Conditional Empirical Likelihood Research Articles

Empirical Likelihood Ratio Tests of Conditional Moment Restrictions With Unknown Functions

This article introduces empirical likelihood ratio tests for conditional moment models in which the unknown parameter contains infinite-dimensional components. We allow unknown functions to be included in the conditional moment restrictions. We discusses (1) the limiting distribution of the sieve conditional empirical likelihood ratio (SCELR) test statistic for functionals of parameters under the null hypothesis and local alternatives; and (2) the limiting distribution of the SCELR test statistic for conditional moment restrictions (a consistent specification test) under the null hypothesis and local alternatives. A Monte Carlo study examines finite sample performance. We then apply these tests in an empirical application to construct confidence intervals for Engel curves and test restrictions on the curves.

On the finite-sample properties of conditional empirical likelihood estimators

ABSTRACTWe provide Monte Carlo evidence on the finite-sample behavior of the conditional empirical likelihood (CEL) estimator of Kitamura, Tripathi, and Ahn and the conditional Euclidean empirical likelihood (CEEL) estimator of Antoine, Bonnal, and Renault in the context of a heteroscedastic linear model with an endogenous regressor. We compare these estimators with three heteroscedasticity-consistent instrument-based estimators and the Donald, Imbens, and Newey estimator in terms of various performance measures. Our results suggest that the CEL and CEEL with fixed bandwidths may suffer from the no-moment problem, similarly to the unconditional generalized empirical likelihood estimators studied by Guggenberger. We also study the CEL and CEEL estimators with automatic bandwidths selected through cross-validation. We do not find evidence that these suffer from the no-moment problem. When the instruments are weak, we find CEL and CEEL to have finite-sample properties—in terms of mean squared error and coverage probability of confidence intervals—poorer than the heteroscedasticity-consistent Fuller (HFUL) estimator. In the strong instruments case, the CEL and CEEL estimators with automatic bandwidths tend to outperform HFUL in terms of mean squared error, while the reverse holds in terms of the coverage probability, although the differences in numerical performance are rather small.

Conditional empirical likelihood for quantile regression models

In this paper, we propose a new Bayesian quantile regression estimator using conditional empirical likelihood as the working likelihood function. We show that the proposed estimator is asymptotically efficient and the confidence interval constructed is asymptotically valid. Our estimator has low computation cost since the posterior distribution function has explicit form. The finite sample performance of the proposed estimator is evaluated through Monte Carlo studies.

Doubly robust and locally efficient estimation with missing outcomes

We consider parametric regression where the outcome is subject to miss- ingness. To achieve the semiparametric efficiency bound, most existing estimation methods require the correct modeling of certain second moments of the data, which can be very challenging in practice. We propose an estimation procedure based on the conditional empirical likelihood (CEL) method. Our method does not require us to model any second moments. We study the CEL-based inverse probability weighted (CEL-IPW) and augmented inverse probability weighted (CEL-AIPW) estimators in detail. Under some regularity conditions and the missing at ran- dom (MAR) mechanism, the CEL-IPW estimator is consistent if the missingness mechanism is correctly modeled, and the CEL-AIPW estimator is consistent if either the missingness mechanism or the conditional mean of the outcome is cor- rectly modeled. When both quantities are correctly modeled, the CEL-AIPW es- timator attains the semiparametric efficiency bound without modeling any second moments. The asymptotic distributions are derived. Numerical implementation through nested optimization routines using the Newton-Raphson algorithm is dis- cussed. We study the problem of parametric regression when the outcome is subject to missingness. The central interest is the estimation and inference of the re- gression coefficients. In practice there are many reasons that can lead to missing outcomes: budget or technique restrictions, subjects' failure to comply with the protocol, or simply the study design. Missing data usually bring big challenges to estimation and inference, as the application of statistical methods developed for data without missing values can lead to biased estimation and misleading conclusions. In addition to the outcome and covariates, we assume that some auxiliary variables are available for all subjects. Although the auxiliary variables are not of direct statistical interest, they may help to explain the missingness mecha- nism, and thus reduce the impact of missing data on estimation and inference.

Achieving semiparametric efficiency bound in longitudinal data analysis with dropouts

In longitudinal data analysis with dropouts, despite its local efficiency in theory, the augmented inverse probability weighted (AIPW) estimator hardly achieves the semiparametric efficiency bound in practice, even if the variance–covariance of the longitudinal outcomes is correctly modeled. In this paper, we propose a method based on conditional empirical likelihood. Assuming missing at random (MAR) mechanism, our estimator is doubly robust and locally efficient. Unlike the AIPW estimator, our estimator does not require to model any second moments, including the variance–covariance of the longitudinal outcomes, in order to achieve the semiparametric efficiency bound.

