Moment Condition Models Research Articles

Improving confidence set estimation when parameters are weakly identified

We consider inference in weakly identified moment condition models when additional partially identifying moment inequality constraints are available. We detail the limiting distribution of the estimation criterion function and consequently propose a confidence set estimator for the true parameter.

Conditional Inference With a Functional Nuisance Parameter

This paper shows that the problem of testing hypotheses in moment condition models without any assumptions about identification may be considered as a problem of testing with an infinite-dimensional nuisance parameter. We introduce a sufficient statistic for this nuisance parameter in a Gaussian problem and propose conditional tests. These conditional tests have uniformly correct asymptotic size for a large class of models and test statistics. We apply our approach to construct tests based on quasi-likelihood ratio statistics, which we show are efficient in strongly identified models and perform well relative to existing alternatives in two examples.

Asymptotic Refinements of a Misspecification-Robust Bootstrap for GEL Estimators

I propose a nonparametric iid bootstrap procedure for the empirical likelihood, the exponential tilting, and the exponentially tilted empirical likelihood estimators that achieves sharp asymptotic refinements for t tests and confidence intervals based on such estimators. Furthermore, the proposed bootstrap is robust to model misspecification, i.e., it achieves asymptotic refinements regardless of whether the assumed moment condition model is correctly specified or not. This result is new, because asymptotic refinements of the bootstrap based on these estimators have not been established in the literature even under correct model specification. Monte Carlo experiments are conducted in dynamic panel data setting to support the theoretical finding. As an application, bootstrap confidence intervals for the returns to schooling of Hellerstein and Imbens (1999) are calculated. The returns to schooling may be higher.

Asymptotic refinements of a misspecification-robust bootstrap for GEL estimators

I propose a nonparametric iid bootstrap procedure for the empirical likelihood, the exponential tilting, and the exponentially tilted empirical likelihood estimators that achieves asymptotic refinements for t tests and confidence intervals, and Wald tests and confidence regions based on such estimators. Furthermore, the proposed bootstrap is robust to model misspecification, i.e., it achieves asymptotic refinements regardless of whether the assumed moment condition model is correctly specified or not. This result is new, because asymptotic refinements of the bootstrap based on these estimators have not been established in the literature even under correct model specification. Monte Carlo experiments are conducted in dynamic panel data setting to support the theoretical finding. As an application, bootstrap confidence intervals for the returns to schooling of Hellerstein and Imbens (1999) are calculated. The result suggests that the returns to schooling may be higher.

IDENTIFICATION AND INFERENCE ON REGRESSIONS WITH MISSING COVARIATE DATA

This paper examines the problem of identification and inference on a conditional moment condition model with missing data, with special focus on the case when the conditioning covariates are missing. We impose no assumption on the distribution of the missing data and we confront the missing data problem by using a worst case scenario approach.We characterize the sharp identified set and argue that this set is usually too complex to compute or to use for inference. Given this difficulty, we consider the construction of outer identified sets (i.e. supersets of the identified set) that are easier to compute and can still characterize the parameter of interest. Two different outer identification strategies are proposed. Both of these strategies are shown to have nontrivial identifying power and are relatively easy to use and combine for inferential purposes.

EXISTENCE AND CHARACTERIZATION OF CONDITIONAL DENSITY PROJECTIONS

In this paper we propose primitive conditions under which a projection of a conditional density onto a set defined by conditional moment restrictions exists and is unique. Moreover, we provide an analytic expression of the obtained projection. The range of applications where conditional density projections are used is wide. The derived results are potentially useful in a variety of areas including: semiparametric efficient estimation and optimal testing in (conditional) moment models, Bayesian prior determination and inference in semiparametric models, density forecasting, and simulation-based econometric analysis.Regarding existence, we propose three different combinations of assumptions that are all sufficient to show that the projection exists and is unique. The proposed conditions exhibit a clear trade off between restrictions put on the divergence between the conditional densities and on the moment function which defines the projection set. Depending on the nature of the application, the researcher can pick and choose which set of conditions to use. Our second set of results characterizes the projection. The expression for the projected density is new though not surprising given the previously obtained results for the unconditional case. The projection is characterized by the dual of the original projection problem. In establishing the strong duality, however, we work with a constraint qualification condition that is weaker than that used by Borwein and Lewis (1991a, 1992a, 1993 in their seminal work concerning the unconditional case.

