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

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Author Info

  • Richard A. Ashley.
  • Randall J. Verbrugge

Abstract

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://ashleymac.econ.vt.edu/working_papers/freq_depend.pdf
File Function: First version, 2005
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Bibliographic Info

Paper provided by Virginia Polytechnic Institute and State University, Department of Economics in its series Working Papers with number e06-7.

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Length: 18 pages
Date of creation: 2006
Date of revision:
Handle: RePEc:vpi:wpaper:e06-7

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Keywords: Phillips Curve; spectral regression; time series analysis;

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References

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  1. 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.
  2. Jushan Bai & Pierre Perron, 2003. "Computation and analysis of multiple structural change models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 18(1), pages 1-22.
  3. Den Haan, Wouter & Sumner, Steven, 2001. "The Comovements between Real Activity and Prices in the G7," CEPR Discussion Papers 2801, C.E.P.R. Discussion Papers.
  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.
  5. Jushan Bai & Pierre Perron, 1998. "Estimating and Testing Linear Models with Multiple Structural Changes," Econometrica, Econometric Society, vol. 66(1), pages 47-78, January.
  6. 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.
  7. 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.
  8. repec:cup:etheor:v:13:y:1997:i:3:p:315-52 is not listed on IDEAS
  9. 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.
  10. Engle, Robert F, 1978. "Testing Price Equations for Stability across Spectral Frequency Bands," Econometrica, Econometric Society, vol. 46(4), pages 869-81, July.
  11. Stock, James H. & Watson, Mark W., 1999. "Business cycle fluctuations in us macroeconomic time series," Handbook of Macroeconomics, in: J. B. Taylor & M. Woodford (ed.), Handbook of Macroeconomics, edition 1, volume 1, chapter 1, pages 3-64 Elsevier.
  12. 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.
  13. Marianne Baxter & Robert G. King, 1999. "Measuring Business Cycles: Approximate Band-Pass Filters For Economic Time Series," The Review of Economics and Statistics, MIT Press, vol. 81(4), pages 575-593, November.
  14. Engle, Robert F, 1974. "Band Spectrum Regression," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 15(1), pages 1-11, February.
  15. Diego Comin & Mark Gertler, 2003. "Medium Term Business Cycles," NBER Working Papers 10003, National Bureau of Economic Research, Inc.
  16. Edmund S. Phelps, 1968. "Money-Wage Dynamics and Labor-Market Equilibrium," Journal of Political Economy, University of Chicago Press, vol. 76, pages 678.
  17. 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.
  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. Bai, Jushan, 1997. "Estimating Multiple Breaks One at a Time," Econometric Theory, Cambridge University Press, vol. 13(03), pages 315-352, June.
  20. repec:cup:macdyn:v:3:y:1999:i:1:p:69-83 is not listed on IDEAS
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Citations

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Cited by:
  1. Richard A. Ashley & Kwok Ping Tsang & Randal J. Verbrugge, 2013. "Frequency Dependence in a Real-Time Monetary Policy Rule," Working Papers e07-43, Virginia Polytechnic Institute and State University, Department of Economics.
  2. Richard A. Ashley & Kwok Ping Tsang, 2013. "International Evidence On The Oil Price-Real Output Relationship: Does Persistence Matter?," Working Papers e07-42, Virginia Polytechnic Institute and State University, Department of Economics.
  3. Thesia I. Garner & Randal Verbrugge, 2007. "Puzzling Divergence of U.S. Rents and User Costs, 1980-2004: Summary and Extensions," Working Papers 409, U.S. Bureau of Labor Statistics.
  4. Ciner, Cetin, 2011. "Commodity prices and inflation: Testing in the frequency domain," Research in International Business and Finance, Elsevier, vol. 25(3), pages 229-237, September.
  5. 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.
  6. Wei Yanfeng, 2013. "The Dynamic Relationships between Oil Prices and the Japanese Economy: A Frequency Domain Analysis," Review of Economics & Finance, Better Advances Press, Canada, vol. 3, pages 57-67, May.
  7. Joanna Bruzda, 2011. "The Haar Wavelet Transfer Function Model and Its Applications," Dynamic Econometric Models, Uniwersytet Mikolaja Kopernika, vol. 11, pages 141-154.
  8. Sinha, Pankaj & Agnihotri, Shalini, 2014. "Sensitivity of Value at Risk estimation to NonNormality of returns and Market capitalization," MPRA Paper 56307, University Library of Munich, Germany, revised 26 May 2014.
  9. Richard A. Ashley & Christopher F. Parmeter, 2013. "Sensitivity Analysis For Inference In 2SLS Estimation With Possibly-Flawes Instruments," Working Papers e07-38, Virginia Polytechnic Institute and State University, Department of Economics.
  10. Richard A. Ashley. & Randall J. Verbrugge., 2006. "Mis-Specification and Frequency Dependence in a New Keynesian Phillips Curve," Working Papers e06-12, Virginia Polytechnic Institute and State University, Department of Economics.

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