Generalized Empirical Likelihood Research Articles

Portfolio Efficiency Tests with Conditioning Information-Comparing GMM and GEL Estimators.

We evaluate the use of generalized empirical likelihood (GEL) estimators in portfolio efficiency tests for asset pricing models in the presence of conditional information. The use of conditional information is relevant to portfolio management as it allows for checking whether asset allocations are efficiently exploiting all the information available in the market. Estimators from the GEL family present some optimal statistical properties, such as robustness to misspecifications and better properties in finite samples. Unlike generalized method of moments (GMM) estimators, the bias for GEL estimators does not increase with the number of moment conditions included, which is expected in conditional efficiency analysis. Due to these better properties in finite samples, our main hypothesis is that portfolio efficiency tests using GEL estimators may have better properties in terms of size, power, and robustness. Using Monte Carlo experiments, we show that GEL estimators have better performance in the presence of data contaminations, especially under heavy tails and outliers. Extensive empirical analyses show the properties of the estimators for different sample sizes and portfolio types for two asset pricing models.

Generalized empirical likelihood for nonsmooth estimating equations with missing data

In missing data analysis, it is challenging to estimate the propensity score (PS) function. Traditional parametric, nonparametric or semiparametric approaches to estimate the PS function may be subject to model misspecification or lead to inefficient estimation. To address the aforementioned issues, we here assume that the PS function is unknown, and it is estimated by a series estimation method. To our knowledge, there is little work developed in utilizing the series estimation approach to estimate PS function even though it is widely used to estimate a regression function in the literature. The augmented inverse probability weighted (AIPW) estimating equations (EEs) are constructed via the estimated PS functions. Based on the constructed AIPW EEs, we propose generalized empirical likelihood (GEL) estimators of parameters, which are semiparametric efficient. Under some regularity conditions, we establish asymptotic properties of GEL estimators of parameters and GEL ratio statistics for parametric restrictions enjoying the Wilks’ phenomenon. In particular, we construct a pseudo-residual-based stochastic process to assess the plausibility of the posited nonsmooth EEs, their asymptotic properties are shown under null and local alternative hypotheses. A resampling procedure is proposed to evaluate the p-values of the proposed goodness-of-fit test statistics. Simulation studies and real data are used to illustrate the proposed methodologies.

Portfolio Efficiency Tests with Conditioning Information - Comparing GMM and GEL Estimators

We evaluate the use of Generalized Empirical Likelihood (GEL) estimators in portfolio efficiency tests for asset pricing models in the presence of conditional information. Estimators from GEL family present some optimal statistical properties, such as robustness to misspecification and better properties in finite samples. Unlike GMM, the bias for GEL estimators do not increase with the number of moment conditions included, which is expected in conditional efficiency analysis. By means of Monte Carlo experiments, we show that GEL estimators have better performance in the presence of data contaminations, especially under heavy tails and outliers. An extensive empirical analysis shows the properties of the estimators for different sample sizes and portfolios types for two asset pricing models.

EFFICIENT TWO-STEP GENERALIZED EMPIRICAL LIKELIHOOD ESTIMATION AND TESTS WITH MARTINGALE DIFFERENCES

This paper considers two-step generalized empirical likelihood (GEL) estimation and tests with martingale differences when there is a computationally simple $\sqrt n$-consistent estimator of nuisance parameters or the nuisance parameters can be eliminated with an estimating function of parameters of interest. As an initial estimate might have asymptotic impact on final estimates, we propose general $C(\alpha )$-type transformed moments to eliminate the impact, and use them in the GEL framework to construct estimation and tests robust to initial estimates. This two-step approach can save computational burden as the numbers of moments and parameters are reduced. A properly constructed two-step GEL (TGEL) estimator of parameters of interest is asymptotically as efficient as the corresponding joint GEL estimator. TGEL removes several higher-order bias terms of a corresponding two-step generalized method of moments. Our moment functions at the true parameters are martingales, thus they cover some spatial and time series models. We investigate tests for parameter restrictions in the TGEL framework, which are locally as powerful as those in the joint GEL framework when the two-step estimator is efficient.

Score tests in GMM: Why use implied probabilities?

