Inference on power law spatial trends

TL;DR

This paper develops methods for inference on power law spatial trends with unknown exponents, establishing consistency and asymptotic normality of nonlinear least-squares estimates under weak dependence, and discusses extensions to more complex models.

Contribution

It introduces a novel inference framework for power law models with unknown parameters, including a generic consistency result for complex estimators.

Findings

Establishes consistency and asymptotic normality of parameter estimates.

Provides a basis for inference on exponents or coefficients in power law models.

Discusses potential extensions to irregular data and heteroscedastic errors.

Abstract

Power law or generalized polynomial regressions with unknown real-valued exponents and coefficients, and weakly dependent errors, are considered for observations over time, space or space--time. Consistency and asymptotic normality of nonlinear least-squares estimates of the parameters are established. The joint limit distribution is singular, but can be used as a basis for inference on either exponents or coefficients. We discuss issues of implementation, efficiency, potential for improved estimation and possibilities of extension to more general or alternative trending models to allow for irregularly spaced data or heteroscedastic errors; though it focusses on a particular model to fix ideas, the paper can be viewed as offering machinery useful in developing inference for a variety of models in which power law trends are a component. Indeed, the paper also makes a contribution that is potentially relevant to many other statistical models: Our problem is one of many in which consistency of a vector of parameter estimates (which converge at different rates) cannot be established by the usual techniques for coping with implicitly-defined extremum estimates, but requires a more delicate treatment; we present a generic consistency result.