INDEX TO VOLUME 17 (1999)

We study how to select or combine estimators of the average treatment effect (ATE) and the average treatment effect on the treated (ATT) in the presence of multiple sets of covariates. We consider two cases: (1) all sets of covariates satisfy the unconfoundedness assumption and (2) some sets of covariates violate the unconfoundedness assumption locally. For both cases, we propose a data-driven covariate selection criterion (CSC) to minimize the asymptotic mean squared errors (AMSEs). Based on our CSC, we propose new average estimators of ATE and ATT, which include the selected estimators based on a single set of covariates as a special case. We derive the asymptotic distributions of our new estimators and propose how to construct valid confidence intervals. Our Monte Carlo simulations show that in finite samples, our new average estimators achieve substantial efficiency gains over the estimators based on a single set of covariates. We apply our new estimators to study the impact of inherited control on firm performance.     

Efficient Estimation of Data Combination Models by the Method of Auxiliary-to-Study Tilting (AST)

We propose a locally efficient estimator for a class of semiparametric data combination problems. A leading estimand in this class is the average treatment effect on the treated (ATT). Data combination problems are related to, but distinct from, the class of missing data problems with data missing at random (of which the average treatment effect (ATE) estimand is a special case). Our estimator also possesses a double robustness property. Our procedure may be used to efficiently estimate, among other objects, the ATT, the two-sample instrumental variables model (TSIV), counterfactual distributions, poverty maps, and semiparametric difference-in-differences. In an empirical application, we use our procedure to characterize residual Black–White wage inequality after flexibly controlling for “premarket” differences in measured cognitive achievement. Supplementary materials for this article are available online.     

Testing the information matrix equality with robust estimators

The information matrix (IM) equality can be used to test for misspecification of a parametric model. We study the behavior of the IM test when the maximum-likelihood (ML) estimators used in the construction of this test are replaced with robust estimators. The latter do not suffer from the masking effect in the presence of outliers and can improve the power of the IM test. At the normal location-scale model, the IM test using the ML estimators is known as the Jarque–Bera test, and uses skewness and kurtosis to detect deviations from normality. When robust estimators are employed to test the IM equality, a robust version of the Jarque–Bera test emerges. We investigate in detail the local asymptotic power of the IM test, for various estimators and under a variety of local alternatives. For the normal regression model, it is shown by simulations under fixed alternatives that in many cases the use of robust estimators substantially increases the power of the IM test.     

The Finite Sample Performance of Inference Methods for Propensity Score Matching and Weighting Estimators

ABSTRACT

This article investigates the finite sample properties of a range of inference methods for propensity score-based matching and weighting estimators frequently applied to evaluate the average treatment effect on the treated. We analyze both asymptotic approximations and bootstrap methods for computing variances and confidence intervals in our simulation designs, which are based on German register data and U.S. survey data. We vary the design w.r.t. treatment selectivity, effect heterogeneity, share of treated, and sample size. The results suggest that in general, theoretically justified bootstrap procedures (i.e., wild bootstrapping for pair matching and standard bootstrapping for “smoother” treatment effect estimators) dominate the asymptotic approximations in terms of coverage rates for both matching and weighting estimators. Most findings are robust across simulation designs and estimators.     

Quantile regression estimation of partially linear additive models

In this paper, we consider the estimation of partially linear additive quantile regression models where the conditional quantile function comprises a linear parametric component and a nonparametric additive component. We propose a two-step estimation approach: in the first step, we approximate the conditional quantile function using a series estimation method. In the second step, the nonparametric additive component is recovered using either a local polynomial estimator or a weighted Nadaraya–Watson estimator. Both consistency and asymptotic normality of the proposed estimators are established. Particularly, we show that the first-stage estimator for the finite-dimensional parameters attains the semiparametric efficiency bound under homoskedasticity, and that the second-stage estimators for the nonparametric additive component have an oracle efficiency property. Monte Carlo experiments are conducted to assess the finite sample performance of the proposed estimators. An application to a real data set is also illustrated.     