While simple to implement and thus attractive in practice, the GMM score test of Newey and West (1987) often displays upward size distortion under common scenarios involving skewed moment vectors or models with weak identification. Inference based on the Generalized Empirical Likelihood (GEL) is seen as a general solution to this problem. Kleibergen (2005) and, more generally, Guggenberger and Smith (2005) devised an elegant theory for the GEL score tests. However, strictly speaking, the GEL score tests do not nest the Newey–West score test. Our paper provides a unified framework for score tests in GMM that nests all of the above as special cases and helps us to understand the precise mechanism by which the standard first order asymptotic theory on size and power well approximates the finite sample behavior of some score tests (namely, a subset of the GEL score tests) but not others. Special attention is paid to models with weak identification. We also argue that the apparent computational burden of GEL can be overcome in practice by recognizing the fundamental common role played by the GEL implied probabilities under all special cases of our framework. In particular, we show that all the GEL implied probabilities are asymptotically equivalent at a higher order – both under the null and under appropriate sequences of alternatives – and thus are exchangeable across computationally burdensome (e.g. Empirical Likelihood) and easy (e.g. Euclidean Empirical Likelihood) GEL score tests without affecting the first order asymptotics. Extensive simulation evidence is provided to corroborate our theoretical results. The simulation results also support a simple and yet important insight on the power of the tests: the use of implied probabilities to efficiently estimate the components of the score statistic, namely, the Jacobian and the asymptotic variance of the moment vector, can significantly improve the power of the score test in finite samples.

Robust causality test of infinite variance processes

This paper develops a robust causality test for time series with infinite variance innovation processes. First, we introduce a measure of dependence for vector nonparametric linear processes, and derive the asymptotic distribution of the test statistic by Taniguchi et al. (1996) in the infinite variance case. Second, we construct a weighted version of the generalized empirical likelihood (GEL) test statistic, called the self-weighted GEL statistic in the time domain. The limiting distribution of the self-weighted GEL test statistic is shown to be the usual chi-squared one regardless of whether the model has finite variance or not. Some simulation experiments illustrate satisfactory finite sample performances of the proposed test.

A new class of tests for overidentifying restrictions in moment condition models

In this study, we propose a new class of tests for overidentifying restrictions in moment condition models, extending Neyman’s (1959) test for parameter hypotheses in maximum likelihood to generalized empirical likelihood (GEL). These tests lack the complicated saddle point problem seen in GEL estimation; only a consistent estimator, where n is the sample size, is needed. In addition to discussing their first-order properties, we establish that under some regularity conditions, these tests share the same higher-order properties as GEL overidentifying tests, given proper consistent estimators. A Monte Carlo simulation study shows that the new class of tests of overidentifying restrictions has better finite sample performance than the two-step GMM overidentification test, and compares well to several potential alternatives in terms of overall performance.

Penalized sieve GEL for weighted average derivatives of nonparametric quantile IV regressions

This paper considers estimation and inference for a weighted average derivative (WAD) of a nonparametric quantile instrumental variables regression (NPQIV). NPQIV is a non-separable and nonlinear ill-posed inverse problem, which might be why there is no published work on the asymptotic properties of any estimator of its WAD. We first characterize the semiparametric efficiency bound for a WAD of a NPQIV, which, unfortunately, depends on an unknown conditional derivative operator and hence an unknown degree of ill-posedness, making it difficult to know if the information bound is singular or not. In either case, we propose a penalized sieve generalized empirical likelihood (GEL) estimation and inference procedure, which is based on the unconditional WAD moment restriction and an increasing number of unconditional moments that are implied by the conditional NPQIV restriction, where the unknown quantile function is approximated by a penalized sieve. Under some regularity conditions, we show that the self-normalized penalized sieve GEL estimator of the WAD of a NPQIV is asymptotically standard normal. We also show that the quasi likelihood ratio statistic based on the penalized sieve GEL criterion is asymptotically chi-square distributed regardless of whether or not the information bound is singular.

