Share equations in econometrics: A story of repression,trustration and dead ends

Share equations play an important role in applied economic research, notably in marketing and demand analysis. Both market shares and budget shares have been used as dependent variables in econometric models which were partly motivated by microeconomic theory. However attempts of econometricians (and other statisticians) to treat share equations adequately led mostly to unsatisfactory approaches: Some researchers although admitting that shares satisfy a sum constraint simply repressed the fact that shares cannot be normally distributed. Some researchers looked in vain for a stochastic specification which at the same time is consistent and allows a flexible covariance structure. Last not least almost nobody has properly taken care of additional problems arising from dynamic share models. The paper discusses these three issues and proposes a possible way out of this dilemma which was first suggested by Aitchison (1982) and has been applied to econometric demand analysis by Considine and Mount (1984). Demandtheoretic implications as well as methods of estimation are discussed. An example using German import data illustrates some of the results. Research for this paper is financially supported by Deutsche Forschungsgemeinschaft, SFB 178 “Internationalisierung der Wirtschaft” at the University of Konstanz. Some results reported in this paper have been achieved jointly with Karl Ringwald whose permission for using unpublished joint work is gratefully acknowledged. Critical remarks by an anonymous referee helped to improve the exposition of the paper.     

Most Likely Transformations

We propose and study properties of maximum likelihood estimators in the class of conditional transformation models. Based on a suitable explicit parameterization of the unconditional or conditional transformation function, we establish a cascade of increasingly complex transformation models that can be estimated, compared and analysed in the maximum likelihood framework. Models for the unconditional or conditional distribution function of any univariate response variable can be set up and estimated in the same theoretical and computational framework simply by choosing an appropriate transformation function and parameterization thereof. The ability to evaluate the distribution function directly allows us to estimate models based on the exact likelihood, especially in the presence of random censoring or truncation. For discrete and continuous responses, we establish the asymptotic normality of the proposed estimators. A reference software implementation of maximum likelihood‐based estimation for conditional transformation models that allows the same flexibility as the theory developed here was employed to illustrate the wide range of possible applications.     

Nonresponse Assessment of a Consumer Price Index

A household budget survey often suffers from a high nonresponse rate and a selective response. The bias that may be introduced in the estimation of budget shares because of this nonresponse can affect the estimate of a consumer price index, which is a weighted sum of partial price index numbers (weighted with the estimated budget shares). The bias is especially important when related to the standard error of the estimate. Because of the impossibility of subsampling nonrespondents to the budget survey, no exact information on the bias can be obtained. To evaluate the nonresponse bias, bounds for this bias are calculated using linear programming methods for several assumptions. The impact on a price index of a high nonresponse rate among people with a high income can also be assessed by using the elasticity with respect to total expenditure. Attention is also given to the possible nonresponse bias in a time series of price index numbers. The possible nonresponse bias is much larger than the standard error of the estimate.     

Estimation and Welfare Analysis With Large Demand Systems

We develop an approach for estimating individual or household level preferences for a large set of quality-differentiated goods and for constructing Hicksian welfare measures within the demand system framework. Our approach uses a maximum simulated likelihood procedure to recover estimates of the structural parameters and a multistage, Monte Carlo Markov chain algorithm for constructing Hicksian consumer surplus estimates. We illustrate our approach with a recreation dataset consisting of day trips to 62 Mid-Atlantic beaches.     

中美贸易摩擦、全球价值链分工与福利效应

基于全球价值链分工,本文利用2018年中美贸易摩擦加征关税清单、世界投入产出数据库(WIOD)和中美家庭消费支出调查等数据,使用Taylor一阶展开法和考虑消费异质性的Creedy方法测算了中美贸易摩擦给全球价值链参与者带来的福利效应。研究发现:第一,无论基于Taylor一阶展开法还是Creedy方法,本轮贸易摩擦给中美两国自身带来的福利损失最大,且美国受损程度明显高于中国;第二,全球价值链上加拿大、墨西哥等北美自由贸易区(NAFTA)成员以及日本、德国、英国等经济体受中美贸易摩擦影响的福利损失较大;第三,国别责任分解表明中美福利损失主要来源于自身加税行为,而其他绝大多数经济体的损失主要由美国加税引致;第四,收入分配方面,中美贸易摩擦引致的中国福利损失是累退性的,而美国福利受损程度与收入呈现非线性的“倒U型”关系,表明中美贸易摩擦不利于收入分配格局的改善。全球价值链各参与者应秉持人类命运共同体理念,积极主动维护多边贸易规则和国际治理秩序。     

