Testing the Number of Components in Finite Mixture Normal Regression Model with Panel Data

TL;DR

This paper extends likelihood ratio-based tests for finite mixture models to panel data, addressing unboundedness issues with penalized likelihood, and demonstrates its effectiveness through simulations and real data application to production functions.

Contribution

It develops a penalized likelihood ratio test for mixture models with panel data, extending existing methods and deriving their asymptotic distributions.

Findings

Good finite sample performance of the EM test.

Evidence of heterogeneity in production elasticities.

Production function heterogeneity beyond Hicks-neutral productivity.

Abstract

This paper develops the likelihood ratio-based test of the null hypothesis of a M0-component model against an alternative of (M0 + 1)-component model in the normal mixture panel regression by extending the Expectation-Maximization (EM) test of Chen and Li (2009a) and Kasahara and Shimotsu (2015) to the case of panel data. We show that, unlike the cross-sectional normal mixture, the first-order derivative of the density function for the variance parameter in the panel normal mixture is linearly independent of its second-order derivatives for the mean parameter. On the other hand, like the cross-sectional normal mixture, the likelihood ratio test statistic of the panel normal mixture is unbounded. We consider the Penalized Maximum Likelihood Estimator to deal with the unboundedness, where we obtain the data-driven penalty function via computational experiments. We derive the asymptotic distribution of the Penalized Likelihood Ratio Test (PLRT) and EM test statistics by expanding the log-likelihood function up to five times for the reparameterized parameters. The simulation experiment indicates good finite sample performance of the proposed EM test. We apply our EM test to estimate the number of production technology types for the finite mixture Cobb-Douglas production function model studied by Kasahara et al. (2022) used the panel data of the Japanese and Chilean manufacturing firms. We find the evidence of heterogeneity in elasticities of output for intermediate goods, suggesting that production function is heterogeneous across firms beyond their Hicks-neutral productivity terms.