Robust methods in econometrics

This survey of recent developments in robust estimation and inference is directed primarily toward econometricians. It is argued that many of the techniques in common use in econometrics are highly sensitive to unverified hypotheses. Recent progress in designing alternative robust procedures is described and some prospects for future developments are discussed.     

Properties of robust m-estimators for poisson and negative binomial data ∗

We investigate robust M-estimators of location and over-dispersion for independent and identically distributed samples from Poisson and Negative Binomial (NB)distributions. We focus on asymptotic and small-sample efficiencies, outlier-induced biases, and biases caused by model mis-specification. This is important information for assessing the practical utility of the estimation method. Our results demonstrate that resonably efficient estimation of location and over-dispersion parameters for count data is possible with samples sizes as small as n=25. The sensitivity of these stimators, especially when the amount of over-dispersion is small. We aslo conclude that serious biases result when using robust Poisson M-estimation with NB data. The biases are less serious when using robust NB M-estimation with Poisson data.     

A Robust Test for Non-nested Hypotheses

We propose a robust version of Cox-type test statistics for the choice between two non-nested hypotheses. We first show that the influence of small amounts of contamination in the data on the test decision can be very large. Secondly, we build a robust test statistic by using the results on robust parametric tests that are available in the literature and show that the level of the robust test is stable. Finally, we show numerically not only the robustness of this new test statistic but also that its asymptotic distribution is a good approximation of its sample distribution, unlike for the classical test statistic. We apply our results to the choice between a Pareto and an exponential distribution as well as between two competing regressors in the simple linear regression model without intercept.     

Focused Information Criterion and Model Averaging in Quantile Regression

In this article, we study model selection and model averaging in quantile regression. Under general conditions, we develop a focused information criterion and a frequentist model average estimator for the parameters in quantile regression model, and examine their theoretical properties. The new procedures provide a robust alternative to the least squares method or likelihood method, and a major advantage of the proposed procedures is that when the variance of random error is infinite, the proposed procedure works beautifully while the least squares method breaks down. A simulation study and a real data example are presented to show that the proposed method performs well with a finite sample and is easy to use in practice.     

Fast and robust bootstrap

In this paper we review recent developments on a bootstrap method for robust estimators which is computationally faster and more resistant to outliers than the classical bootstrap. This fast and robust bootstrap method is, under reasonable regularity conditions, asymptotically consistent. We describe the method in general and then consider its application to perform inference based on robust estimators for the linear regression and multivariate location-scatter models. In particular, we study confidence and prediction intervals and tests of hypotheses for linear regression models, inference for location-scatter parameters and principal components, and classification error estimation for discriminant analysis.     

Higher-order asymptotics under model misspecification

Most of the higher-order asymptotic results in statistical inference available in the literature assume model correctness. The aim of this paper is to develop higher-order results under model misspecification. The density functions to O(n^?3/2) of the robust score test statistic and the robust Wald test statistic are derived under the null hypothesis, for the scalar as well as the multiparameter case. Alternate statistics which are robust to O(n^?3/2) are also proposed.     

Likelihood inference for correlated binary data without any information about the joint distributions

We propose a universal robust likelihood that is able to accommodate correlated binary data without any information about the underlying joint distributions. This likelihood function is asymptotically valid for the regression parameter for any underlying correlation configurations, including varying under- or over-dispersion situations, which undermines one of the regularity conditions ensuring the validity of crucial large sample theories. This robust likelihood procedure can be easily implemented by using any statistical software that provides naïve and sandwich covariance matrices for regression parameter estimates. Simulations and real data analyses are used to demonstrate the efficacy of this parametric robust method.     

Robust two-level regular fractional factorial designs

Abstract

In this paper, we introduce the concept of model quality for two-level regular fractional factorial designs. Under the effect hierarchy principle, this paper raises the definition of model quality and introduces robust model-number pattern (RP) to choose the optimal robust design. Some theoretical results on this optimality and comparisons with GMC and MEC criterion are given.     

