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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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  • 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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Suggested Citation

  • Richard A. Ashley. & Randall J. Verbrugge, 2006. "Frequency Dependence in Regression Model Coefficients: An Alternative Approach for Modeling Nonlinear Dynamic Relationships in Time Series," Working Papers e06-7, Virginia Polytechnic Institute and State University, Department of Economics.
  • Handle: RePEc:vpi:wpaper:e06-7
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    File URL: http://ashleymac.econ.vt.edu/working_papers/freq_depend.pdf
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    References listed on IDEAS

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    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. 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, May.
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    Citations

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    Cited by:

    1. Verbrugge, Randal & Higgins, Amy, 2015. "Tracking Trend Inflation: Nonseasonally Adjusted Variants of the Median and Trimmed-Mean CPI," Working Paper 1527, Federal Reserve Bank of Cleveland.
    2. 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.
    3. 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.
    4. Ciner, Cetin, 2015. "Are equities good inflation hedges? A frequency domain perspective," Review of Financial Economics, Elsevier, vol. 24(C), pages 12-17.
    5. Richard Ashley & Kwok Ping Tsang & Randal J. Verbrugge, 2010. "Frequency Dependence in a Real-Time Monetary Policy Rule," Working Papers e07-21, Virginia Polytechnic Institute and State University, Department of Economics.
    6. repec:spr:jecfin:v:41:y:2017:i:3:d:10.1007_s12197-016-9378-2 is not listed on IDEAS
    7. 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.
    8. 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.
    9. Joanna Bruzda, 2011. "The Haar Wavelet Transfer Function Model and Its Applications," Dynamic Econometric Models, Uniwersytet Mikolaja Kopernika, vol. 11, pages 141-154.
    10. Ashley, Richard & Li, Guo, 2014. "Re-examining the impact of housing wealth and stock wealth on retail sales: Does persistence in wealth changes matter?," Journal of Housing Economics, Elsevier, vol. 26(C), pages 109-118.
    11. 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.
    12. 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.
    13. 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.
    14. Ashley, Richard & Verbrugge, Randal, 2015. "Persistence Dependence in Empirical Relations: The Velocity of Money," Working Paper 1530, Federal Reserve Bank of Cleveland.
    15. 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.

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    Keywords

    Phillips Curve; spectral regression; time series analysis;

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