A new discussion paper by Marine Carrasco and Guy Tchuente, KDPE 1608, September 2016
In many empirical works in economics, the aim of the researcher is to establish a causal or a noncausal relationship between two variables. Because unobserved variables affect most economics variables, identification and estimation of parameters of interest suffer from the endogeneity problem. In presence of endogeneity, identification of the causal parameter of interest is achieved using instrumental variables. The instrumental variables are assumed to be highly correlated with the right-hand side endogenous variables (strong) and uncorrelated with the structural error (valid or respecting the exclusion restriction).
In empirical applications, finding a valid instrumental variable that is at the same time strong is very difficult. This difficulty to have perfect instruments has encouraged the study of inference and estimation in presence of weak (weakly correlated with the right-hand side endogenous variables) instruments. We assume, in the present paper, that we have many valid (but potentially weak) instrumental variables. We use them along side with robust to weak instruments robust statistics to improve the quality of the inference. The Anderson-Rubin (AR) (Anderson and Rubin (1949)) test is an example of such robust to weak identification procedure.
We examine the regularization of the AR test when the number of instruments is large. The regularized AR tests use information-reduction methods to provide robust inference in instrumental variable estimation for data-rich environments. We derive the asymptotic properties of the regularized AR tests. Their asymptotic distributions depend on unknown eigenvalues and a regularization parameter. A bootstrap method is used to obtain more reliable inference. The regularized tests are robust to many moment conditions in the sense that they are valid for both few and many instruments, and even for more instruments than the sample size. We perform a limited set of Monte Carlo experiments. Our simulations show that the proposed AR tests work well and have better performance than competing AR tests when the number of instruments is very large.