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A Parametric Approach to Flexible Nonlinear Inference

James D. Hamilton

Abstract

This paper proposes a new framework for determining whether a given relationship is nonlinear, what the nonlinearity looks like, and whether it is adequately described by a particular parametric model. The paper studies a regression or forecasting model of the form yt=μ(xt)+st where the functional form of μ(⋅) is unknown. We propose viewing μ(⋅) itself as the outcome of a random process. The paper introduces a new stationary random field m(⋅) that generalizes finite-differenced Brownian motion to a vector field and whose realizations could represent a broad class of possible forms for μ(⋅). We view the parameters that characterize the relation between a given realization of m(⋅) and the particular value of μ(⋅) for a given sample as population parameters to be estimated by maximum likelihood or Bayesian methods. We show that the resulting inference about the functional relation also yields consistent estimates for a broad class of deterministic functions μ(⋅). The paper further develops a new test of the null hypothesis of linearity based on the Lagrange multiplier principle and small-sample confidence intervals based on numerical Bayesian methods. An empirical application suggests that properly accounting for the nonlinearity of the inflation-unemployment trade-off may explain the previously reported uneven empirical success of the Phillips Curve.