Longitudinal data analysis using the conditional empirical likelihood method

AbstractThis paper studies a new approach to longitudinal data analysis using the conditional empirical likelihood (CEL) method within the framework of marginal models. The possible unbalanced follow‐up visits are dealt with via stratification according to distinctive follow‐up patterns. The CEL method does not require any explicit modelling of the variance–covariance of the longitudinal outcomes. Instead, it implicitly incorporates a consistently estimated variance–covariance matrix in a nonparametric fashion. The proposed CEL estimator is connected to the generalized estimating equations (GEE) estimator, and achieves the same efficiency as the GEE estimator employing the true variance–covariance. The asymptotic distribution of the CEL estimator is derived, and simulation studies are conducted to assess the finite sample performance. Data collected from a longitudinal nutrition study are analysed as an application. The Canadian Journal of Statistics 42: 404–422; 2014 © 2014 Statistical Society of Canada

EMPIRICAL LIKELIHOOD ESTIMATION OF CONDITIONAL MOMENT RESTRICTION MODELS WITH UNKNOWN FUNCTIONS

This paper proposes an empirical likelihood-based estimation method for conditional moment restriction models with unknown functions, which include several semiparametric models. Our estimator is called the sieve conditional empirical likelihood (SCEL) estimator, which is based on the methods of conditional empirical likelihood and sieves. We derive (i) the consistency and a convergence rate of the SCEL estimator for the whole parameter, and (ii) the asymptotic normality and efficiency of the SCEL estimator for the parametric component. As an illustrating example, we consider a partially linear regression model with nonparametric endogeneity and heteroskedasticity.

TESTING FOR NONNESTED CONDITIONAL MOMENT RESTRICTIONS VIA CONDITIONAL EMPIRICAL LIKELIHOOD

We propose nonnested tests for competing conditional moment restriction models using the method of conditional empirical likelihood, recently developed by Kitamura, Tripathi, and Ahn (2004) and Zhang and Gijbels (2003). To define the test statistics, we use the implied conditional probabilities from conditional empirical likelihood, which take into account the full implications of conditional moment restrictions. We propose three types of nonnested tests: the moment-encompassing, Cox-type, and efficient score-encompassing tests. We derive the asymptotic null distributions and investigate their power properties against a sequence of local alternatives and a fixed global alternative. Our tests have distinct global power properties from some of the existing tests based on finite-dimensional unconditional moment restrictions. Simulation experiments show that our tests have reasonable finite sample properties and dominate some of the existing nonnested tests in terms of size-corrected powers.

Conditional empirical likelihood estimation and inference for quantile regression models

Abstract This paper considers two empirical likelihood-based estimation, inference, and specification testing methods for quantile regression models. First, we apply the method of conditional empirical likelihood (CEL) by Kitamura et al. [2004. Empirical likelihood-based inference in conditional moment restriction models. Econometrica 72, 1667–1714] and Zhang and Gijbels [2003. Sieve empirical likelihood and extensions of the generalized least squares. Scandinavian Journal of Statistics 30, 1–24] to quantile regression models. Second, to avoid practical problems of the CEL method induced by the discontinuity in parameters of CEL, we propose a smoothed counterpart of CEL, called smoothed conditional empirical likelihood (SCEL). We derive asymptotic properties of the CEL and SCEL estimators, parameter hypothesis tests, and model specification tests. Important features are (i) the CEL and SCEL estimators are asymptotically efficient and do not require preliminary weight estimation; (ii) by inverting the CEL and SCEL ratio parameter hypothesis tests, asymptotically valid confidence intervals can be obtained without estimating the asymptotic variances of the estimators; and (iii) in contrast to CEL, the SCEL method can be implemented by some standard Newton-type optimization. Simulation results demonstrate that the SCEL method in particular compares favorably with existing alternatives.

Penalized empirical likelihood estimation of semiparametric models

We propose an empirical likelihood-based estimation method for conditional estimating equations containing unknown functions, which can be applied for various semiparametric models. The proposed method is based on the methods of conditional empirical likelihood and penalization. Thus, our estimator is called the penalized empirical likelihood (PEL) estimator. For the whole parameter including infinite-dimensional unknown functions, we derive the consistency and a convergence rate of the PEL estimator. Furthermore, for the finite-dimensional parametric component, we show the asymptotic normality and efficiency of the PEL estimator. We illustrate the theory by three examples. Simulation results show reasonable finite sample properties of our estimator.