Essays on Identification and Weak Identification /

My doctoral dissertation aims to study several issues on identification and weak identification, with applications in linear instrumental variables (IV) models and transformation models. Chapter 1 is joint with Patrik Guggenberger, Frank Kleibergen and Sophocles Mavroeidis, which considers tests of a simple null hypothesis on a subset of the coefficients of the exogenous and endogenous regressors in a single-equation linear IV model with potentially weak identification. Existing methods of subset inference (i) rely on the assumption that the parameters not under test are strongly identified, or (ii) are based on projection-type arguments. We show that under homoskedasticity the subset Anderson and Rubin (1949) test, which replaces unknown parameters by limited information maximum likelihood (LIML) estimates has correct asymptotic size without imposing additional identification assumptions, but that the corresponding subset Lagrange multiplier (LM) test is size distorted asymptotically. Subsequently, Chapter 2, joint with Qihui Chen and Patrik Guggenberger, derives the asymptotic size of the corresponding subset LM test, and shows it is size distorted. We provide the smallest nonrandom size corrected (SC) critical value that ensures that the resulting "SC subset LM test" has correct asymptotic size. We introduce an easy to implement generalized moment selection plug-in SC subset LM test ("GMS-PSC subset LM test" from now on) that uses a data-dependent critical value that gives correct asymptotic size. Chapter 3 focuses on transformation models. It provides sufficient conditions for transformation models with endogenous regressors H(Y) =X[beta]+U to be identified under conditional moment restrictions, E(U/Z)=0, where Z is the IV for X. Allowing observables (X, Y, Z) and unobservable U to be high-dimensional, we show that the assumption of completeness suffices for identification. Based on the identification results, we propose to apply the penalized sieve minimum distance estimator (ĥ[beta]̂ in Chen and Pouzo (2009) with possible shape constraints to estimate (ĥ[beta]̂. Demand model of differentiated products markets is considered as an application of transformation models, and we provide the identification and PSMD estimation results for its parameter