A Dirichlet random coefficient regression model for quality indicators

We present a random coefficient regression model in which a response is linearly related to some explanatory variables with random coefficients following a Dirichlet distribution. These coefficients can be interpreted as weights because they are nonnegative and add up to one. The proposed estimation procedure combines iteratively reweighted least squares and the maximization on an approximated likelihood function. We also present a diagnostic tool based on a residual Q–Q plot and two procedures for estimating individual weights. The model is used to construct an index for measuring the quality of the railroad system in Spain.     

A robust coefficient of determination for regression

To assess the quality of the fit in a multiple linear regression, the coefficient of determination or R² is a very simple tool, yet the most used by practitioners. Indeed, it is reported in most statistical analyzes, and although it is not recommended as a final model selection tool, it provides an indication of the suitability of the chosen explanatory variables in predicting the response. In the classical setting, it is well known that the least-squares fit and coefficient of determination can be arbitrary and/or misleading in the presence of a single outlier. In many applied settings, the assumption of normality of the errors and the absence of outliers are difficult to establish. In these cases, robust procedures for estimation and inference in linear regression are available and provide a suitable alternative.     

Estimation of semi-varying coefficient models with nonstationary regressors

We study a semivarying coefficient model where the regressors are generated by the multivariate unit root I(1) processes. The influence of the explanatory vectors on the response variable satisfies the semiparametric partially linear structure with the nonlinear component being functional coefficients. A semiparametric estimation methodology with the first-stage local polynomial smoothing is applied to estimate both the constant coefficients in the linear component and the functional coefficients in the nonlinear component. The asymptotic distribution theory for the proposed semiparametric estimators is established under some mild conditions, from which both the parametric and nonparametric estimators are shown to enjoy the well-known super-consistency property. Furthermore, a simulation study is conducted to investigate the finite sample performance of the developed methodology and results.     

A Monte Carlo-based pseudo-coefficient of determination for generalized linear models with binary outcome

In this article, we focus on a pseudo-coefficient of determination for generalized linear models with binary outcome. Although there are numerous coefficients of determination proposed in the literature, none of them is identified as the best in terms of estimation accuracy, or incorporates all desired characteristics of a precise coefficient of determination. Considering this, we propose a new coefficient of determination by using a computational Monte Carlo approach, and exhibit main characteristics of the proposed coefficient of determination both analytically and numerically. We evaluate and compare performances of the proposed and nine existing coefficients of determination by a comprehensive Monte Carlo simulation study. The proposed measure is found superior to the existent measures when dependent variable is balanced or moderately unbalanced for probit, logit, and complementary log–log link functions and a wide range of sample sizes. Due to the extensive design space of our simulation study, we identify new conditions in which previously recommended coefficients of determination should be used carefully.     

Single-index modelling of conditional probabilities in two-way contingency tables

When analysing a contingency table, it is often worth relating the probabilities that a given individual falls into different cells from a set of predictors. These conditional probabilities are usually estimated using appropriate regression techniques. In particular, in this paper, a semiparametric model is developed. Essentially, it is only assumed that the effect of the vector of covariates on the probabilities can entirely be captured by a single index, which is a linear combination of the initial covariates. The estimation is then twofold: the coefficients of the linear combination and the functions linking this index to the related conditional probabilities have to be estimated. Inspired by the estimation procedures already proposed in the literature for single-index regression models, four estimators of the index coefficients are proposed and compared, from a theoretical point-of-view, but also practically, with the aid of simulations. Estimation of the link functions is also addressed.     

