Simple inference in multinomial and ordered logit

This paper provides some simple methods of interpreting the coefficients in multinomial logit and ordered logit models. These methods are summarized in Propositions concerning the magnitudes, signs, and patterns of partial derivatives of the outcome probabilities with respect to the exogenousvariables. The paper also provides an empirical example illustrating the use of these Propositions.     

Discrimination between two binary data models: sequentially designed experiments

In this paper we propose a sequential procedure to design optimum experiments for discriminating between two binary data models. For the problem to be fully specified, not only the mode1link functions should be provided but also their associated linear predictor structures. Further, we suppose that one of the models is true, albeit it is not known which of them. Under these assumptions the procedure consists of making sequential choices of single experimental units to discriminate between the rival models as efficiently as possible. Depending on whether the models are nested or not, alternative methods are proposed.

To illustrate the procedure, a simulation study for the classical case of pro bit versus logit model is presented. It enables us to estimate the total sample sizes required to gain a certain power of discrimination and compare them to sample sizes for methods that were previously suggested in the literature.     

A Polynomial Logistic Distribution and its Applications in Finance

The importance of Logistic distribution has been widely recognized in many applied areas such as, demography, population studies, finance, agriculture, etc. Since its introduction as a model, much attention has been paid to the study of several generalizations of it, which would offer additional flexibility when data fitting is chased. In the present paper we introduce and develop a natural generalization of the Logistic distribution by considering a probability model whose logit cumulative distribution function transformation is of polynomial type. The performance of the model's fitting to financial data, using different parameter estimation methods, is also investigated.     

Probit and Logit Model Selection

Monte Carlo experiments are conducted to compare the Bayesian and sample theory model selection criteria in choosing the univariate probit and logit models. We use five criteria: the deviance information criterion (DIC), predictive deviance information criterion (PDIC), Akaike information criterion (AIC), weighted, and unweighted sums of squared errors. The first two criteria are Bayesian while the others are sample theory criteria. The results show that if data are balanced none of the model selection criteria considered in this article can distinguish the probit and logit models. If data are unbalanced and the sample size is large the DIC and AIC choose the correct models better than the other criteria. We show that if unbalanced binary data are generated by a leptokurtic distribution the logit model is preferred over the probit model. The probit model is preferred if unbalanced data are generated by a platykurtic distribution. We apply the model selection criteria to the probit and logit models that link the ups and downs of the returns on S&P500 to the crude oil price.     

A Note on Confidence Bands for the Logistic Response Curve

A problem in logit analysis is the interval estimation of the logistic response curve. Scheffé's method is used to obtain confidence bands for the logistic response function for any number of explanatory variables. This method is computationally easier and more general than a previously reported method.     

The effects of sampling strategies on the small sample properties of the logit estimator

Empirical researchers face a trade-off between the lower resource costs associated with smaller samples and the increased confidence in the results gained from larger samples. Choice of sampling strategy is one tool researchers can use to reduce costs yet still attain desired confidence levels. This study uses Monte Carlo simulation to examine the impact of nine sampling strategies on the finite sample performance of the maximum likelihood logit estimator. The results show stratified random sampling with balanced strata sizes and a bias correction for choice-based sampling outperforms all other sampling strategies with respect to four small-sample performance measures.     

Book Reviews

Aggregate forecasts using McFadden's conditional logit model of discrete choice harbor the unrealistic implicit assumption of a random-utility distribution that is homogeneous across both alternatives and individuals. This article relaxes that assumption. A choice model is developed that describes the random-utility component as a probability-mixture model. Some numerical results illustrate that the derived model is not constrained by the independence-of-irrelevant-alternatives property. An experimental test of visual perceptions demonstrates the potential superiority of the model.     

Overview of multinomial models for ordinal data

The purpose of this paper is to relate a number of multinomial models currently in use for ordinal response data in a unified manner. By studying generalized logit models, proportional generalized odds ratio models and proportional generalized hazard models under different parameterizations, we conclude that there are only four different models and they can be specified genericaUy in a uniform way. These four models all possess the same stochastic ordering property and we compare them graphically in a simple case. Data from the NHLBI TYPE II study (Brensike et al (1984)) is used to illustrate these models. We show that the BMDP programs LE and PR can be employed in computing maximum likelihood estimators for these four models.     

