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 = Âµ(xt) + et where the functional form of Âµ(.) is unknown. We propose viewing Âµ(.) 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 Âµ(.). We view the parameters that characterize the relation between a given realization of m(.) and the particular value of Âµ(.) 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 Âµ(.). 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.
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Volume (Year): 69 (2001)
Issue (Month): 3 (May)
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