Using Lagrange multiplier type tests to detect structural intra-person heterogeneity in composite marginal likelihood estimation in panel data sets

Abstract

Gradient-based Lagrange multiplier-type tests represent a valuable tool for discriminating between nested models, obviating the necessity to estimate the unrestricted model. This is particularly advantageous when testing for pooling in panel data sets, as it permits the testing of multiple groupings without the necessity of re-estimating the model for each grouping. This makes the process considerably faster and more flexible in comparison to Wald or likelihood ratio type tests.

In this paper, we demonstrate that the use of pairwise composite marginal likelihood (CML) estimation enables the comparison of gradients between different CML contributions of pairs of observations for individuals. This allows for the testing of pooling over time, as well as the identification of neglected temporal correlation. The CML approach thus offers a degree of flexibility that is not present in the classical likelihood setting.

Theoretical derivations of the asymptotic distribution of the test statistics under the null hypothesis are provided for the special case of multinomial probit models, thereby forming the basis for the statistical interpretation of the test statistic.

Moreover, a comprehensive simulation study was conducted to assess the finite-sample performance of the test statistics. In particular, the distribution of the test statistic under the null hypothesis and the rejection rates of the tests under various types and degrees of violations of the null hypothesis were evaluated using synthetic panel data sets of varying sizes. This empirical evaluation provides insights into the effectiveness and reliability of the proposed tests in detecting intra-personal heterogeneity and into causes of misspecifications in the deterministic utility structure.