Robust Estimation of Multi-response Surfaces Considering Correlation Structure

Response surfaces express the behavior of responses and can be used for both single and multi-response problems. A common approach to estimate a response surface using experimental results is the ordinary least squares (OLS) method. Since OLS is very sensitive to outliers, some robust approaches have been discussed in the literature. Although there are many methods available in the literature for multiple response optimizations, there are a few studies in model building especially robust models. Assuming correlated responses, in this paper, a robust coefficient estimation method is proposed for multi response problem based on M-estimators. In order to illustrate the performance of the proposed procedure, a contaminated experimental design using a numerical example available in the literature with some modifications is used. Both the classical multivariate least squares method and the proposed robust multivariate approach are used to estimate regression coefficients of multi-response surfaces based on this example. Moreover, a comparison of the proposed robust multi response surface (RMRS) approach with separate robust estimation of single response show that the proposed approach is more efficient.     

Simulation-based model comparison methodology with application to road accident models

This article assumes the goal of proposing a simulation-based theoretical model comparison methodology with application to two time series road accident models. The model comparison exercise helps to quantify the main differences and similarities between the two models and comprises of three main stages: (1) simulation of time series through a true model with predefined properties; (2) estimation of the alternative model using the simulated data; (3) sensitivity analysis to quantify the effect of changes in the true model parameters on alternative model parameter estimates through analysis of variance, ANOVA. The proposed methodology is applied to two time series road accident models: UCM (unobserved components model) and DRAG (Demand for Road Use, Accidents and their Severity). Assuming that the real data-generating process is the UCM, new datasets approximating the road accident data are generated, and DRAG models are estimated using the simulated data. Since these two methodologies are usually assumed to be equivalent, in a sense that both models accurately capture the true effects of the regressors, we are specifically addressing the modeling of the stochastic trend, through the alternative model. Stochastic trend is the time-varying component and is one of the crucial factors in time series road accident data. Theoretically, it can be easily modeled through UCM, given its modeling properties. However, properly capturing the effect of a non-stationary component such as stochastic trend in a stationary explanatory model such as DRAG is challenging. After obtaining the parameter estimates of the alternative model (DRAG), the estimates of both true and alternative models are compared and the differences are quantified through experimental design and ANOVA techniques. It is observed that the effects of the explanatory variables used in the UCM simulation are only partially captured by the respective DRAG coefficients. This a priori, could be due to multicollinearity but the results of both simulation of UCM data and estimating of DRAG models reveal that there is no significant static correlation among regressors. Moreover, in fact, using ANOVA, it is determined that this regression coefficient estimation bias is caused by the presence of the stochastic trend present in the simulated data. Thus, the results of the methodological development suggest that the stochastic component present in the data should be treated accordingly through a preliminary, exploratory data analysis.     

时变三因子模型风险系数的动态研究

由Fama和French提出的三因子模型能够较好地解释股票的收益率风险溢价。文章以状态空间模型为框架,将风险因子系数作为状态变量,市场风险溢价作为观测变量,构建时变三因子模型来应对股票市场价格的时变特征。研究结果显示,利用卡尔曼滤波来估计时变风险因子系数,增强了估计结果的准确性与连贯性;风险因子系数变化规律与中国A股市场政策和环境影响相吻合,消除非理性噪声后的时变三因子模型更具有解释力度。     

Credit risk assessment with Bayesian model averaging

Many credit risk models are based on the selection of a single logistic regression model, on which to base parameter estimation. When many competing models are available, and without enough guidance from economical theory, model averaging represents an appealing alternative to the selection of single models. Despite model averaging approaches have been present in statistics for many years, only recently they are starting to receive attention in economics and finance applications. This contribution shows how Bayesian model averaging can be applied to credit risk estimation, a research area that has received a great deal of attention recently, especially in the light of the global financial crisis of the last few years and the correlated attempts to regulate international finance. The paper considers the use of logistic regression models under the Bayesian Model Averaging paradigm. We argue that Bayesian model averaging is not only more correct from a theoretical viewpoint, but also slightly superior, in terms of predictive performance, with respect to single selected models.     

