Estimation with weak instruments: Accuracy of higher‐order bias and MSE approximations

Summary In this paper, we consider parameter estimation in a linear simultaneous equations model. It is well known that two‐stage least squares (2SLS) estimators may perform poorly when the instruments are weak. In this case 2SLS tends to suffer from the substantial small sample biases. It is also known that LIML and Nagar‐type estimators are less biased than 2SLS but suffer from large small sample variability. We construct a bias‐corrected version of 2SLS based on the Jackknife principle. Using higher‐order expansions we show that the MSE of our Jackknife 2SLS estimator is approximately the same as the MSE of the Nagar‐type estimator. We also compare the Jackknife 2SLS with an estimator suggested by Fuller (Econometrica 45, 933–54) that significantly decreases the small sample variability of LIML. Monte Carlo simulations show that even in relatively large samples the MSE of LIML and Nagar can be substantially larger than for Jackknife 2SLS. The Jackknife 2SLS estimator and Fuller's estimator give the best overall performance. Based on our Monte Carlo experiments we conduct informal statistical tests of the accuracy of approximate bias and MSE formulas. We find that higher‐order expansions traditionally used to rank LIML, 2SLS and other IV estimators are unreliable when identification of the model is weak. Overall, our results show that only estimators with well‐defined finite sample moments should be used when identification of the model is weak.     

ECF estimation of Markov models where the transition density is unknown

Summary In this paper, we consider the estimation of Markov models where the transition density is unknown. The approach we propose is based on the empirical characteristic function estimation procedure with an approximate optimal weight function. The approximate optimal weight function is obtained through an Edgeworth/Gram–Charlier expansion of the logarithmic transition density of the Markov process. We derive the estimating equations and demonstrate that they are similar to the approximate maximum likelihood estimation (AMLE). However, in contrast to the conventional AMLE our approach ensures the consistency of the estimator even with the approximate likelihood function. We illustrate our approach with examples of various Markov processes. Monte Carlo simulations are performed to investigate the finite sample properties of the proposed estimator in comparison with other methods.     

Two‐stage estimation of limited dependent variable models with errors‐in‐variables

Summary This paper deals with censored or truncated regression models where the explanatory variables are measured with additive errors. We propose a two‐stage estimation procedure that combines the instrumental variable method and the minimum distance estimation. This approach produces consistent and asymptotically normally distributed estimators for model parameters. When the predictor and instrumental variables are normally distributed, we also propose a maximum likelihood based estimator and a two‐stage moment estimator. Simulation studies show that all proposed estimators perform satisfactorily for relatively small samples and relatively high degree of censoring. In addition, the maximum likelihood based estimators are fairly robust against non‐normal and /or heteroskedastic random errors in our simulations. The method can be generalized to panel data models.     

Efficient inference in multivariate fractionally integrated time series models

Summary A computationally simple maximum likelihood procedure for multivariate fractionally integrated time series models is introduced. This allows, e.g., efficient estimation of the memory parameters of fractional models or efficient testing of the hypothesis that two or more series are integrated of the same possibly fractional order. In particular, we show the existence of a local time domain maximum likelihood estimator and its asymptotic normality, and under Gaussianity asymptotic efficiency. The likelihood‐based test statistics (Wald, likelihood ratio and Lagrange multiplier) are derived and shown to be asymptotically equivalent and chi‐squared distributed under local alternatives, and under Gaussianity locally most powerful. The finite sample properties of the likelihood ratio test are evaluated by Monte Carlo experiments, which show that rejection frequencies are very close to the asymptotic local power for samples as small as n= 100 .     

Heterogeneity in dynamic discrete choice models

Summary We consider dynamic discrete choice models with heterogeneity in both the levels parameter and the state dependence parameter. We first present an empirical analysis that motivates the theoretical analysis which follows. The theoretical analysis considers a simple two‐state, first‐order Markov chain model without covariates in which both transition probabilities are heterogeneous. Using such a model we are able to derive exact small sample results for bias and mean squared error (MSE). We discuss the maximum likelihood approach and derive two novel estimators. The first is a bias corrected version of the Maximum Likelihood Estimator (MLE) although the second, which we term MIMSE, minimizes the integrated mean square error. The MIMSE estimator is always well defined, has a closed‐form expression and inherits the desirable large sample properties of the MLE. Our main finding is that in almost all short panel contexts the MIMSE significantly outperforms the other two estimators in terms of MSE. A final section extends the MIMSE estimator to allow for exogenous covariates.     

