicago Press, vol. 76, pages 678.
  18. Ashley, Richard, 1984. "A Simple Test for Regression Parameter Instability," Economic Inquiry, Western Economic Association International, vol. 22(2), pages 253-68, April.
  19. repec:cup:etheor:v:13:y:1997:i:3:p:315-52 is not listed on IDEAS
  20. Farley, John U. & Hinich, Melvin & McGuire, Timothy W., 1975. "Some comparisons of tests for a shift in the slopes of a multivariate linear time series model," Journal of Econometrics, Elsevier, vol. 3(3), pages 297-318, August.
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