## Essays in Econometrics /

- Author(s): Moon, Jong-Myun
- et al.

## Abstract

Chapter 1 studies transformation models T₀(Y)=Xʹ[beta]₀ + [epsilon] with an unknown monotone transformation T₀. Our focus is on the identification and estimation of [beta]₀, leaving the specification of T₀ and the distribution of [epsilon] nonparametric. We identify [beta]₀ under a new set of conditions; specifically, we demonstrate that identification may be achieved even when the regressor X has bounded support and contains discrete random variables. Our identification is constructive and leads to sieve extremum estimator. The empirical criterion of our estimator has a U-process structure, and therefore does not conform to existing results in the sieve estimation literature. We derive the convergence rate of the estimator and demonstrate its asymptotic normality. For inference, the weighted bootstrap is proved to be consistent. The estimator is simple to implement with standard optimization algorithms. A simulation study provides insight on its finite-sample performance. Chapter 2 is concerned with the problem of maximum likelihood estimation and inference in parametric models that are finitely identified. That identification is finite means that there is a finite set of parameter values that are observationally equivalent, i.e. they generate the same distribution of the observed variables. This implies in particular that no amount of sample information (data) can allow the econometrician to distinguish between those parameter values. Under finite identification, the asymptotic distribution of the maximum likelihood estimator is nonstandard and we study its properties. In particular, we show that bootstrap can be used to uncover the identified set. Inference on a simple null hypothesis is conducted using likelihood ratio, Lagrange multiplier Wald test statistics, which in this situation are no longer asymptotically equivalent. In Chapter 3, we study a family of nonparametric tests of density ratio ordering between two continuous probability distributions on the real line. Density ratio ordering is satisfied when the two distributions admit a nonincreasing density ratio. Equivalently, density ratio ordering is satisfied when the ordinal dominance curve associated with the two distributions is concave. To test this property, we consider statistics based on the Lp-distance between an empirical ordinal dominance curve and its least concave majorant. We derive the limit distribution of these statistics when density ratio ordering is satisfied. Further, we establish that, when 1<̲ p <̲ 2, the limit distribution is stochastically largest when the two distributions are equal. When 2< p<̲ [infinity], this is not the case, and in fact the limit distribution diverges to infinity along a suitably chosen sequence of concave ordinal dominance curves. Our results serve to clarify, extend and amend assertions appearing previously in the literature for the cases p = 1 and p = [infinity]. We provide numerical evidence confirming their relevance in finite samples