Partial Identification of State Dependence

This paper is about the empirical measurement of state dependence in dynamic binary outcomes. I propose a nonparametric dynamic potential outcomes (DPO) model and develop an array of parameters and identifying assumptions that can be considered in this model. I show how to construct sharp identified sets in the DPO model by using a flexible linear programming procedure that is valid for any of these parameters under any combination of identifying assumptions. Confidence regions for these identified sets are obtained by applying recent results from the literature on partially identified moment condition models. I apply the analysis to study state dependence in the labor force participation of married women using a well-known extract from the 1986 Panel Study of Income Dynamics (PSID), and to study state dependence in unemployment for working age high school educated men using a new extract from the 2008 Survey of Income and Program Participation (SIPP). Using conservative, non-parametric assumptions, I estimate that state dependence accounts for at least 28.1% of the observed persistence in yearly non-employment for women in the PSID sample. In the SIPP data, I find that state dependence accounts for at least 33.1% of the four-month persistence in unemployment among high school educated men. This contrasts with the findings of recent experimental studies based on fictitious resumes, which have found no evidence of short-term state dependence.

GMM with Multiple Missing Variables

We consider efficient estimation in moment conditions models with non-monotonically missing-at-random (MAR) variables. A version of MAR point-identifies the parameters of interest and gives a closed-form efficient influence function that can be used directly to obtain efficient semi-parametric generalized method of moments (GMM) estimators under standard regularity conditions. A small-scale Monte Carlo experiment with MAR instrumental variables demonstrates that the asymptotic superiority of these estimators over the standard methods carries over to finite samples. An illustrative empirical study of the relationship between a child's years of schooling and number of siblings indicates that these GMM estimators can generate results with substantive differences from standard methods. Copyright © 2015 John Wiley & Sons, Ltd.

Semiparametric estimation of models with conditional moment restrictions in the presence of nonclassical measurement errors

This paper develops a framework for the analysis of semiparametric conditional moment models with endogenous and mismeasured causes, which is of empirical importance. We show that one set of valid instruments is sufficient to control for both endogeneity and measurement errors of the causes of interest, which has been observed in linear parametric models. Two-step consistent estimators of the parameters of interest are proposed. We also show that the proposed estimators are consistent with a rate faster than n−1/4 under a certain metric, and the proposed estimators of the finite-dimensional unknown parameters obtain root-n asymptotic normality. Monte Carlo evidences show that the proposed estimators perform well under a variety of identification conditions. An application to instrumental variables estimation of Engel curves illustrates the usefulness of our method. It supports that correcting for both endogeneity and measurement errors on total expenditure is substantial in estimating economically meaningful Engel curves.

Using Implied Probabilities to Improve the Estimation of Unconditional Moment Restrictions for Weakly Dependent Data

In this article, we investigate the use of implied probabilities (Back and Brown, 1993) to improve estimation in unconditional moment conditions models. Using the seminal contributions of Bonnal and Renault (2001) and Antoine et al. (2007), we propose two three-step Euclidian empirical likelihood (3S-EEL) estimators for weakly dependent data. Both estimators make use of a control variates principle that can be interpreted in terms of implied probabilities in order to achieve higher-order improvements relative to the traditional two-step GMM estimator. A Monte Carlo study reveals that the finite and large sample properties of the three-step estimators compare favorably to the existing approaches: the two-step GMM and the continuous updating estimator.

