Neyman's C(\alpha) Test for Unobserved Heterogeneity

TL;DR

This paper introduces a unified Neyman's $C(\alpha)$ testing framework for unobserved heterogeneity in parametric models, extending to multi-dimensional cases and improving power without restrictive regularity conditions.

Contribution

It develops a general $C(\alpha)$ test approach for heterogeneity, removing common regularity assumptions and extending the theory to multi-dimensional heterogeneity.

Findings

Established local asymptotic optimality for the tests.

Extended the framework to multi-dimensional heterogeneity.

Suggested modifications for existing tests to improve power.

Abstract

A unified framework is proposed for tests of unobserved heterogeneity in parametric statistic models based on Neyman's $C(\alpha)$ approach. Such tests are irregular in the sense that the first order derivative of the log likelihood with respect to the heterogeneity parameter is identically zero, and consequently the conventional Fisher information about the parameter is zero. Nevertheless, local asymptotic optimality of the $C(\alpha)$ tests can be established via LeCam's differentiability in quadratic mean and the limit experiment approach. This leads to local alternatives of order $n^{-1/4}$. The scalar case result is already familiar from existing literature and we extend it to the multi-dimensional case. The new framework reveals that certain regularity conditions commonly employed in earlier developments are unnecessary, i.e. the symmetry or third moment condition imposed on the heterogeneity distribution. Additionally, the limit experiment for the multi-dimensional case suggests modifications on existing tests for slope heterogeneity in cross sectional and panel data models that lead to power improvement. Since the $C(\alpha)$ framework is not restricted to the parametric model and the test statistics do not depend on the particular choice of the heterogeneity distribution, it is useful for a broad range of applications for testing parametric heterogeneity.