Consistent estimation of regression coefficients in ultrastructural measurement error model using stochastic prior information

The problem of consistent estimation of regression coefficients in a multivariate linear ultrastructural measurement error model is considered in this article when some additional information on regression coefficients is available a priori. Such additional information is expressible in the form of stochastic linear restrictions. Utilizing stochastic restrictions given a priori, some methodologies are presented to obtain the consistent estimators of regression coefficients under two types of additional information separately, viz., covariance matrix of measurement errors and reliability matrix associated with explanatory variables. The measurement errors are assumed to be not necessarily normally distributed. The asymptotic properties of the proposed estimators are derived and analyzed analytically as well as numerically through a Monte Carlo simulation experiment.     

Backfitting Random Varying-Coefficient Models with Time-Dependent Smoothing Covariates

Abstract.  In this paper, we propose a random varying-coefficient model for longitudinal data. This model is different from the standard varying-coefficient model in the sense that the time-varying coefficients are assumed to be subject-specific, and can be considered as realizations of stochastic processes. This modelling strategy allows us to employ powerful mixed-effects modelling techniques to efficiently incorporate the within-subject and between-subject variations in the estimators of time-varying coefficients. Thus, the subject-specific feature of longitudinal data is effectively considered in the proposed model. A backfitting algorithm is proposed to estimate the coefficient functions. Simulation studies show that the proposed estimation methods are more efficient in finite-sample performance compared with the standard local least squares method. An application to an AIDS clinical study is presented to illustrate the proposed methodologies.     

Factor analysis regression

In the presence of multicollinearity the literature points to principal component regression (PCR) as an estimation method for the regression coefficients of a multiple regression model. Due to ambiguities in the interpretation, involved by the orthogonal transformation of the set of explanatory variables, the method could not yet gain wide acceptance. Factor analysis regression (FAR) provides a model-based estimation method which is particularly tailored to overcome multicollinearity in an errors-in-variables setting. In this paper two feasible versions of a FAR estimator are compared with the OLS estimator and the PCR estimator by means of Monte Carlo simulation. While the PCR estimator performs best in cases of strong and high multicollinearity, the Thomson-based FAR estimator proves to be superior when the regressors are moderately correlated.     

Analytical closed-form solution for binary logit regression by categorical predictors

In contrast to the common belief that the logit model has no analytical presentation, it is possible to find such a solution in the case of categorical predictors. This paper shows that a binary logistic regression by categorical explanatory variables can be constructed in a closed-form solution. No special software and no iterative procedures of nonlinear estimation are needed to obtain a model with all its parameters and characteristics, including coefficients of regression, their standard errors and t-statistics, as well as the residual and null deviances. The derivation is performed for logistic models with one binary or categorical predictor, and several binary or categorical predictors. The analytical formulae can be used for arithmetical calculation of all the parameters of the logit regression. The explicit expressions for the characteristics of logit regression are convenient for the analysis and interpretation of the results of logistic modeling.     

Functional Mixed Effects Model for Small Area Estimation

Functional data analysis has become an important area of research because of its ability of handling high‐dimensional and complex data structures. However, the development is limited in the context of linear mixed effect models and, in particular, for small area estimation. The linear mixed effect models are the backbone of small area estimation. In this article, we consider area‐level data and fit a varying coefficient linear mixed effect model where the varying coefficients are semiparametrically modelled via B‐splines. We propose a method of estimating the fixed effect parameters and consider prediction of random effects that can be implemented using a standard software. For measuring prediction uncertainties, we derive an analytical expression for the mean squared errors and propose a method of estimating the mean squared errors. The procedure is illustrated via a real data example, and operating characteristics of the method are judged using finite sample simulation studies.     