Conditional moment models under semi-strong identification

We consider conditional moment models under semi-strong identification. Identification strength is directly defined through the conditional moments that flatten as the sample size increases. Our new minimum distance estimator is consistent, asymptotically normal, robust to semi-strong identification, and does not rely on the choice of a user-chosen parameter, such as the number of instruments or some smoothing parameter. Heteroskedasticity-robust inference is possible through Wald testing without prior knowledge of the identification pattern. Simulations show that our estimator is competitive with alternative estimators based on many instruments, being well-centered with better coverage rates for confidence intervals.

Conditional Inference with a Functional Nuisance Parameter

This paper shows that the problem of testing hypotheses in moment condition models without any assumptions about identification may be considered as a problem of testing with an infinite-dimensional nuisance parameter. We introduce a sufficient statistic for this nuisance parameter and propose conditional tests. These conditional tests have uniformly correct asymptotic size for a large class of models and test statistics. We apply our approach to construct tests based on quasi-likelihood ratio statistics, which we show are efficient in strongly identified models and perform well relative to existing alternatives in two examples.

GMM ESTIMATION AND UNIFORM SUBVECTOR INFERENCE WITH POSSIBLE IDENTIFICATION FAILURE

This paper determines the properties of standard generalized method of moments (GMM) estimators, tests, and confidence sets (CSs) in moment condition models in which some parameters are unidentified or weakly identified in part of the parameter space. The asymptotic distributions of GMM estimators are established under a full range of drifting sequences of true parameters and distributions. The asymptotic sizes (in a uniform sense) of standard GMM tests and CSs are established.The paper also establishes the correct asymptotic sizes of “robust” GMM-based Wald,t, and quasi-likelihood ratio tests and CSs whose critical values are designed to yield robustness to identification problems.The results of the paper are applied to a nonlinear regression model with endogeneity and a probit model with endogeneity and possibly weak instrumental variables.

Bayesian regression with heteroscedastic error density and parametric mean function

In this paper we consider Bayesian estimation of restricted conditional moment models with the linear regression as a particular example. A common practice in the Bayesian literature for linear regression and other semi-parametric models is to use flexible families of distributions for the errors and to assume that the errors are independent from covariates. However, a model with flexible covariate dependent error distributions should be preferred for the following reason. Assuming that the error distribution is independent of predictors might lead to inconsistent estimation of the parameters of interest when errors and covariates are dependent. To address this issue, we develop a Bayesian regression model with a parametric mean function defined by a conditional moment condition and flexible predictor dependent error densities. Sufficient conditions to achieve posterior consistency of the regression parameters and conditional error densities are provided. In experiments, the proposed method compares favorably with classical and alternative Bayesian estimation methods for the estimation of the regression coefficients and conditional densities.

Asymptotic Refinements of a Misspecification-Robust Bootstrap for Generalized Method of Moments Estimators

I propose a nonparametric iid bootstrap that achieves asymptotic refinements for t tests and confidence intervals based on the generalized method of moments (GMM) estimators even when the model is misspecified. In addition, my bootstrap does not require recentering the bootstrap moment function, which has been considered as critical for GMM. Regardless of model misspecification, the proposed bootstrap achieves the same sharp magnitude of refinements as the conventional bootstrap methods which establish asymptotic refinements by recentering in the absence of misspecification. The key idea is to link the misspecified bootstrap moment condition to the large sample theory of GMM under misspecification of Hall and Inoue (2003, Journal of Econometrics 114, 361-394). Examples of possibly misspecified moment condition models with Monte Carlo simulation results are provided: (i) Combining data sets, and (ii) invalid instrumental variables.