一个新的稳健ARCH检验和YJ-GARCH模型

 众所周知,Engle (1982) 的ARCH检验对于条件均值模型误设并不稳健,特别地,当条件均值是非线性过程而我们仅对之建立线性模型时,它过度地拒绝真实的原假设,导致出现严重的水平扭曲 (size distortion)。因此,本文在文献当中首次利用Yeo-Johnson变换方法来转换均值模型的因变量以排除ARCH 过程中均值部分的非线性,进而提出一个新的稳健ARCH检验以及一个新的GARCH模型——Yeo-Johnson (YJ) GARCH模型。蒙特卡罗模拟结果表明,稳健的ARCH检验在水平 (size) 和势 (power) 方面的表现要显著优于Engle (1982) 的ARCH检验。对上证综指收益率的实证研究结果表明,YJ-GARCH模型的拟合效果要显著优于线性GARCH模型。     

A semi-parametric approach to robust parameter design

Parameter design or robust parameter design (RPD) is an engineering methodology intended as a cost-effective approach for improving the quality of products and processes. The goal of parameter design is to choose the levels of the control variables that optimize a defined quality characteristic. An essential component of RPD involves the assumption of well estimated models for the process mean and variance. Traditionally, the modeling of the mean and variance has been done parametrically. It is often the case, particularly when modeling the variance, that nonparametric techniques are more appropriate due to the nature of the curvature in the underlying function. Most response surface experiments involve sparse data. In sparse data situations with unusual curvature in the underlying function, nonparametric techniques often result in estimates with problematic variation whereas their parametric counterparts may result in estimates with problematic bias. We propose the use of semi-parametric modeling within the robust design setting, combining parametric and nonparametric functions to improve the quality of both mean and variance model estimation. The proposed method will be illustrated with an example and simulations.     

D-optimal minimax design criterion for two-level fractional factorial designs

A D-optimal minimax design criterion is proposed to construct two-level fractional factorial designs, which can be used to estimate a linear model with main effects and some specified interactions. D-optimal minimax designs are robust against model misspecification and have small biases if the linear model contains more interaction terms. When the D-optimal minimax criterion is compared with the D-optimal design criterion, we find that the D-optimal design criterion is quite robust against model misspecification. Lower and upper bounds derived for the loss functions of optimal designs can be used to estimate the efficiencies of any design and evaluate the effectiveness of a search algorithm. Four algorithms to search for optimal designs for any run size are discussed and compared through several examples. An annealing algorithm and a sequential algorithm are particularly effective to search for optimal designs.     

On minimax designs when there are two candidate models

This work is motivated by the need to find experimental designs which are robust under different model assumptions. We measure robustness by calculating a measure of design efficiency with respect to a design optimality criterion and say that a design is robust if it is reasonably efficient under different model scenarios. We discuss two design criteria and an algorithm which can be used to obtain robust designs. The first criterion employs a Bayesian-type approach by putting a prior or weight on each candidate model and possibly priors on the corresponding model parameters. We define the first criterion as the expected value of the design efficiency over the priors. The second design criterion we study is the minimax design which minimizes the worst value of a design criterion over all candidate models. We establish conditions when these two criteria are equivalent when there are two candidate models. We apply our findings to the area of accelerated life testing and perform sensitivity analysis of designs with respect to priors and misspecification of planning values.     

Implications of Model Misspecification in Robust Tests for Recurrent Events

Chronic disease processes often feature transient recurrent adverse clinical events. Treatment comparisons in clinical trials of such disorders must be based on valid and efficient methods of analysis. We discuss robust strategies for testing treatment effects with recurrent events using methods based on marginal rate functions, partially conditional rate functions, and methods based on marginal failure time models. While all three approaches lead to valid tests of the null hypothesis when robust variance estimates are used, they differ in power. Moreover, some approaches lead to estimators of treatment effect which are more easily interpreted than others. To investigate this, we derive the limiting value of estimators of treatment effect from marginal failure time models and illustrate their dependence on features of the underlying point process, as well as the censoring mechanism. Through simulation, we show that methods based on marginal failure time distributions are shown to be sensitive to treatment effects delaying the occurrence of the very first recurrences. Methods based on marginal or partially conditional rate functions perform well in situations where treatment effects persist or in settings where the aim is to summarizee long-term data on efficacy.     

Bayesian Longitudinal Data Analysis with Mixed Models and Thick-tailed Distributions using MCMC

Linear mixed effects models are frequently used to analyse longitudinal data, due to their flexibility in modelling the covariance structure between and within observations. Further, it is easy to deal with unbalanced data, either with respect to the number of observations per subject or per time period, and with varying time intervals between observations. In most applications of mixed models to biological sciences, a normal distribution is assumed both for the random effects and for the residuals. This, however, makes inferences vulnerable to the presence of outliers. Here, linear mixed models employing thick-tailed distributions for robust inferences in longitudinal data analysis are described. Specific distributions discussed include the Student-t, the slash and the contaminated normal. A Bayesian framework is adopted, and the Gibbs sampler and the Metropolis-Hastings algorithms are used to carry out the posterior analyses. An example with data on orthodontic distance growth in children is discussed to illustrate the methodology. Analyses based on either the Student-t distribution or on the usual Gaussian assumption are contrasted. The thick-tailed distributions provide an appealing robust alternative to the Gaussian process for modelling distributions of the random effects and of residuals in linear mixed models, and the MCMC implementation allows the computations to be performed in a flexible manner.     