GEL estimation and tests of spatial autoregressive models

This paper considers the generalized empirical likelihood (GEL) estimation and tests of high order spatial autoregressive (SAR) models by exploring an inherent martingale structure. The GEL estimator has the same asymptotic distribution as the generalized method of moments estimator explored with same moment conditions for estimation, but circumvents a first step estimation of the optimal weighting matrix with a preliminary estimator, and thus can be robust to unknown heteroskedasticity and non-normality. While the GEL removes the asymptotic bias from the preliminary estimator and partially removes the bias due to the correlation between the moment conditions and their Jacobian, the empirical likelihood as a special member of GELs further partially removes the bias from estimating the second moment matrix. We also formulate the GEL overidentification test, Moran’s I test, and GEL ratio tests for parameter restrictions and non-nested hypotheses. While some of the conventional tests might not be robust to non-normality and/or unknown heteroskedasticity, the corresponding GEL tests can be robust.

Generalized empirical likelihood specification test robust to local misspecification

It is well known that many of the standard specification tests may not be robust when the alternative is misspecified. This study analyzes a robust specification test for generalized empirical likelihood (GEL) estimators in a weakly dependent time series setting. We show that the usual score test statistic asymptotically follows a non-central chi-square distribution under the local misspecification in the GEL framework. Thus, it spuriously rejects the null hypothesis too frequently. We propose a robust score specification test that asymptotically follows a central chi-square distribution under the local misspecification. A Monte Carlo simulation verifies the usefulness of the proposed tests.

Self-weighted generalized empirical likelihood methods for hypothesis testing in infinite variance ARMA models

This paper develops the generalized empirical likelihood (GEL) method for infinite variance ARMA models, and constructs a robust testing procedure for general linear hypotheses. In particular, we use the GEL method based on the least absolute deviations and self-weighting, and construct a natural class of statistics including the empirical likelihood and the continuous updating-generalized method of moments for infinite variance ARMA models. The self-weighted GEL test statistic is shown to converge to a \(\chi ^2\)-distribution, although the model may have infinite variance. Therefore, we can make inference without estimating any unknown quantity of the model such as the tail index or the density function of unobserved innovation processes. We also compare the finite sample performance of the proposed test with the Wald-type test by Pan et al. (Econom Theory 23:852–879, 2007) via some simulation experiments.

Generalized empirical likelihood inference in partially linear model for longitudinal data with missing response variables and error-prone covariates

ABSTRACTIn this article, we consider statistical inference for longitudinal partial linear models when the response variable is sometimes missing with missingness probability depending on the covariate that is measured with error. A generalized empirical likelihood (GEL) method is proposed by combining correction attenuation and quadratic inference functions. The method that takes into consideration the correlation within groups is used to estimate the regression coefficients. Furthermore, residual-adjusted empirical likelihood (EL) is employed for estimating the baseline function so that undersmoothing is avoided. The empirical log-likelihood ratios are proven to be asymptotically Chi-squared, and the corresponding confidence regions for the parameters of interest are then constructed. Compared with methods based on NAs, the GEL does not require consistent estimators for the asymptotic variance and bias. The numerical study is conducted to compare the performance of the EL and the normal approximation-based method, and a real example is analysed.

Structural change tests for GEL criteria

ABSTRACTThis article examines structural change tests based on generalized empirical likelihood methods in the time series context, allowing for dependent data. Standard structural change tests for the Generalized method of moments (GMM) are adapted to the generalized empirical likelihood (GEL) context. We show that when moment conditions are properly smoothed, these test statistics converge to the same asymptotic distribution as in the GMM, in cases with known and unknown breakpoints. New test statistics specific to GEL methods, and that are robust to weak identification, are also introduced. A simulation study examines the small sample properties of the tests and reveals that GEL-based robust tests performed well, both in terms of the presence and location of a structural change and in terms of the nature of identification.

A Simulation-Based Comparison of Covariate Adjustment Methods for the Analysis of Randomized Controlled Trials.

Covariate adjustment methods are frequently used when baseline covariate information is available for randomized controlled trials. Using a simulation study, we compared the analysis of covariance (ANCOVA) with three nonparametric covariate adjustment methods with respect to point and interval estimation for the difference between means. The three alternative methods were based on important members of the generalized empirical likelihood (GEL) family, specifically on the empirical likelihood (EL) method, the exponential tilting (ET) method, and the continuous updated estimator (CUE) method. Two criteria were considered for the comparison of the four statistical methods: the root mean squared error and the empirical coverage of the nominal 95% confidence intervals for the difference between means. Based on the results of the simulation study, for sensitivity analysis purposes, we recommend the use of ANCOVA (with robust standard errors when heteroscedasticity is present) together with the CUE-based covariate adjustment method.