An Empirical Investigation of the “Dynamic McFadden” Model of Purchase Timing and Brand Choice: Implications for Market Structure

Purchase timing and brand-choice decisions of households are jointly investigated using the “dynamic McFadden” model of Heckman and Singer. The hazard of brand purchase is decomposed into the category purchase hazard and the probability of brand choice conditional on category purchase. The former is modeled using the hazard-function approach and the latter using a logit model. Unobserved heterogeneity in brand preferences, marketing effects, and baseline hazard parameters is accounted for in the empirical analysis. The distribution of preference heterogeneity identifies the locations of brands in multiattribute perceptual space and the distribution of attribute importance weights across households.     

带异质线性趋势的动态二元选择面板模型估计

本文提出了带异质线性趋势的动态二元面板模型的极大似然偏误纠正估计量和近似条件Logit估计量。我们给出了通常极大似然估计量偏误的解析形式,并提供了相应的估计方法。小样本实验表明近似条件似然函数可以很好的消除异质性参数的影响,而偏误纠正估计量可以显著的修正极大似然估计量的偏误。最后我们将本文提出的方法应用到现金红利支付模型。     

中国出口产品质量提高了吗

本文基于Khandelwal(2010)的出口产品质量测度思路及Berry（1994）的嵌套Logit模型,构建了测度中国出口产品质量的理论框架。测度结果表明：我国制造业总体的质量水平呈缓慢上升趋势,但各行业表现出不同的特征。在此基础上,对我国出口产品质量决定因素进行了多维度的经验考察,得到了以下基本结论：我国人均国民收入较高时,它与出口产品质量显著正相关;随着一般贸易份额的增加,其对出口质量的提升已超过了加工贸易;企业研发对劳动密集型出口产品质量贡献最大,对资本密集型产品质量贡献最小;人力资本对出口产品质量的贡献刚好跟国内研发相反;外资出资方式对出口产品质量影响与其采取的贸易方式有关。另外,企业中间品投入、资本强度及政府的研发补贴都会对出口质量产生影响。据此,本文认为在引资导向上应逐步引导外资与一般贸易方式的结合力度;适度加大对传统劳动密集型行业的研发力度是促进产业结构优化升级的重要途径;需加强企业中间投入品的投入,优化要素投入比例,积极培育高质量、适合各行业实际发展需要的人力资本。     

The Effectiveness of Risk Scores: the Logit Rank Plot

A risk score s for event E is a function of covariates with the property that P ( E | s ) is an increasing function of s . Motivated by applications in medicine and in criminology, we suggest the logit rank plot as a good way of summarizing the effectiveness of such a score. Explicitly, plot logit{ P ( E | s )} against logit( r ), where r is the proportional rank of s in a sample or population. The slope of this plot gives an overall measure of effectiveness, and the logit rank transformation provides a common basis on which different risk scores can be compared. Some practical and theoretical aspects are discussed.     

Thurstone scaling via binary response regression

Thurstone scaling is a widely used tool in marketing research, as well as in areas of applied psychology. The positions of the compared items, or stimuli on a Thurstone scale are estimated by averaging the quantiles corresponding to frequencies of each stimulus’s preference over the other stimuli. We consider maximum likelihood estimation for Thurstone scaling that utilizes paired comparison data. From this perspective we obtain a binary response regression with a probit or logit link. In addition to the levels on a psychological scale, the suggested approach produces standard errors, t-statistics, and other characteristics of regression quality. This approach can help in both the theoretical interpretation and the practical application of Thurstone modeling.     

Estimation and specification testing in female labor participation models: parametric and semiparametric methods

Female labor participation models have been usually studied through probit and logit specifications. Little attention has been paid to verify the assumptions that are used in these sort of models, basically distributional assumptions and homoskedasticity. In this paper we apply semiparametirc methods in order to test the previous hypothesis. We also estimate a Spanish female labor participation model using both parametric and semiparametirc approaches. The parametirc model includes fixed and random coefficients probit specification. The estimation procedures are parametric maximum likelihood for both probit and logit models, and semiparametric quasi maximum likelihood following Klein and Spady (1993). The results depend cricially in the assumed model.     

