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)+εt 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.

This paper models occasional, discrete shifts in the growth rate of a nonstationary series. Algorithms for inferring these unobserved shifts are presented, a byproduct of which permits estimation of parameters by maximum likelihood. An empirical application of this technique suggests that the periodic shift from a positive growth rate to a negative growth rate is a recurrent feature of the U.S. business cycle, and indeed could be used as an objective criterion for defining and measuring economic recessions. The estimated parameter values suggest that a typical economic recession is associated with a 3 percent permanent drop in the level of GNP. Copyright 1989 by The Econometric Society.

This paper makes the following original contributions to the literature. (i) We develop a simpler analytical characterization and numerical algorithm for Bayesian inference in structural vector autoregressions (VARs) that can be used for models that are overidentified, just-identified, or underidentified. (ii) We analyze the asymptotic properties of Bayesian inference and show that in the underidentified case, the asymptotic posterior distribution of contemporaneous coefficients in an n-variable VAR is confined to the set of values that orthogonalize the population variance–covariance matrix of ordinary least squares residuals, with the height of the posterior proportional to the height of the prior at any point within that set. For example, in a bivariate VAR for supply and demand identified solely by sign restrictions, if the population correlation between the VAR residuals is positive, then even if one has available an infinite sample of data, any inference about the demand elasticity is coming exclusively from the prior distribution. (iii) We provide analytical characterizations of the informative prior distributions for impulse-response functions that are implicit in the traditional sign-restriction approach to VARs, and we note, as a special case of result (ii), that the influence of these priors does not vanish asymptotically. (iv) We illustrate how Bayesian inference with informative priors can be both a strict generalization and an unambiguous improvement over frequentist inference in just-identified models. (v) We propose that researchers need to explicitly acknowledge and defend the role of prior beliefs in influencing structural conclusions and we illustrate how this could be done using a simple model of the U.S. labor market