“Nonparametric Estimation of Triangular Simultaneous Equations Models under Weak Identification” (Latest version: Feb 15, 2014)
Abstract: This paper analyzes the problem of weak instruments on identification, estimation, and inference in a simple nonparametric model of a triangular system. The paper derives a necessary and sufficient rank condition for identification, based on which weak identification is established. Then nonparametric weak instruments are defined as a sequence of reduced form functions where the associated rank shrinks to zero. The problem of weak instruments is characterized to be similar to the ill-posed inverse problem, which motivates the introduction of a regularization scheme. The paper proposes a penalized series estimation method to alleviate the effects of weak instruments. The rate of convergence of the resulting estimator is given, and it is shown that weak instruments slow down the rate and penalization derives a faster rate. Consistency and asymptotic normality results are also derived. Monte Carlo results are presented, and an empirical example is given, where the effect of class size on test scores is estimated nonparametrically.
“Identification in a Generalization of Bivariate Probit Models with Endogenous Regressors” with Edward Vytlacil (Submitted)
“Robust Inference in Bivariate Threshold Crossing Models under Weak Identification”
Summary: This paper analyzes the properties of ML estimators under weak identification in bivariate threshold crossing models of triangular system and develops a robust inference procedure. Weak identification is resulted in due to weak instruments or more generally as the reduced form relationship becomes weak. Drifting sequences of coefficients are introduced to model the full range of strength of identification, and asymptotic theory of ML estimators are followed. Inference procedures are introduced so that tests and confidence sets are robust to weak identification. As a related work, Andrews and Cheng (2012) develop asymptotic theory and robust inference procedures in a particular extremum estimation framework, where the criterion function does not depend on the non-identified parameters under the lack of identification. This framework is not directly applicable to our problem, and we develop a novel reparametrization scheme that allows us to separate out parameters that are subject to identification problems.
“Reexamining The Secular Trend in the Standard of Living During Industrialization in Britain”
Summary: This note studies the long-term trend in the standard of living during the Industrial Revolution in Britain. In particular, we investigate methods and findings in Komlos's (1993) study. As a proxy for the living standard, he estimates the trend in the mean heights of subgroups of the population, and concludes that the living standard has deteriorated throughout the period. In this note, we examine his findings by analyzing the two-step procedure that Komlos employs to deal with the sample that is truncated due to institutional policies. We find that, in each step of the procedure, Komlos requires a set of strong assumptions that is not consistent with the data. We show, however, that the first step procedure is justified under weaker assumptions, which implies that the result obtained from this procedure is robust. In doing so, we develop a generalized version of the main theorem based on which Komlos employs his procedure. Despite the validity of the first step, we show that a fairly general distributional assumption that justifies the first step creates bias in the second step. We also find that, even with the same data and the two-step procedure that Komlos uses, one of his reported graphs, which is most supportive to his conclusions, cannot be replicated. Lastly, by decomposing the first and second step of Komlos's procedure, we find that his final results are mainly driven by the second step procedure. We calculate an alternative trend using a method that does not create bias in the second step, which does not present a downward trend. According to the result, the living standard during the period has not deteriorated.
“Invalidity of the Bootstrap and the m out of n Bootstrap for Confidence Interval Endpoints Defined by Moment Inequalities,” with Donald Andrews, Econometrics Journal (2009), Volume 12, pp. S172–S199
Abstract: This paper analyses the finite-sample and asymptotic properties of several bootstrap and m out of n bootstrap methods for constructing confidence interval (CI) endpoints in models defined by moment inequalities. In particular, we consider using these methods directly to construct CI endpoints. By considering two very simple models, the paper shows that neither the bootstrap nor the m out of n bootstrap is valid in finite samples or in a uniform asymptotic sense in general when applied directly to construct CI endpoints.