Solving endogeneity problems in multilevel estimation: an example using education production functions

This paper explores endogeneity problems in multilevel estimation of education production functions. The focus is on level 2 endogeneity which arises from correlations between student characteristics and omitted school variables. Theses correlations are mainly the result of student stratification between schools. From an econometric point of view, the correlations between student and school characteristics imply that the omission of some variables may generate endogeneity bias. Therefore, an estimation approach based on the Mundlak [20 Mundlak, Y. 1978. On the pooling of time series and cross-sectional data. Econometrica, 46: 69–86. [Crossref], [Web of Science ®] , [Google Scholar]] technique is developed in order to tackle bias and to generate consistent estimates. Note that our analysis can be extended to any multilevel-structured data (students nested within schools, employees within firms, firms within regions, etc). The entire analysis is undertaken in a comparative context between three countries: Germany, Finland and the UK. Each one of them represents a particular system. For instance, Finland is known for its extreme comprehensiveness, Germany for early selection and the UK for its liberalism. These countries are used to illustrate the theory and to prove that the level of bias arising from omitted variables varies according to the characteristics of education systems.     

On MSE of EBLUP

We consider Best Linear Unbiased Predictors (BLUPs) and Empirical Best Linear Unbiased Predictors (EBLUPs) under the general mixed linear model. The BLUP was proposed by Henderson (Ann Math Stat 21:309–310, 1950). The formula of this BLUP includes unknown elements of the variance-covariance matrix of random variables. If the elements in the formula of the BLUP proposed by Henderson (Ann Math Stat 21:309–310, 1950) are replaced by some type of estimators, we obtain the two-stage predictor called the EBLUP which is model-unbiased (Kackar and Harville in Commun Stat A 10:1249–1261, 1981). Kackar and Harville (J Am Stat Assoc 79:853–862, 1984) show an approximation of the mean square error (the MSE) of the predictor and propose an estimator of the MSE. The MSE and estimators of the MSE are also studied by Prasad and Rao (J Am Stat Assoc 85:163–171, 1990), Datta and Lahiri (Stat Sin 10:613–627, 2000) and Das et al. (Ann Stat 32(2):818–840, 2004). In the paper we consider the BLUP proposed by Royall (J Am Stat Assoc 71:657–473, 1976. Ża̧dło (On unbiasedness of some EBLU predictor. Physica-Verlag, Heidelberg, pp 2019–2026, 2004) shows that the BLUP proposed by Royall (J Am Stat Assoc 71:657–473, 1976) may be treated as a generalisation of the BLUP proposed by Henderson (Ann Math Stat 21:309–310, 1950) and proves model unbiasedness of the EBLUP based on the formula of the BLUP proposed by Royall (J Am Stat Assoc 71:657–473, 1976) under some assumptions. In this paper we derive the formula of the approximate MSE of the EBLUP and its estimators. We prove that the approximation of the MSE is accurate to terms o(D ⁻¹) and that the estimator of the MSE is approximately unbiased in the sense that its bias is o(D ⁻¹) under some assumptions, where D is the number of domains. The proof is based on the results obtained by Datta and Lahiri (Stat Sin 10:613–627, 2000). Using our results we show some EBLUP based on the special case of the general linear model. We also present the formula of its MSE and estimators of its MSE and their performance in Monte Carlo simulation study.      

Asymptotic Normality of a Kernel Conditional Quantile Estimator Under Strong Mixing Hypothesis and Left-Truncation

We consider the estimation of the conditional quantile when the interest variable is subject to left truncation. Under regularity conditions, it is shown that the kernel estimate of the conditional quantile is asymptotically normally distributed, when the data exhibit some kind of dependence. We use asymptotic normality to construct confidence bands for predictors based on the kernel estimate of the conditional median.     

Histogram estimation of radon-nikodym derivatives for strong mixing data

Nonparametric inference for point processes is discussed by way of histograms, which provide a nice tool for the analysis of on-line data. The construction of histograms depends on a sequence of partitions, which we take tc be nonenibedded to allow partitions with sets of equal measure. This presents some theoretical problems, which are addressed with an assumption on the decomposition of second order moments. In another direction, we drop the usual independence assumption on the sample, replacing it by a strong mixing assumption. Under this setting, we study the convergence of the histogram in probability, which depends on approximation conditions between the distributions of random pairs and the product of their marginal distributions, and^almost completely, which is based on the decomposition of the second order moments. This last convergence is stated on two versions according to the assumption of Laplace transforms or the Cramer moment conditions. These are somewhat stronger, but enable us to recover the usual condition on the decrease rate of sets on each partition. In the final section we prove that the finite dimensional distributions converge in distribution to a Gaussian centered vector with a specified covariance.     