Assessing Response Profiles from Incomplete Longitudinal Clinical Trial Data Under Regulatory Considerations

Total cholesterol (TC) measurements are subject to errors primarily because of temporal variations in cholesterol levels within each individual. These errors make it difficult to estimate the proportion of study subjects with true (error-free) TC in a specific range, an extremely important parameter to policy makers in health care management. To properly address this issue, it is key to accurately estimate the distribution function of the true TC, which typically deviates from the normal distribution. To better approximate the distribution function of the true TC, we propose a constrained maximum likelihood estimator based on a mixture-of-normals model. A simulation study illustrates that the proposed estimator performs better than an estimator based on the normality assumption that is frequently used in the literature to address the same issue. Finally, the proposed estimator is applied to data from a study, and its performance is once again compared with that of an estimator based on the normality assumption.     

Identification and estimation of semi‐parametric censored dynamic panel data models of short time periods

In this paper, we present a semi‐parametric identification and estimation method for censored dynamic panel data models of short time periods and their average partial effects with only two periods of data. The proposed method transforms the semi‐parametric specification of censored dynamic panel data models into a parametric family of distribution functions of observables without specifying the distribution of the initial condition. Then the censored dynamic panel data models are globally identified under a standard maximum likelihood estimation framework. The identifying assumptions are related to the completeness of the families of known parametric distribution functions corresponding to censored dynamic panel data models. Dynamic tobit models and two‐part dynamic regression models satisfy the key assumptions. We propose a sieve maximum likelihood estimator and we investigate the finite sample properties of these sieve‐based estimators using Monte Carlo analysis. Our empirical application using the Medical Expenditure Panel Survey shows that individuals consume more health care when their incomes increase, after controlling for past health expenditures.     

Do you know your total cholesterol (TC) number?

Total cholesterol (TC) measurements are subject to errors primarily because of temporal variations in cholesterol levels within each individual. These errors make it difficult to estimate the proportion of study subjects with true (error-free) TC in a specific range, an extremely important parameter to policy makers in health care management. To properly address this issue, it is key to accurately estimate the distribution function of the true TC, which typically deviates from the normal distribution. To better approximate the distribution function of the true TC, we propose a constrained maximum likelihood estimator based on a mixture-of-normals model. A simulation study illustrates that the proposed estimator performs better than an estimator based on the normality assumption that is frequently used in the literature to address the same issue. Finally, the proposed estimator is applied to data from a study, and its performance is once again compared with that of an estimator based on the normality assumption.     

Guest Editors' Notes on Statistical Issues in Design and Analysis of Thorough QTc Studies

In generalized linear models, such as the logistic regression model, maximum likelihood estimators are well known to be biased at smaller sample sizes. When the number of dose levels or replications per dose is small, bias in the maximum likelihood estimates can lead to very misleading results and the model often fails to converge. In order to correct the bias present in the maximum likelihood estimates and the problem of nonconvergence, the penalized maximum likelihood estimator is considered. Simulations compare the fit and empirical confidence levels of inferences made from the maximum likelihood and penalized maximum likelihood based models.     

Bias reduction in logistic dose-response models

In generalized linear models, such as the logistic regression model, maximum likelihood estimators are well known to be biased at smaller sample sizes. When the number of dose levels or replications per dose is small, bias in the maximum likelihood estimates can lead to very misleading results and the model often fails to converge. In order to correct the bias present in the maximum likelihood estimates and the problem of nonconvergence, the penalized maximum likelihood estimator is considered. Simulations compare the fit and empirical confidence levels of inferences made from the maximum likelihood and penalized maximum likelihood based models.     

Partially adaptive estimation via the maximum entropy densities

Summary We propose a partially adaptive estimator based on information theoretic maximum entropy estimates of the error distribution. The maximum entropy (maxent) densities have simple yet flexible functional forms to nest most of the mathematical distributions. Unlike the non-parametric fully adaptive estimators, our parametric estimators do not involve choosing a bandwidth or trimming, and only require estimating a small number of nuisance parameters, which is desirable when the sample size is small. Monte Carlo simulations suggest that the proposed estimators fare well with non-normal error distributions. When the errors are normal, the efficiency loss due to redundant nuisance parameters is negligible as the proposed error densities nest the normal. The proposed partially adaptive estimator compares favourably with existing methods, especially when the sample size is small. We apply the estimator to a stochastic frontier model, whose error distribution is usually non-normal.     

