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 y_t=μ( x _t)+ε_t 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.