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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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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. Engle, Robert F, 1978. "Testing Price Equations for Stability across Spectral Frequency Bands," Econometrica, Econometric Society, vol. 46(4), pages 869-81, July.
  2. Dean Corbae & Sam Ouliaris & Peter C. B. Phillips, 2002. "Band Spectral Regression with Trending Data," Econometrica, Econometric Society, vol. 70(3), pages 1067-1109, May.
  3. Lawrence J. Christiano & Terry J. Fitzgerald, 2003. "The Band Pass Filter," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 44(2), pages 435-465, 05.
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  9. R. F. Engle, 1972. "Band Spectrum Regressions," Working papers 96, Massachusetts Institute of Technology (MIT), Department of Economics.
  10. 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.
  11. 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.
  12. 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.
  13. Ashley, Richard, 1984. "A Simple Test for Regression Parameter Instability," Economic Inquiry, Western Economic Association International, vol. 22(2), pages 253-68, April.
  14. 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.
  15. 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.
  16. Bai, Jushan, 1997. "Estimating Multiple Breaks One at a Time," Econometric Theory, Cambridge University Press, vol. 13(03), pages 315-352, June.
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  18. BAI, Jushan & PERRON, Pierre, 1998. "Computation and Analysis of Multiple Structural-Change Models," Cahiers de recherche 9807, Universite de Montreal, Departement de sciences economiques.
  19. Perron, P. & Bai, J., 1995. "Estimating and Testing Linear Models with Multiple Structural Changes," Cahiers de recherche 9552, Universite de Montreal, Departement de sciences economiques.
  20. Edmund S. Phelps, 1968. "Money-Wage Dynamics and Labor-Market Equilibrium," Journal of Political Economy, University of Chicago Press, vol. 76, pages 678.
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