Dose-Finding in Drug Development. N. Ting (Ed.)

In stratified matched-pair studies, risk difference between two proportions is one of the most frequently used indices in comparing efficiency between two treatments or diagnostic tests. This article presents five simultaneous confidence intervals and two bootstrap simultaneous confidence intervals for risk differences in stratified matched-pair designs. The proposed confidence intervals are evaluated with respect to their coverage probabilities, expected widths, and ratios of the mesial noncoverage to noncoverage probability. Empirical results show that (1) hybrid simultaneous confidence intervals outperform nonhybrid simultaneous confidence intervals; (2) hybrid simultaneous confidence intervals based on median estimator outperform those based on maximum likelihood estimator; and (3) hybrid simultaneous confidence intervals incorporated with Wilson score and Agresti coull intervals and the bootstrap t-percentile simultaneous interval based on median unbiased estimators behave satisfactorily for small to large sample sizes in the sense that their empirical coverage probabilities are close to the prespecified nominal confidence level, and their ratios of the mesial noncoverage to noncoverage probabilities lie in [0.4,0.6] and are hence recommended. Real examples from clinical studies are used to illustrate the proposed methodologies.     

Estimation of the risk difference under a noncompliance randomized clinical trial with missing outcomes

In a randomized clinical trial (RCT), we often come across the situations in which there are patients who do not comply with their assigned treatments or whose outcomes are missing due to their refusal or loss to follow-up. Because noncompliance and missing outcomes do not generally occur completely at random, analyzing data as treated or excluding patients with missing outcomes from our analysis can produce a biased estimate of a treatment effect. In this paper, we consider estimation of the risk difference (RD) in the presence of both noncompliance and missing outcomes under a RCT. On the basis of a constant risk additive model proposed elsewhere, we derive the maximum likelihood estimator (MLE) and develop three asymptotic interval estimators in closed form for the RD when we have outcome missing at random. We apply Monte Carlo simulation to evaluate and compare the performance of these estimators in a variety of situations. We note that all interval estimators developed here can perform well with respect to the coverage probability in all the situations considered here. We find that the interval estimator using tanh-1(x) transformation is generally more precise than the other two estimators with respect to the average length. Finally, we use the data taken from a randomized trial studying the association between flu vaccine and the risk of flu-related hospitalization to illustrate the practical use of these interval estimators.     

Lag‐augmented two‐ and three‐stage least squares estimators for integrated structural dynamic models

Summary We consider a lag‐augmented two‐ or three‐stage least‐squares estimator for a structural dynamic model of non‐stationary and possibly cointegrated variables without the prior knowledge of unit roots or rank of cointegration. We show that the conventional two‐and three‐stage least‐squares estimators are consistent but contain non‐standard distributions without the strict exogeneity assumption; hence the conventional Wald type test statistics may not be chi‐square distributed. We propose a lag order augmented two‐ or three‐stage least‐squares estimator that is consistent and asymptotically normally distributed. Limited Monte Carlo studies are conducted to shed light on the finite sample properties of various estimators.     

A class of indirect inference estimators: higher‐order asymptotics and approximate bias correction

In this paper, we define a set of indirect inference estimators based on moment approximations of the auxiliary estimators. Their introduction is motivated by reasons of analytical and computational facilitation. Their definition provides an indirect inference framework for some classical bias correction procedures. We derive higher‐order asymptotic properties of these estimators. We demonstrate that under our assumption framework, and in the special case of deterministic weighting and affinity of the binding function, these are second‐order unbiased. Moreover, their second‐order approximate mean square errors do not depend on the cardinality of the Monte Carlo or bootstrap samples that our definition might involve. Consequently, the second‐order mean square error of the auxiliary estimator is not altered. We extend this to a class of multistep indirect inference estimators that have zero higher‐order bias without increasing the approximate mean square error, up to the same order. Our theoretical results are also validated by three Monte Carlo experiments.     

