We provide general formulation of weak identification in semiparametric models and an efficiency concept. Weak identification occurs when a parameter is weakly regular, i.e., when it is locally homogeneous of degree zero. When this happens, consistent or equivariant estimation is shown to be impossible. We then show that there exists an underlying regular parameter that fully characterizes the weakly regular parameter. While this parameter is not unique, concepts of sufficiency and minimality help pin down a desirable one. If estimation of minimal sufficient underlying parameters is inefficient, it introduces noise in the corresponding estimation of weakly regular parameters, whence we can improve the estimators by local asymptotic Rao-Blackwellization. We call an estimator weakly efficient if it does not admit such improvement. New weakly efficient estimators are presented in linear IV and nonlinear regression models. Simulation of a linear IV model demonstrates how 2SLS and optimal IV estimators are improved.
This paper presents a new perspective on the identification at infinity for the intercept of the sample selection model as identification at the boundary via a transformation of the selection index. This perspective suggests generalizations of estimation at infinity to kernel regression estimation at the boundary and further to local linear estimation at the boundary. The proposed kernel-type estimators with an estimated transformation are proven to be nonparametric-rate consistent and asymptotically normal under mild regularity conditions. A fully data-driven method of selecting the optimal b…
This paper analyzes a semiparametric model of network formation in the presence of unobserved agent-specific heterogeneity. The objective is to identify and estimate the preference parameters associated with homophily on observed attributes when the distributions of the unobserved factors are not parametrically specified. This paper offers two main contributions to the literature on network formation. First, it establishes a new point identification result for the vector of parameters that relies on the existence of a special repressor. The identification proof is constructive and characterize…
Parameter estimation via M- and Z-estimation is equally powerful in semiparametric models for one-dimensional functionals due to a one-to-one relation between corresponding loss and identification functions via integration and differentiation. For multivariate functionals such as multiple moments, quantiles, or the pair (Value at Risk, Expected Shortfall), this one-to-one relation fails and not every identification function possesses an antiderivative. The most important implication is an efficiency gap: The most efficient Z-estimator often outperforms the most efficient M-estimator. We theore…
Nonseparable panel models are important in a variety of economic settings, including discrete choice. This paper gives identification and estimation results for nonseparable models under time homogeneity conditions that are like "time is randomly assigned" or "time is an instrument." Partial identification results for average and quantile effects are given for discrete regressors, under static or dynamic conditions, in fully nonparametric and in semiparametric models, with time effects. It is shown that the usual, linear, fixed-effects estimator is not a consistent estimator of the identified …
We study a linear random coefficient model where slope parameters may be correlated with some continuous covariates. Such a model specification may occur in empirical research, for instance, when quantifying the effect of a continuous treatment observed at two time periods. We show one can carry identification and estimation without instruments. We propose a semiparametric estimator of average partial effects and of average treatment effects on the treated. We showcase the small sample properties of our estimator in an extensive simulation study. Among other things, we reveal that it compares …
We propose a new semiparametric approach for modelling nonlinear univariate diffusions, where the observed process is a nonparametric transformation of an underlying parametric diffusion (UPD). This modelling strategy yields a general class of semiparametric Markov diffusion models with parametric dynamic copulas and nonparametric marginal distributions. We provide primitive conditions for the identification of the UPD parameters together with the unknown transformations from discrete samples. Likelihood-based estimators of both parametric and nonparametric components are developed and we anal…
Wang and Tchetgen Tchetgen (2017) studied identification and estimation of the average treatment effect when some confounders are unmeasured. Under their identification condition, they showed that the semiparametric efficient influence function depends on five unknown functionals. They proposed to parameterize all functionals and estimate the average treatment effect from the efficient influence function by replacing the unknown functionals with estimated functionals. They established that their estimator is consistent when certain functionals are correctly specified and attains the semiparame…
Triangular systems with nonadditively separable unobserved heterogeneity provide a theoretically appealing framework for the modelling of complex structural relationships. However, they are not commonly used in practice due to the need for exogenous variables with large support for identification, the curse of dimensionality in estimation, and the lack of inferential tools. This paper introduces two classes of semiparametric nonseparable triangular models that address these limitations. They are based on distribution and quantile regression modelling of the reduced form conditional distributio…
This paper provides new uniform rate results for kernel estimators of absolutely regular stationary processes that are uniform in the bandwidth and in infinite-dimensional classes of dependent variables and regressors. Our results are useful for establishing asymptotic theory for two-step semiparametric estimators in time series models. We apply our results to obtain nonparametric estimates and their rates for Expected Shortfall processes.
This paper studies the estimation of causal parameters in the generalized local average treatment effect (GLATE) model, a generalization of the classical LATE model encompassing multi-valued treatment and instrument. We derive the efficient influence function (EIF) and the semiparametric efficiency bound (SPEB) for two types of parameters: local average structural function (LASF) and local average structural function for the treated (LASF-T). The moment condition generated by the EIF satisfies two robustness properties: double robustness and Neyman orthogonality. Based on the robust moment con…
The purpose of this paper is to provide guidelines for empirical researchers who use a class of bivariate threshold crossing models with dummy endogenous variables. A common practice employed by the researchers is the specification of the joint distribution of the unobservables as a bivariate normal distribution, which results in a bivariate probit model. To address the problem of misspecification in this practice, we propose an easy-to-implement semiparametric estimation framework with parametric copula and nonparametric marginal distributions. We establish asymptotic theory, including root-n…
The widespread co-existence of misspecification and weak identification in asset pricing has led to an overstated performance of risk factors. Because the conventional Fama and MacBeth (1973) methodology is jeopardized by misspecification and weak identification, we infer risk premia by using a double robust Lagrange multiplier test that remains reliable in the presence of these two empirically relevant issues. Moreover, we show how the identification, and the resulting appropriate interpretation, of the risk premia is governed by the relative magnitudes of the misspecification J-statistic and…
We study the impact of weak identification in discrete choice models, and provide insights into the determinants of identification strength in these models. Using these insights, we propose a novel test that can consistently detect weak identification in commonly applied discrete choice models, such as probit, logit, and many of their extensions. Furthermore, we demonstrate that when the null hypothesis of weak identification is rejected, Wald-based inference can be carried out using standard formulas and critical values. A Monte Carlo study compares our proposed testing approach against commo…
The identification of the network effect is based on either group size variation, the structure of the network or the relative position in the network. I provide easy-to-verify necessary conditions for identification of undirected network models based on the number of distinct eigenvalues of the adjacency matrix. Identification of network effects is possible; although in many empirical situations existing identification strategies may require the use of many instruments or instruments that could be strongly correlated with each other. The use of highly correlated instruments or many instrument…
This paper studies optimal decision rules, including estimators and tests, for weakly identified GMM models. We derive the limit experiment for weakly identified GMM, and propose a theoretically-motivated class of priors which give rise to quasi-Bayes decision rules as a limiting case. Together with results in the previous literature, this establishes desirable properties for the quasi-Bayes approach regardless of model identification status, and we recommend quasi-Bayes for settings where identification is a concern. We further propose weighted average power-optimal identification-robust freq…
We develop a concept of weak identification in linear IV models in which the number of instruments can grow at the same rate or slower than the sample size. We propose a jackknifed version of the classical weak identification-robust Anderson-Rubin (AR) test statistic. Large-sample inference based on the jackknifed AR is valid under heteroscedasticity and weak identification. The feasible version of this statistic uses a novel variance estimator. The test has uniformly correct size and good power properties. We also develop a pre-test for weak identification that is related to the size property…
