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

Abstract

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 frequenciesjust as predicted by Friedman and Phelps in the 1960's. JEL Classification Codes: C22, E00, E31 ⋆ We thank

Tracking Trend Inflation: Nonseasonally Adjusted Variants of the Median and Trimmed-Mean CPI

“…16 These uses are misguided. Two-sided filters distort econometric inference (see Ashley and Verbrugge, 2009); and their use as a stand-in trend inflation indicator is conceptually misguided since, by design, information from the future is required to compute them in the first place. Market participants would not use the Hodrick-Prescott filter in a onesided manner (i.e.…”

Persistence Dependence in Empirical Relations: The Velocity of Money

“…But recent developments in econometric theory allow one to properly circumvent these difficulties and obtain correct inferences, despite the presence of persistence dependence in a relationship. In particular, here we demonstrate how the nonlinear tools of Ashley and Verbrugge (2009a) can be used to obtain a more complete (and richer) depiction of the relationship between the velocity of money and the interest rate, and to formally test for frequency-dependence in this relationship. We find that the relationship between velocity and interest rates is strong at low frequencies, but essentially nonexistent at high frequencies, even though the data needed to be differenced in this case, since the levels of these two variables are not covariance stationary.…”

“…Ashley and Verbrugge (2009a) observes that a linear relationship between   and  −1 implies that the relationship between   and  −1 is the same at all frequencies; that is, if  −1 experiences a one-standard-deviation increase, then   responds in the same way regardless of whether that one-standard-deviation increase is part of an unusually persistent movement in  −1 or whether it is part of an unusually transient movement in  −1 . Now consider the following hypothetical consumption function:…”

“…This filter was introduced in Ashley (1999a and1999b) and is also discussed in detail in Ashley and Verbrugge (2009a), so we only briefly review it here.…”

Frequency Dependence in a Real-Time Monetary Policy Rule

“…We use a 36-month moving window in the analysis, which implies that persistence can vary in 19 steps, from completely "permanent" -referring to a fluctuation with an average reversion period of more than 36 months -to "temporary," referring to a fluctuation with an average reversion period of just two months. 4 Explicitly allowing for varying persistence in the central bank's responses to fluctuations in the unemployment rate renders explicit modeling of a time-varying NAIRU unnecessary.…”