On the performance of block-bootstrap continuously updated GMM for a class of non-linear conditional moment models

In the context of the continuously updated generalized-methods-of-moments (GMM), this study evaluates the finite sample properties of Wald- and criterion-based bootstrap inference for a class of models defined by non-linear conditional moment functions. This work provides simulation evidence that validates the moving block-bootstrap (MBB) as an alternative to asymptotic approximation for robust finite sample GMM inference. The study considers data generating processes with highly non-linear conditional moment functions, weak instruments, and near failure of the identification condition. In the absence of a consensus on best practice when identification is weak, Monte Carlo results of this study are encouraging to the empirical researchers. For criterion-based tests, the MBB performs fairly well in reducing the error in the rejection frequency that occurs when first-order asymptotic critical values are used. In particular, it is possible to improve finite sample inference by inverting bootstrap Wald-type statistics which are commonly used in practice The bootstrap percentile- $$t$$ t confidence intervals performed better than the asymptotic confidence intervals but only marginally in weakly identified specifications with high non-linear moment functions.

Neglected heterogeneity in moment condition models

The central concern of this paper is parameter heterogeneity in models specified by a number of unconditional or conditional moment conditions and thereby the provision of a framework for the development of apposite optimal m-tests against its potential presence. We initially consider the unconditional moment restrictions framework. Optimal m-tests against moment condition parameter heterogeneity are derived with the relevant Jacobian matrix obtained in terms of the second order own derivatives of the moment indicator in a leading case. GMM and GEL tests of specification based on generalized information matrix equalities appropriate for moment-based models are described and their relation to optimal m-tests against moment condition parameter heterogeneity examined. A fundamental and important difference is noted between GMM and GEL constructions. The paper is concluded by a generalization of these tests to the conditional moment context and the provision of a limited set of simulation experiments to illustrate the efficacy of the proposed tests.

GMM Estimation and Uniform Subvector Inference with Possible Identification Failure

This paper determines the properties of standard generalized method of moments (GMM) estimators, tests, and confidence sets (CS's) in moment condition models in which some parameters are unidentified or weakly identified in part of the parameter space. The asymptotic distributions of GMM estimators are established under a full range of drifting sequences of true parameters and distributions. The asymptotic sizes (in a uniform sense) of standard GMM tests and CS's are established.The paper also establishes the correct asymptotic sizes of robust GMM-based Wald, t; and quasi-likelihood ratio tests and CS's whose critical values are designed to yield robustness to identification problems.The results of the paper are applied to a nonlinear regression model with endogeneity and a probit model with endogeneity and possibly weak instrumental variables.

GMM Estimation and Uniform Subvector Inference with Possible Identification Failure

This paper determines the properties of standard generalized method of moments (GMM) estimators, tests, and confidence sets (CS's) in moment condition models in which some parameters are unidentified or weakly identified in part of the parameter space. The asymptotic distributions of GMM estimators are established under a full range of drifting sequences of true parameters and distributions. The asymptotic sizes (in a uniform sense) of standard GMM tests and CS's are established. The paper also establishes the correct asymptotic sizes of robust GMM-based Wald, t, and quasi-likelihood ratio tests and CS's whose critical values are designed to yield robustness to identification problems. The results of the paper are applied to a nonlinear regression model with endogeneity and a probit model with endogeneity and possibly weak instrumental variables.

Testing for Common Conditionally Heteroskedastic Factors

This paper proposes a test for common conditionally heteroskedastic (CH) features in asset returns. Following Engle and Kozicki (1993), the common CH features property is expressed in terms of testable overidentifying moment restrictions. However, as we show, these moment conditions have a degenerate Jacobian matrix at the true parameter value and therefore the standard asymptotic results of Hansen (1982) do not apply. We show in this context that Hansen's (1982) J-test statistic is asymptotically distributed as the minimum of the limit of a certain random process with a markedly nonstandard distribution. If two assets are considered, this asymptotic distribution is a fifty-fifty mixture of ... and ..., where H is the number of moment conditions, as opposed to a .... With more than two assets, this distribution lies between the ... and ... (p denotes the number of parameters). These results show that ignoring the lack of first-order identification of the moment condition model leads to oversized tests with a possibly increasing overrejection rate with the number of assets. A Monte Carlo study illustrates these findings. (ProQuest: ... denotes formulae/symbols omitted.)