Essays on bootstrap inference under weakly identified models

Instrumental Variables (IV) are widely used in econometrics to overcome endogeneity problem in regression models, which occurs when regressors are correlated with the stochastic component. Nonetheless, in applied works, practitioners face with instruments that are collectively ``weak'', i.e. poorly correlated with endogenous regressors. Under weak instruments, conventional estimators are no longer consistent and asymptotically normal. Furthermore, bootstrap methods could be useful to improve inference in IV estimation. However, under poorly relevant instruments, the bootstrap is deemed invalid and its use is generally discouraged in applied papers. In this work, we propose a new derivation of bootstrapped IV estimators under weak instruments asymptotics (Stock and Yogo, 2005) using residual--based bootstrap method involving fixed or resampled instruments. We prove that bootstrap counterpart of estimators, conditionally on the data, converges to a random distribution preserving some patterns (non--normality) of weak and irrelevant instruments scenarios. These issues may be also reflected in bootstrap--based confidence sets and hypothesis testing. In this sense, we explore the usefulness of bootstrap methods to provide information on the weakness (or the strength) of the instruments. We consider descriptive indicators and develop new bootstrap-based tests useful to detect weak instruments in IV framework. The method basically relies on Angelini et al. (2016) and allows to test normality of a certain number of (possibly standardized) bootstrap replications. Since conventional normality tests can lose power in presence of more instruments and high endogeneity, we propose new test statistics with the aim to test standard normality on the bootstrap replications. These tests are based on the moments of standard normal and are asymptotically chi--square distributed under the null hypothesis. In conclusion, we find that, in some cases, bootstrapped estimators may be used to test weak identification.