纵向部分线性变系数EV模型的估计

纵向数据是随着时间变化对个体进行重复观测而得到的一种相关性数据,广泛出现在诸多科学研究领域。在对个体进行观测时,测量误差不可避免,忽略测量误差往往会导致有偏估计。本文利用二次推断函数方法研究关于纵向数据的参数部分和非参数部分协变量均含有测量误差的部分线性变系数测量误差(errors-in-variables, EV)模型的估计问题。利用B样条逼近模型中的未知系数函数,构造关于回归参数和B样条系数的偏差修正的二次推断函数以处理个体内相关性和测量误差,得到回归参数和变系数的偏差修正的二次推断函数估计,然后证明了估计方法和结果的渐近性质。数值模拟和实例数据分析结果显示本文提出的方法具有一定的实用价值。     

Estimation of Some Nonlinear Panel Data Models With Both Time-Varying and Time-Invariant Explanatory Variables

The so-called “fixed effects” approach to the estimation of panel data models suffers from the limitation that it is not possible to estimate the coefficients on explanatory variables that are time-invariant. This is in contrast to a “random effects” approach, which achieves this by making much stronger assumptions on the relationship between the explanatory variables and the individual-specific effect. In a linear model, it is possible to obtain the best of both worlds by making random effects-type assumptions on the time-invariant explanatory variables while maintaining the flexibility of a fixed effects approach when it comes to the time-varying covariates. This article attempts to do the same for some popular nonlinear models.     

Sieve Estimation of Time-Varying Panel Data Models With Latent Structures

We propose a heterogeneous time-varying panel data model with a latent group structure that allows the coefficients to vary over both individuals and time. We assume that the coefficients change smoothly over time and form different unobserved groups. When treated as smooth functions of time, the individual functional coefficients are heterogeneous across groups but homogeneous within a group. We propose a penalized-sieve-estimation-based classifier-Lasso (C-Lasso) procedure to identify the individuals’ membership and to estimate the group-specific functional coefficients in a single step. The classification exhibits the desirable property of uniform consistency. The C-Lasso estimators and their post-Lasso versions achieve the oracle property so that the group-specific functional coefficients can be estimated as well as if the individuals’ membership were known. Several extensions are discussed. Simulations demonstrate excellent finite sample performance of the approach in both classification and estimation. We apply our method to study the heterogeneous trending behavior of GDP per capita across 91 countries for the period 1960–2012 and find four latent groups.     

A general condition for avoiding effect reversal after marginalization

Summary.  The paper examines the effect of marginalizing over a possibly unobserved background variable on the conditional relation between a response and an explanatory variable. In particular it is shown that some conclusions derived from least squares regression theory apply in general to testing independence for arbitrary distributions. It is also shown that the general condition of independence of the explanatory variable and the background ensures that mono- tonicity of dependence is preserved after marginalization. Relations with effect reversal and with collapsibility are sketched.     

Location and Scale Estimation with Correlation Coefficients

This article shows how to use any correlation coefficient to produce an estimate of location and scale. It is part of a broader system, called a correlation estimation system (CES), that uses correlation coefficients as the starting point for estimations. The method is illustrated using the well-known normal distribution. This article shows that any correlation coefficient can be used to fit a simple linear regression line to bivariate data and then the slope and intercept are estimates of standard deviation and location. Because a robust correlation will produce robust estimates, this CES can be recommended as a tool for everyday data analysis. Simulations indicate that the median with this method using a robust correlation coefficient appears to be nearly as efficient as the mean with good data and much better if there are a few errant data points. Hypothesis testing and confidence intervals are discussed for the scale parameter; both normal and Cauchy distributions are covered.     

Weighting methods for ties between event times and covariate change times

Recent work has shown that the presence of ties between an outcome event and the time that a binary covariate changes or jumps can lead to biased estimates of regression coefficients in the Cox proportional hazards model. One proposed solution is the Equally Weighted method. The coefficient estimate of the Equally Weighted method is defined to be the average of the coefficient estimates of the Jump Before Event method and the Jump After Event method, where these two methods assume that the jump always occurs before or after the event time, respectively. In previous work, the bootstrap method was used to estimate the standard error of the Equally Weighted coefficient estimate. However, the bootstrap approach was computationally intensive and resulted in overestimation. In this article, two new methods for the estimation of the Equally Weighted standard error are proposed. Three alternative methods for estimating both the regression coefficient and the corresponding standard error are also proposed. All the proposed methods are easy to implement. The five methods are investigated using a simulation study and are illustrated using two real datasets.