Testing underidentification in linear models, with applications to dynamic panel and asset pricing models

This paper develops the links between overidentification tests, underidentification tests, score tests and the Cragg and Donald (1993, 1997) and Kleibergen and Paap (2006) rank tests in linear instrumental variable (IV) models. For the structural linear model $y=X\beta +u$, with the endogenous explanatory variables partitioned as $X=\left(x_1{X}_2\right)$, this general framework shows that standard underidentification tests are tests for overidentification in an auxiliary linear model, $x_1=X_2\delta +\varepsilon$, estimated by IV estimation methods using the same instruments as for the original model. This simple structure makes it possible to establish valid robust underidentification tests for linear IV models where these have not been proposed or used before, like clustered dynamic panel data models estimated by GMM. The framework also applies to tests for the rank of general parameter matrices. Invariant rank tests are based on the LIML or continuously updated GMM estimators of both structural and first-stage parameters. This insight leads to the proposal of new two-step invariant asymptotically efficient GMM estimators, and a new iterated GMM estimator that, if it converges, converges to the continuously updated GMM estimator.