Robust likelihood functions in Bayesian inference

In order to deal with mild deviations from the assumed parametric model, we propose a procedure for accounting for model uncertainty in the Bayesian framework. In particular, in the derivation of posterior distributions, we discuss the use of robust pseudo-likelihoods, which offer the advantage of preventing the effects caused by model misspecifications, i.e. when the underlying distribution lies in a neighborhood of the assumed model. The influence functions of posterior summaries, such as the posterior mean, are investigated as well as the asymptotic properties of robust posterior distributions. Although the use of a pseudo-likelihood cannot be considered orthodox in the Bayesian perspective, it is shown that, also through some illustrative examples, how a robust pseudo-likelihood, with the same asymptotic properties of a genuine likelihood, can be useful in the inferential process in order to prevent the effects caused by model misspecifications.     

Robust estimation in growth curve models

The growth curve model introduced by potthoff and Roy 1964 is a general statistical model which includes as special cases regression models and both univariate and multivariate analysis of variance models. The methods currently available for estimating the parameters of this model assume an underlying multivariate normal distribution of errors. In this paper, we discuss tw robst estimators of the growth curve loction and scatter parameters based upon M-estimation techniques and the work done by maronna 1976. The asymptotic distribution of these robust estimators are discussed and a numerical example given.     

城市森林对经济发展影响的实证检验——基于281个城市的面板数据

文章基于2001—2017年中国281个地级及以上城市的森林资源及地区经济发展的数据,运用空间面板SARAR模型,系统考察了城市森林资源状况对中国经济发展的影响及其作用机制。结果发现,城市森林建设可以显著提升中国地区的经济发展;城市森林资源影响中国经济发展的途径可以通过人力资本集聚等途径来实现;城市森林资源的经济效益是广泛存在的,不因城市规模大小而消失或减退。对以上结论的进一步检验发现其仍然是稳健的。     

缺失数据下的逆概率多重加权分位回归估计及其应用

数据缺失问题普遍存在于应用研究中。在随机缺失机制假定下,本文从模型推断角度出发,针对线性缺失分位回归模型,提出一种新的有效估计方法——逆概率多重加权（IPMW）估计。该方法是在逆概率加权（IPW）估计的基础上,结合倾向得分匹配及模型平均思想,经过多次估计,加权确定最终参数估计结果。该方法适用于响应变量是独立同分布或独立非同分布的情形,并适用于绝大多数缺失场景。经过理论推导及模拟研究发现,IPMW估计量在继承IPW估计量的优势上具有更稳健的性质。最后,将该方法应用于含有缺失数据的微观调查数据中,研究了经济较发达的准一线城市中等收入群体消费水平的影响因素,对比两种估计方法的估计结果及置信带,发现逆概率多重加权估计量的标准偏差更小,估计结果更稳健。     

Model averaging for M-estimation

M-estimation is a widely used technique for robust statistical inference. In this paper, we study model selection and model averaging for M-estimation to simultaneously improve the coverage probability of confidence intervals of the parameters of interest and reduce the impact of heavy-tailed errors or outliers in the response. Under general conditions, we develop robust versions of the focused information criterion and a frequentist model average estimator for M-estimation, and we examine their theoretical properties. In addition, we carry out extensive simulation studies as well as two real examples to assess the performance of our new procedure, and find that the proposed method produces satisfactory results.     

Recent Developments and Their Implications for the Future of Academic Statistical Consulting Centers

I describe how developments over the past 25 years in computing, funding, personnel, purpose, and training have affected academic statistical consulting centers and discuss how these developments and trends point to a range of potential futures. At one extreme, academic statistical consulting centers fail to adapt to competition from other disciplines in an increasingly fragmented market for statistical consulting and spiral downward toward irrelevancy and extinction. At the other extreme, purpose-driven academic statistical consulting centers constantly increase their impact in a virtuous cycle, leading the way toward the profession of statistics having greater positive impact on society. I conclude with actions to take to assure a robust future and increased impact for academic statistical consulting centers.