Testing Over- and Underidentification in Linear Models, with Applications to Dynamic Panel Data and Asset-Pricing Models

This paper develops the links between overidentification tests, underidentification tests, score tests and the Cragg-Donald (1993, 1997) and Kleibergen-Paap (2006) rank tests in linear instrumental variables (IV) models. This general framework shows that standard underidentification tests are (robust) score tests for overidentification in an auxiliary linear model, x_1 = X_2 δ + ε_1, where X = [x_1 X_2] are the endogenous explanatory variables in the original model, 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 general tests of rank, including the I test of Arellano, Hansen and Sentana (2012), and, outside the IV setting, for tests of rank of parameter matrices estimated by OLS. Invariant rank tests are based on LIML or continuously updated GMM estimators of the first-stage parameters. This insight leads to the proposal of a new two-step invariant asymptotically efficient GMM estimator, and a new iterated GMM estimator that converges to the continuously updated GMM estimator.