Closed testing procedures for all pairwise comparisons in a randomized block design

We consider multiple comparison test procedures among treatment effects in a randomized block design. We propose closed testing procedures based on maximum values of some two-sample t test statistics and based on F test statistics. It is shown that the proposed procedures are more powerful than single-step procedures and the REGW (Ryan/Einot–Gabriel/Welsch)-type tests. Next, we consider the randomized block design under simple ordered restrictions of treatment effects. We propose closed testing procedures based on maximum values of two-sample one-sided t test statistics and based on Batholomew’s statistics for all pairwise comparisons of treatment effects. Although single-step multiple comparison procedures are utilized in general, the power of these procedures is low for a large number of groups. The closed testing procedures stated in the present article are more powerful than the single-step procedures. Simulation studies are performed under the null hypothesis and some alternative hypotheses. In this studies, the proposed procedures show a good performance.     

Partial identification of average treatment effects on the treated through difference-in-differences

ABSTRACT

The difference-in-differences (DID) method is widely used as a tool for identifying causal effects of treatments in program evaluation. When panel data sets are available, it is well-known that the average treatment effect on the treated (ATT) is point-identified under the DID setup. If a panel data set is not available, repeated cross sections (pretreatment and posttreatment) may be used, but may not point-identify the ATT. This paper systematically studies the identification of the ATT under the DID setup when posttreatment treatment status is unknown for the pretreatment sample. This is done through a novel application of an extension of a continuous version of the classical monotone rearrangement inequality which allows for general copula bounds. The identifying power of an instrumental variable and of a ‘matched subsample’ is also explored. Finally, we illustrate our approach by estimating the effect of the Americans with Disabilities Act of 1991 on employment outcomes of the disabled.     

Using entropy distance measures for testing odds ratio parameters under logistic regression models based on case–control data

We propose two retrospective test statistics for testing the vector of odds ratio parameters under the logistic regression model based on case–control data by exploiting the density ratio structure under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. The proposed test statistics are based on Kullback–Leibler entropy distance and are particularly relevant to the case–control sampling plan. These two test statistics have identical asymptotic chi-squared distributions under the null hypothesis and identical asymptotic noncentral chi-squared distributions under local alternatives to the null hypothesis. Moreover, the proposed test statistics require computation of the maximum semiparametric likelihood estimators of the underlying parameters, but are otherwise easily computed. We present some results on simulation and on the analysis of two real data sets.     

Local Linear Estimation of Second-order Jump-diffusion Model

In this article, we propose the local linear estimators of the drift coefficient and diffusion coefficient in the second-order jump-diffusion model. We also show the consistency and asymptotic normality of these estimators under mild conditions.     

Boundary and Bias Correction in Kernel Hazard Estimation

A new class of local linear hazard estimators based on weighted least square kernel estimation is considered. The class includes the kernel hazard estimator of Ramlau-Hansen (1983), which has the same boundary correction property as the local linear regression estimator (see Fan & Gijbels, 1996). It is shown that all the local linear estimators in the class have the same pointwise asymptotic properties. We derive the multiplicative bias correction of the local linear estimator. In addition we propose a new bias correction technique based on bootstrap estimation of additive bias. This latter method has excellent theoretical properties. Based on an extensive simulation study where we compare the performance of competing estimators, we also recommend the use of the additive bias correction in applied work.     

