This paper proposes a new framework for determining whether a given relatio
nship is nonlinear, what the nonlinearity looks like, and whether it is ade
quately described by a particular parametric model. The paper studies a reg
ression or forecasting model of the form y(t) = mu (x(t)) + epsilon (t) whe
re the functional form of mu(.) is unknown. We propose viewing mu(.) 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 v
ector field and whose realizations could represent a broad class of possibl
e forms for mu(.). We view the parameters that characterize the relation be
tween a given realization of m(.) and the particular value of ILL(.) for a
given sample as population parameters to be estimated by maximum likelihood
or Bayesian methods. We show that the resulting inference about the functi
onal relation also yields consistent estimates for a broad class of determi
nistic functions mu(.). The paper further develops a new test of the null h
ypothesis of linearity based on the Lagrange multiplier principle and small
-sample confidence intervals based on numerical Bayesian methods, An empiri
cal application suggests that properly accounting for the nonlinearity of t
he inflation-unemployment trade-off may explain the previously reported une
ven empirical success of the Phillips Curve.