Estimation of the Risk Difference Under a Noncompliance Randomized Clinical Trial with Missing Outcomes

In a randomized clinical trial (RCT), we often come across the situations in which there are patients who do not comply with their assigned treatments or whose outcomes are missing due to their refusal or loss to follow-up. Because noncompliance and missing outcomes do not generally occur completely at random, analyzing data as treated or excluding patients with missing outcomes from our analysis can produce a biased estimate of a treatment effect. In this paper, we consider estimation of the risk difference (RD) in the presence of both noncompliance and missing outcomes under a RCT. On the basis of a constant risk additive model proposed elsewhere, we derive the maximum likelihood estimator (MLE) and develop three asymptotic interval estimators in closed form for the RD when we have outcome missing at random. We apply Monte Carlo simulation to evaluate and compare the performance of these estimators in a variety of situations. We note that all interval estimators developed here can perform well with respect to the coverage probability in all the situations considered here. We find that the interval estimator using tanh  ? 1 (x) transformation is generally more precise than the other two estimators with respect to the average length. Finally, we use the data taken from a randomized trial studying the association between flu vaccine and the risk of flu-related hospitalization to illustrate the practical use of these interval estimators.     

Hypothesis Testing and Estimation in Ordinal Data Under a Simple Crossover Design

Since each patient serves as his/her own control, the crossover design can be of use to improve power as compared with the parallel-groups design in studying noncurative treatments to certain chronic diseases. Although the research studies on the crossover design have been quite intensive, the discussions on analyzing ordinal data under such a design are truly limited. We propose using the generalized odds ratio (GOR) for paired sample data to measure the relative effect on patient responses for both treatment and period in ordinal data under a simple crossover trial. Assuming the treatment and period effects are multiplicative, we note that one can easily derive the maximum likelihood estimator (LE) in closed forms for the GOR of treatment and period effects. We develop asymptotic and exact procedures for testing treatment and period effects. We further derive asymptotic and exact interval estimators for the GOR of treatment and period effects. We use the data taken from a crossover trial to assess the clarity of leaflet instructions between two devices among asthma patients to illustrate the use of these test procedures and estimators developed here.     

The log-Burr XII regression model for grouped survival data

The log-Burr XII regression model for grouped survival data is evaluated in the presence of many ties. The methodology for grouped survival data is based on life tables, where the times are grouped in k intervals, and we fit discrete lifetime regression models to the data. The model parameters are estimated by maximum likelihood and jackknife methods. To detect influential observations in the proposed model, diagnostic measures based on case deletion, so-called global influence, and influence measures based on small perturbations in the data or in the model, referred to as local influence, are used. In addition to these measures, the total local influence and influential estimates are also used. We conduct Monte Carlo simulation studies to assess the finite sample behavior of the maximum likelihood estimators of the proposed model for grouped survival. A real data set is analyzed using a regression model for grouped data.     

A new technique for simulating the likelihood of stochastic differential equations

This article presents a new simulation‐based technique for estimating the likelihood of stochastic differential equations. This technique is based on a result of Dacunha‐Castelle and Florens‐Zmirou. These authors proved that the transition densities of a nonlinear diffusion process with a constant diffusion coefficient can be written in a closed form involving a stochastic integral. We show that this stochastic integral can be easily estimated through simulations and we prove a convergence result. This simulator for the transition density is used to obtain the simulated maximum likelihood (SML) estimator. We show through some Monte Carlo experiments that our technique is highly computationally efficient and the SML estimator converges rapidly to the maximum likelihood estimator.     

Non‐standard rates of convergence of criterion‐function‐based set estimators for binary response models

This paper establishes consistency and non‐standard rates of convergence for set estimators based on contour sets of criterion functions for a semi‐parametric binary response model under a conditional median restriction. The model can be partially identified due to potentially limited‐support regressors and an unknown distribution of errors. A set estimator analogous to the maximum score estimator is essentially cube‐root consistent for the identified set when a continuous but possibly bounded regressor is present. Arbitrarily fast convergence occurs when all regressors are discrete. We also establish the validity of a subsampling procedure for constructing confidence sets for the identified set. As a technical contribution, we provide more convenient sufficient conditions on the underlying empirical processes for cube‐root convergence and a sufficient condition for arbitrarily fast convergence, both of which can be applied to other models. Finally, we carry out a series of Monte Carlo experiments, which verify our theoretical findings and shed light on the finite‐sample performance of the proposed procedures.