Semiparametric estimation and inference for distributional and general treatment effects

Summary.  There is a large literature on methods of analysis for randomized trials with noncompliance which focuses on the effect of treatment on the average outcome. The paper considers evaluating the effect of treatment on the entire distribution and general functions of this effect. For distributional treatment effects, fully non-parametric and fully parametric approaches have been proposed. The fully non-parametric approach could be inefficient but the fully parametric approach is not robust to the violation of distribution assumptions. We develop a semiparametric instrumental variable method based on the empirical likelihood approach. Our method can be applied to general outcomes and general functions of outcome distributions and allows us to predict a subject's latent compliance class on the basis of an observed outcome value in observed assignment and treatment received groups. Asymptotic results for the estimators and likelihood ratio statistic are derived. A simulation study shows that our estimators of various treatment effects are substantially more efficient than the currently used fully non-parametric estimators. The method is illustrated by an analysis of data from a randomized trial of an encouragement intervention to improve adherence to prescribed depression treatments among depressed elderly patients in primary care practices.     

Bias reduction in kernel density estimation

In this paper, we propose two kernel density estimators based on a bias reduction technique. We study the properties of these estimators and compare them with Parzen–Rosenblatt's density estimator and Mokkadem, A., Pelletier, M., and Slaoui, Y. (2009, ‘The stochastic approximation method for the estimation of a multivariate probability density’, J. Statist. Plann. Inference, 139, 2459–2478) is density estimators. It turns out that, with an adequate choice of the parameters of the two proposed estimators, the rate of convergence of two estimators will be faster than the two classical estimators and the asymptotic MISE (Mean Integrated Squared Error) will be smaller than the two classical estimators. We corroborate these theoretical results through simulations.     

Prognostic score matching methods for estimating the average effect of a non-reversible binary time-dependent treatment on the survival function

In evaluating the benefit of a treatment on survival, it is often of interest to compare post-treatment survival with the survival function that would have been observed in the absence of treatment. In many practical settings, treatment is time-dependent in the sense that subjects typically begin follow-up untreated, with some going on to receive treatment at some later time point. In observational studies, treatment is not assigned at random and, therefore, may depend on various patient characteristics. We have developed semi-parametric matching methods to estimate the average treatment effect on the treated (ATT) with respect to survival probability and restricted mean survival time. Matching is based on a prognostic score which reflects each patient’s death hazard in the absence of treatment. Specifically, each treated patient is matched with multiple as-yet-untreated patients with similar prognostic scores. The matched sets do not need to be of equal size, since each matched control is weighted in order to preserve risk score balancing across treated and untreated groups. After matching, we estimate the ATT non-parametrically by contrasting pre- and post-treatment weighted Nelson–Aalen survival curves. A closed-form variance is proposed and shown to work well in simulation studies. The proposed methods are applied to national organ transplant registry data.

     

Maximum likelihood estimation for reflected Ornstein-Uhlenbeck processes

In this paper, we investigate the maximum likelihood estimation for the reflected Ornstein-Uhlenbeck (ROU) processes based on continuous observations. Both the cases with one-sided barrier and two-sided barriers are considered. We derive the explicit formulas for the estimators, and then prove their strong consistency and asymptotic normality. Moreover, the bias and mean square errors are represented in terms of the solutions to some PDEs with homogeneous Neumann boundary conditions. We also illustrate the asymptotic behavior of the estimators through a simulation study.     

Semiparametric jump-preserving estimation for single-index models

Estimation of the single-index model with a discontinuous unknown link function is considered in this paper. Existed refined minimum average variance estimation (rMAVE) method can estimate the single-index parameter and unknown link function simultaneously by minimising the average pointwise conditional variance, where the conditional variance can be estimated using the local linear fit method with centred kernel function. When there are jumps in the link function, big biases around jumps can appear. For this reason, we embed the jump-preserving technique in the rMAVE method, then propose an adaptive jump-preserving estimation procedure for the single-index model. Concretely speaking, the conditional variance is obtained by the one among local linear fits with centred, left-sided and right-sided kernel functions who has minimum weighted residual mean squares. The resulting estimators can preserve the jumps well and also give smooth estimates of the continuity parts. Asymptotic properties are established under some mild conditions. Simulations and real data analysis show the proposed method works well.     