High dimensional generalized empirical likelihood for moment restrictions with dependent data

This paper considers the maximum generalized empirical likelihood (GEL) estimation and inference on parameters identified by high dimensional moment restrictions with weakly dependent data when the dimensions of the moment restrictions and the parameters diverge along with the sample size. The consistency with rates and the asymptotic normality of the GEL estimator are obtained by properly restricting the growth rates of the dimensions of the parameters and the moment restrictions, as well as the degree of data dependence. It is shown that even in the high dimensional time series setting, the GEL ratio can still behave like a chi-square random variable asymptotically. A consistent test for the over-identification is proposed. A penalized GEL method is also provided for estimation under sparsity setting.

Saddlepoint expansions for GEL estimators

A simple saddlepoint (SP) approximation for the distribution of generalized empirical likelihood (GEL) estimators is derived. Simulations compare the performance of the SP and other methods such as the Edgeworth and the bootstrap for special cases of GEL: continuous updating, empirical likelihood and exponential tilting estimators.

Near exogeneity and weak identification in generalized empirical likelihood estimators: Many moment asymptotics

This paper investigates the Generalized Empirical Likelihood (GEL) estimators when there are local violations of the exogeneity condition (near exogeneity) in the case of many weak moments. We also examine the tradeoff between the degree of violation of the exogeneity and the number of nearly exogenous instruments. In this respect, this paper extends many weak moment asymptotics of Newey and Windmeijer (2009a). The overidentifying restrictions test can detect both mild and large violations of exogeneity. In the case of minor violations, the Anderson–Rubin (1949) and Wald tests are not size distorted.

The Empirical Likelihood MPEC Approach to Demand Estimation

The family of Generalized Empirical Likelihood (GEL) estimators provide a number of potential advantages relative to Generalized Method of Moments (GMM) estimators. While it is well known these estimators share an asymptotic distribution, the GEL estimators may perform better in nite sample, particularly in the case of many weak instruments. A relatively new literature has documented that nite-sample bias in the demand estimation problem of Berry, Levinsohn, and Pakes (1995) is often large, especially in the absence of exogenous cost shifting instruments. This paper provides a formulation for a computationally tractable GEL estimator based on the MPEC method of Su and Judd (2012) and adapts it to the BLP problem. When compared to GMM, the GEL estimator performs substantially better, reducing the bias by as much as 90%. Furthermore, it is possible to use analytic bias correction to reduce the bias even more and obtain accurate estimates with relatively small numbers of markets.

Estimation for multivariate stable distributions with generalized empirical likelihood

This paper considers the generalized empirical likelihood (GEL) method for estimating the parameters of the multivariate stable distribution. The GEL method is considered to be an extension of the generalized method of moments (GMM). The multivariate stable distributions are widely applicable as they can accommodate both skewness and heavy tails. We treat the spectral measure, which summarizes scale and asymmetry, by discretization. In order to estimate all the model parameters simultaneously, we apply the estimating function constructed by equating empirical and theoretical characteristic functions. The efficacy of the proposed GEL method is demonstrated in Monte Carlo studies. An illustrative example involving daily returns of market indexes is also included.

Hodges–Lehmann optimality for testing moment conditions

This paper studies the Hodges and Lehmann (1956) optimality of tests in a general setup. The tests are compared by the exponential rates of growth to one of the power functions evaluated at a fixed alternative while keeping the asymptotic sizes bounded by some constant. We present two sets of sufficient conditions for a test to be Hodges–Lehmann optimal. These new conditions extend the scope of the Hodges–Lehmann optimality analysis to setups that cannot be covered by other conditions in the literature. The general result is illustrated by our applications of interest: testing for moment conditions and overidentifying restrictions. In particular, we show that (i) the empirical likelihood test does not necessarily satisfy existing conditions for optimality but does satisfy our new conditions; and (ii) the generalized method of moments (GMM) test and the generalized empirical likelihood (GEL) tests are Hodges–Lehmann optimal under mild primitive conditions. These results support the belief that the Hodges–Lehmann optimality is a weak asymptotic requirement.