Small-Sample Methods for Cluster-Robust Variance Estimation and Hypothesis Testing in Fixed Effects Models

ABSTRACT

In panel data models and other regressions with unobserved effects, fixed effects estimation is often paired with cluster-robust variance estimation (CRVE) to account for heteroscedasticity and un-modeled dependence among the errors. Although asymptotically consistent, CRVE can be biased downward when the number of clusters is small, leading to hypothesis tests with rejection rates that are too high. More accurate tests can be constructed using bias-reduced linearization (BRL), which corrects the CRVE based on a working model, in conjunction with a Satterthwaite approximation for t-tests. We propose a generalization of BRL that can be applied in models with arbitrary sets of fixed effects, where the original BRL method is undefined, and describe how to apply the method when the regression is estimated after absorbing the fixed effects. We also propose a small-sample test for multiple-parameter hypotheses, which generalizes the Satterthwaite approximation for t-tests. In simulations covering a wide range of scenarios, we find that the conventional cluster-robust Wald test can severely over-reject while the proposed small-sample test maintains Type I error close to nominal levels. The proposed methods are implemented in an R package called clubSandwich. This article has online supplementary materials.     

Model Averaging for Prediction With Fragmentary Data

ABSTRACT

One main challenge for statistical prediction with data from multiple sources is that not all the associated covariate data are available for many sampled subjects. Consequently, we need new statistical methodology to handle this type of “fragmentary data” that has become more and more popular in recent years. In this article, we propose a novel method based on the frequentist model averaging that fits some candidate models using all available covariate data. The weights in model averaging are selected by delete-one cross-validation based on the data from complete cases. The optimality of the selected weights is rigorously proved under some conditions. The finite sample performance of the proposed method is confirmed by simulation studies. An example for personal income prediction based on real data from a leading e-community of wealth management in China is also presented for illustration.     

Efficiency and Validity Analyses of Two-Stage Estimation Procedures and Derived Testing Procedures in Quantitative Linear Models with AR(1) Errors

Abstract

In a quantitative linear model with errors following a stationary Gaussian, first-order autoregressive or AR(1) process, Generalized Least Squares (GLS) on raw data and Ordinary Least Squares (OLS) on prewhitened data are efficient methods of estimation of the slope parameters when the autocorrelation parameter of the error AR(1) process, ρ, is known. In practice, ρ is generally unknown. In the so-called two-stage estimation procedures, ρ is then estimated first before using the estimate of ρ to transform the data and estimate the slope parameters by OLS on the transformed data. Different estimators of ρ have been considered in previous studies. In this article, we study nine two-stage estimation procedures for their efficiency in estimating the slope parameters. Six of them (i.e., three noniterative, three iterative) are based on three estimators of ρ that have been considered previously. Two more (i.e., one noniterative, one iterative) are based on a new estimator of ρ that we propose: it is provided by the sample autocorrelation coefficient of the OLS residuals at lag 1, denoted r(1). Lastly, REstricted Maximum Likelihood (REML) represents a different type of two-stage estimation procedure whose efficiency has not been compared to the others yet. We also study the validity of the testing procedures derived from GLS and the nine two-stage estimation procedures. Efficiency and validity are analyzed in a Monte Carlo study. Three types of explanatory variable x in a simple quantitative linear model with AR(1) errors are considered in the time domain: Case 1, x is fixed; Case 2, x is purely random; and Case 3, x follows an AR(1) process with the same autocorrelation parameter value as the error AR(1) process. In a preliminary step, the number of inadmissible estimates and the efficiency of the different estimators of ρ are compared empirically, whereas their approximate expected value in finite samples and their asymptotic variance are derived theoretically. Thereafter, the efficiency of the estimation procedures and the validity of the derived testing procedures are discussed in terms of the sample size and the magnitude and sign of ρ. The noniterative two-stage estimation procedure based on the new estimator of ρ is shown to be more efficient for moderate values of ρ at small sample sizes. With the exception of small sample sizes, REML and its derived F-test perform the best overall. The asymptotic equivalence of two-stage estimation procedures, besides REML, is observed empirically. Differences related to the nature, fixed or random (uncorrelated or autocorrelated), of the explanatory variable are also discussed.     