Change Detection in INAR(p) Processes Against Various Alternative Hypotheses

Change in the coefficients or the mean of the innovation of an INAR(p) process is a sign of disturbance that is important to detect. The proposed methods can test for change in any one of these quantities separately, or in any collection of them. They make both one-sided and two-sided tests possible, furthermore, they can be used to test against the “epidemic” alternative. The tests are based on a CUSUM process using CLS estimators of the parameters. Under the one-sided and two-sided alternatives, consistency of the tests is proved and the properties of the change-point estimator are also explored.     

Bayesian inference of three-parameter bathtub-shaped lifetime distribution

We investigate a Bayesian inference in the three-parameter bathtub-shaped lifetime distribution which is obtained by adding a power parameter to the two-parameter bathtub-shaped lifetime distribution suggested by Chen (2000). The Bayes estimators under the balanced squared error loss function are derived for three parameters. Then, we have used Lindley's and Tierney–Kadane approximations (see Lindley 1980; Tierney and Kadane 1986) for computing these Bayes estimators. In particular, we propose the explicit form of Lindley's approximation for the model with three parameters. We also give applications with a simulated data set and two real data sets to show the use of discussed computing methods. Finally, concluding remarks are mentioned.     

Two-way ANOVA when the distribution of the error terms is skew t

In this article, we assume that the distribution of the error terms is skew t in two-way analysis of variance (ANOVA). Skew t distribution is very flexible for modeling the symmetric and the skew datasets, since it reduces to the well-known normal, skew normal, and Student's t distributions. We obtain the estimators of the model parameters by using the maximum likelihood (ML) and the modified maximum likelihood (MML) methodologies. We also propose new test statistics based on these estimators for testing the equality of the treatment and the block means and also the interaction effect. The efficiencies of the ML and the MML estimators and the power values of the test statistics based on them are compared with the corresponding normal theory results via Monte Carlo simulation study. Simulation results show that the proposed methodologies are more preferable. We also show that the test statistics based on the ML estimators are more powerful than the test statistics based on the MML estimators as expected. However, power values of the test statistics based on the MML estimators are very close to the corresponding test statistics based on the ML estimators. At the end of the study, a real life example is given to show the implementation of the proposed methodologies.     

Understanding Estimators of Treatment Effects in Regression Discontinuity Designs

In this paper, we propose two new estimators of treatment effects in regression discontinuity designs. These estimators can aid understanding of the existing estimators such as the local polynomial estimator and the partially linear estimator. The first estimator is the partially polynomial estimator which extends the partially linear estimator by further incorporating derivative differences of the conditional mean of the outcome on the two sides of the discontinuity point. This estimator is related to the local polynomial estimator by a relocalization effect. Unlike the partially linear estimator, this estimator can achieve the optimal rate of convergence even under broader regularity conditions. The second estimator is an instrumental variable estimator in the fuzzy design. This estimator will reduce to the local polynomial estimator if higher order endogeneities are neglected. We study the asymptotic properties of these two estimators and conduct simulation studies to confirm the theoretical analysis.     

A fiducial-based approach to the one-way ANOVA in the presence of nonnormality and heterogeneous error variances

In this study, we propose a new test for testing the equality of the treatment means in one-way ANOVA when the usual normality and the homogeneity of variances assumptions are not met. In developing the proposed test, we benefit from the Fisher's fiducial inference [1–3]. Distribution of the error terms is assumed to be long-tailed symmetric (LTS) which includes the normal distribution as a limiting case. Modified maximum likelihood (MML) estimators are used in the test statistics rather than the traditional least squares (LS) estimators, since LS estimators have very low efficiencies under nonnormal distributions, see Tiku [4] for the details of MML methodology. An extensive Monte Carlo simulation study is done to compare the efficiency of the proposed test with the corresponding test based on normal theory, see Li et al. [5]. Finally, we give a real life example to show the applicability of the proposed methodology.