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**A parametric approach to flexible nonlinear inference.**
*(English)*
Zbl 1017.62076

Summary: 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 \(y_t=\mu(x_t)+\epsilon_t\), where the functional form of \(\mu(\cdot)\) is unknown. We propose viewing \(\mu(\cdot)\) itself as the outcome of a random process.

The paper introduces a new stationary random field \(m(\cdot)\) that generalizes finite-differenced Brownian motion to a vector field and whose realizations could represent a broad class of possible forms for \(\mu(\cdot)\). We view the parameters that characterize the relation between a given realization of \(m(\cdot)\) and the particular value of \(\mu(\cdot)\) 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 \(\mu(\cdot)\).

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.

The paper introduces a new stationary random field \(m(\cdot)\) that generalizes finite-differenced Brownian motion to a vector field and whose realizations could represent a broad class of possible forms for \(\mu(\cdot)\). We view the parameters that characterize the relation between a given realization of \(m(\cdot)\) and the particular value of \(\mu(\cdot)\) 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 \(\mu(\cdot)\).

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.