Risk of Error and the Kappa Coefficient of a Binary Diagnostic Test in the Presence of Partial Verification

The accuracy of a binary diagnostic test is usually measured in terms of its sensitivity and its specificity, or through positive and negative predictive values. Another way to describe the validity of a binary diagnostic test is the risk of error and the kappa coefficient of the risk of error. The risk of error is the average loss that is caused when incorrectly classifying a non-diseased or a diseased patient, and the kappa coefficient of the risk of error is a measure of the agreement between the diagnostic test and the gold standard. In the presence of partial verification of the disease, the disease status of some patients is unknown, and therefore the evaluation of a diagnostic test cannot be carried out through the traditional method. In this paper, we have deduced the maximum likelihood estimators and variances of the risk of error and of the kappa coefficient of the risk of error in the presence of partial verification of the disease. Simulation experiments have been carried out to study the effect of the verification probabilities on the coverage of the confidence interval of the kappa coefficient.     

Understanding exponential smoothing via kernel regression

Exponential smoothing is the most common model-free means of forecasting a future realization of a time series. It requires the specification of a smoothing factor which is usually chosen from the data to minimize the average squared residual of previous one-step-ahead forecasts. In this paper we show that exponential smoothing can be put into a nonparametric regression framework and gain some interesting insights into its performance through this interpretation. We also use theoretical developments from the kernel regression field to derive, for the first time, asymptotic properties of exponential smoothing forecasters.     

人的可靠性与人为差错

分析论述了影响人的可靠性的因素 ,提出了提高人的可靠性、减少人为差错的措施和方法。     

检测系统分辨力定量计算的研究

分辨力是传感器和测量仪器的重要技术参数。数字仪器的分辨力通常取决于内部模数转换器的位数,但如何确定由仪器和传感器组成的检测系统的分辨力,目前尚未见到与之相关的专门性研究成果。该文论证了检测系统的分辨力只与随机误差相关,要提高分辨力就必须减小测量结果的随机波动。通过实验提出了检测系统分辨力的定量计算公式,并证明了其置信概率大于90.9%。     

含m-增生算子的非线性方程带误差的迭代程序

证明了对任意Banach空间E中的Lipschitz m-增生算子T,(A)f∈E.带误差的Ishikawa型迭代程序强收敛于方程x+Tx=f的唯一解.作为直接的应用,给出了有关Ishikawa迭代程序稳定性的若干定理.所得的结论改进和推广了近期相关的结果.     

核电厂人因失误机理研究:QAP分析

为了明确核电厂人因失误机理并为后期人因失误的精准防控提供理论依据,文章采用“组织定向人因失误分析技术”对2010—2020年核电厂208件人因失误事件报告进行分析,获得样本数据,经统计辨识主要的人因失误类型(知识型、规则型和技能型)和影响因素(如教育培训、组织设计、规程、人机界面等),结合影响因素之间的相关性及因子分析构建主要人因失误场景(如知识/技能经验、注意力、压力和态度等)。由于各因素之间存在相互影响,因而采用QAP方法(Quadratic Assignment Procedure,二次指派程序)对人因失误场景进行相关分析和回归分析,建立知识型、规则型、技能型人因失误因果模型,辨识不同的人因失误场景,揭示人因失误机理。结果表明,不同的人因失误类型其产生机理以及场景显著性不同,为人因失误的精准防控提供理论依据。     

感悟化：《东方的太阳》拓展政治抒情诗的新路径

谭仲池先生的《东方的太阳》,把抒情建立在与客体的心灵对话之上,结构穿梭在“求谒”的且歌且行之中,语言紧贴智性化的生命律动,作品以感悟的方式丰富、拓展了政治抒情诗的新路径。