A Note On Beta Inverse-Weibull Distribution

The inverse Weibull distribution is one of the widely applied distribution for problems in reliability theory. In this article, we introduce a generalization—referred to as the Beta Inverse-Weibull distribution—generated from the logit of a beta random variable. We provide a comprehensive treatment of the mathematical properties of the Beta Inverse-Weibull distribution. The shapes of the corresponding probability density function and the hazard rate function have been obtained and graphical illustrations have been given. The distribution is found to be unimodal. Results for the non central moments are obtained. The relationship between the parameters and the mean, variance, skewness, and kurtosis are provided. The method of maximum likelihood is proposed for estimating the model parameters. We hope that this generalization will attract wider applicability to the problems in reliability theory and mechanical engineering.     

基于MixedLogit模型的道路交通统计生命价值评价研究

道路交通统计生命价值是道路交通安全项目经济评价的重要指标。运用意愿选择法和正交试验法设计了出行路径选择的调查问卷。假定时间、死亡风险为常数,费用服从对数正态分布,建立了基于Mixed Logit模型的统计生命价值评价模型。以大连市私家车出行者为调查对象获得调查数据,利用Monte Carlo方法并借助GAUSS软件对模型进行了150次仿真实验。研究表明：模型各参数的估计值具有较强集中性,t检验值具有较强显著性,模型优度比较高。统计生命价值服从参数为0．922和0．814的对数正态分布,其数学期望是35万元,可以作为道路交通安全项目经济评价的参考数据。     

随机效应半参数logit模型的惩罚似然估计研究

 在颇具争议的收入差距和健康关系研究中,为了降低可能存在的模型设定和遗漏变量偏误,本文提出了随机效应半参数logit模型,其中非参数的设定还可用于数据的初探性分析。随后本文提出了模型非参数和参数部分的估计方法。这里涉及的难点是随机效应的存在导致似然函数中的积分没有解析式,而非参数的存在更加大了估计难度。本文基于惩罚样条非参数估计方法和四阶Laplace近似方法建立了惩罚对数似然函数,其最大化采用了Newton_Raphson近似方法。文章还建立了惩罚样条中重要光滑参数的选取准则。模型在收入差距和健康实例中的估计结果表明数据支持收入差距弱假说,且非参数估计结果表明其具有U型形式,与实例估计结果的比较指出本文提出的估计方法是较准确的。     

Locally D-Optimal Design for a Logit Model in Discrete Choice Experiment

Conjoint analysis is concerned with understanding how people make choice between products or services (alternatives) or a combination of product and service (choice set), so that businesses can design new products or services that better meet customers needs. In this situation, logit model (Multinomial Logit Model) has been used to calculate the probability related to choosing an alternative in a choice set with the highest utility. Then I considered several choice sets instead of one. In this article, I have used the locally D-optimal design for the combination of the level of attributes (two attributes each with two levels) to create alternatives. The optimal combination of alternatives in choice sets which help us to have a suitable choice.     

Statistical inference for the availability of repairable system with general repair,reboot delay,and switching railure

We studied the inferences of an availability system with reboot delay and standby switching failures in which the system consisted of two operating units and one warm standby. The system was studied under the assumption that the time-to-failure and the time-to-repair were assumed to follow an exponential and a general distribution. The reboot times are assumed to be exponentially distributed with parameter β. We constructed a consistent and asymptotically normal estimator of availability for such a repairable system. Based on this estimator, interval estimation and testing hypothesis were developed by using logit transformation. To implement the simulation inference for the system availability, we adopted two repair-time distributions—namely, lognormal and Weibull; and three types of Weibull distributions—characterized by their shape parameters—were considered. Finally, appropriate tables and figures of all simulation results have been included.     

Geometric approach to the skewed normal distribution

The representations of the skewed normal distribution given in Propositions 1–4 in Genton (Ed., 2004) are considered here from a unified geometric point of view and are, based upon this, generalized in two respects. On the one hand, the four concrete representations motivate us for a unified and much more general algebraic–geometric representation of the skewed normal distribution (Theorems 1 and 2 as well as Remarks 2 and 3); on the other hand, the mentioned representations are generalized to the elliptically contoured case (Propositions and Corollaries 1c–4c).