A Parametric Approach to Flexible Nonlinear Inference

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) + et 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 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 tradeoff may explain the previously reported uneven empirical success of the Phillips Curve.