Testing for residual correlation of any order in the autoregressive process

We are interested in the implications of a linearly autocorrelated driven noise on the asymptotic behavior of the usual least-squares estimator in a stable autoregressive process. We show that the least-squares estimator is not consistent and we suggest a sharp analysis of its almost sure limiting value as well as its asymptotic normality. We also establish the almost sure convergence and the asymptotic normality of the estimated serial correlation parameter of the driven noise. Then, we derive a statistical procedure enabling to test for correlation of any order in the residuals of an autoregressive modelling, giving clearly better results than the commonly used portmanteau tests of Ljung–Box and Box–Pierce, and appearing to outperform the Breusch–Godfrey procedure on small-sized samples.     

A Bayesian dose‐finding design for drug combination clinical trials based on the logistic model

In early phase dose‐finding cancer studies, the objective is to determine the maximum tolerated dose, defined as the highest dose with an acceptable dose‐limiting toxicity rate. Finding this dose for drug‐combination trials is complicated because of drug–drug interactions, and many trial designs have been proposed to address this issue. These designs rely on complicated statistical models that typically are not familiar to clinicians, and are rarely used in practice. The aim of this paper is to propose a Bayesian dose‐finding design for drug combination trials based on standard logistic regression. Under the proposed design, we continuously update the posterior estimates of the model parameters to make the decisions of dose assignment and early stopping. Simulation studies show that the proposed design is competitive and outperforms some existing designs. We also extend our design to handle delayed toxicities. Copyright © 2014 John Wiley & Sons, Ltd.     

Asymptotic normality of a robust estimator of the regression function for functional time series data

We propose a family of robust nonparametric estimators for a regression function based on the kernel method. We establish the asymptotic normality of the estimator under the concentration property on small balls probability measure of the functional explanatory variable when the observations exhibit some kind of dependence. This approach can be applied in time series analysis to make prediction and build confidence bands. We illustrate our methodology on the US electricity consumption data.     

Hazard Perception, Risk Perception, and the Need for Decontamination by Residents Exposed to Soil Pollution: The Role of Sustainability and the Limits of Expert Knowledge

This case study examines the hazard and risk perception and the need for decontamination according to people exposed to soil pollution. Using an ecological-symbolic approach (ESA), a multidisciplinary model is developed that draws upon psychological and sociological perspectives on risk perception and includes ecological variables by using data from experts' risk assessments. The results show that hazard perception is best predicted by objective knowledge, subjective knowledge, estimated knowledge of experts, and the assessed risks. However, experts' risk assessments induce an increase in hazard perception only when residents know the urgency of decontamination. Risk perception is best predicted by trust in the risk management. Additionally, need for decontamination relates to hazard perception, risk perception, estimated knowledge of experts, and thoughts about sustainability. In contrast to the knowledge deficit model, objective and subjective knowledge did not significantly relate to risk perception and need for decontamination. The results suggest that residents can make a distinction between hazards in terms of the seriousness of contamination on the one hand, and human health risks on the other hand. Moreover, next to the importance of social determinants of environmental risk perception, this study shows that the output of experts' risk assessments—or the objective risks—can create a hazard awareness rather than an alarming risk consciousness, despite residents' distrust of scientific knowledge.     

Incremental exercise tests in master athletes and untrained older adults

This study aimed to analyze the impact of step-duration protocols, 1-min vs. 3-min, on cardiorespiratory responses to exercise, whatever the aerobic-fitness level of sedentary (65.5 +/- 2.3 years, n = 8) or highly fit (63.1 +/- 3.2 years, n = 19) participants. Heart rate and VO2 at the first and second ventilatory thresholds (VT1 ,VT2) and maximal exercise were not significantly different between the two protocols. In master athletes, the 3-min protocol elicited significantly lower ventilation at VT2 and maximal exercise (p < .01). In the latter, breathlessness was also lower at maximal exercise (p < .05) than in sedentary participants. In trained or sedentary older adults, VT1, VT2, and were not influenced by stage duration. According to the lower breathlessness and ventilation, however, the 3-min step protocol could be more appropriate in master athletes. In untrained participants, because the cardiorespiratory responses were similar with the two incremental exercise tests, either of them could be used.     

Modelling Aggregation on the Large Scale and Regularity on the Small Scale in Spatial Point Pattern Datasets

We consider a dependent thinning of a regular point process with the aim of obtaining aggregation on the large scale and regularity on the small scale in the resulting target point process of retained points. Various parametric models for the underlying processes are suggested and the properties of the target point process are studied. Simulation and inference procedures are discussed when a realization of the target point process is observed, depending on whether the thinned points are observed or not. The paper extends previous work by Dietrich Stoyan on interrupted point processes.