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

This paper 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 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 greater than a year, but inverse at higher frequencies – just as predicted by Friedman and Phelps in the 1960’s. JEL Classification Codes: C22, E00, E31 ⋆ We thank Anthony Atkinson, Marianne Baxter, Larry Christiano, Walter Enders, Jonas Fisher, Steve Reed, and John Williams for helpful comments. All errors, misinterpretations, and omissions are ours. All views expressed in this paper are those of the authors and do not reflect the views or policies of the Bureau of Labor Statistics or the views of other BLS staff members. Email addresses: [email protected] (Richard Ashley),