收入风险、金融教育与家庭金融市场参与

运用中国城市居民消费金融调查数据,基于因果推断的分析框架,采用倾向分值匹配法研究了收入风险、金融教育对家庭金融市场参与的影响。研究发现,收入越稳定,家庭开展金融教育的概率越高;金融教育显著提高了家庭的金融市场参与;按收入风险将样本分组后,上述结论依然稳健,且金融教育对不同家庭金融市场参与的影响具有非对称性。在对家庭收入稳定性分类的基础上,有针对性地增加金融教育的供给,提升家庭整体的金融教育水平,有利于提高家庭金融市场参与度,优化家庭金融资产配置,进而改善家庭金融福利。     

农村居民收入差距适度性问卷调查

基于中国31个省份2 852份农村居民家庭问卷调查数据,运用Gini系数等指标,对中国农村居民收入差距现状进行统计测算;从农户家庭基本特征、物质资本、人力资本、政治资本、地理环境和地区差异、家庭人均年收入水平等方面对中国农村居民收入差距及其主要影响因素进行统计分析;对现阶段中国农村居民收入差距适度性相关问题进行问卷调查。研究结果表明:中国农村居民收入差距较大;农户人均年收入及收入差距由于农户家庭特征、人力资本、物质资本、政治资本不同、地理环境和地区差异而存在较大差异;现阶段农村居民收入差距已经超过了合理的界限;"农村劳动力缺少就业技能"和"缺乏农业技术与资金"是造成农民增收难的最主要原因;农村居民对收入差距扩大的承受力较弱,对当前的收入分配状况不满意,但对国家实施的三农政策比较满意;"社会保障"、"教育"和"公共基础设施"是对农村居民不公平的主要表现;"贫富差距过大"是阻碍经济效率的主要风险;"改善城乡二元结构"、"提升农村劳动力整体素质"和"构建城乡均衡的社会保障制度"是缩小城乡收入差距的主要途径。     

Inference for Income Distributions Using Grouped Data

We develop a general approach to estimation and inference for income distributions using grouped or aggregate data that are typically available in the form of population shares and class mean incomes, with unknown group bounds. We derive generic moment conditions and an optimal weight matrix that can be used for generalized method-of-moments (GMM) estimation of any parametric income distribution. Our derivation of the weight matrix and its inverse allows us to express the seemingly complex GMM objective function in a relatively simple form that facilitates estimation. We show that our proposed approach, which incorporates information on class means as well as population proportions, is more efficient than maximum likelihood estimation of the multinomial distribution, which uses only population proportions. In contrast to the earlier work of Chotikapanich, Griffiths, and Rao, and Chotikapanich, Griffiths, Rao, and Valencia, which did not specify a formal GMM framework, did not provide methodology for obtaining standard errors, and restricted the analysis to the beta-2 distribution, we provide standard errors for estimated parameters and relevant functions of them, such as inequality and poverty measures, and we provide methodology for all distributions. A test statistic for testing the adequacy of a distribution is proposed. Using eight countries/regions for the year 2005, we show how the methodology can be applied to estimate the parameters of the generalized beta distribution of the second kind (GB2), and its special-case distributions, the beta-2, Singh–Maddala, Dagum, generalized gamma, and lognormal distributions. We test the adequacy of each distribution and compare predicted and actual income shares, where the number of groups used for prediction can differ from the number used in estimation. Estimates and standard errors for inequality and poverty measures are provided. Supplementary materials for this article are available online.     

Household Consumption Equivalence Scales: Some Estimates from New Zealand Household Expenditure and Income Survey Data

This paper uses data on expenditures and incomes of New Zealand households of different demographic profiles to construct equivalence scales. The scales are useful in estimating the relative levels of spending required by the households to attain a given level of utility. Preference consistent 'complete demand systems' are analyzed to test for the demographic effects on the consumption patterns of households. Equivalence scales for respect to specific items of consumption as well as total consumption are worked out from the observed consumption behaviour of households. The results are largely in line with those found in existing similar studies both in Australia and New Zealand. To the best of our knowledge, this is the first attempt at constructing equivalence scales with New Zealand household budget data in the framework of hypothesized utility maximizing behaviour of households.     

Empirical bayes estimation of a distribution function with a gamma process prior

A sequence of empirical Bayes estimators is given for estimating a distribution function. It is shown that ‘i’ this sequence is asymptotically optimum relative to a Gamma process prior, ‘ii’ the overall expected loss approaches the minimum Bayes risk at a rate of n , and ‘iii’ the estimators form a sequence of proper distribution functions. Finally, the numerical example presented by Susarla and Van Ryzin ‘Ann. Statist., 6, 1978’ reworked by Phadia ‘Ann. Statist., 1, 1980, to appear’ has been analyzed and the results are compared to the numerical results by Phadia     

Modelling skewness and kurtosis with the BCPE density in GAMLSS

This paper illustrates the power of modern statistical modelling in understanding processes characterised by data that are skewed and have heavy tails. Our particular substantive problem concerns film box-office revenues. We are able to show that traditional modelling techniques based on the Pareto–Levy–Mandelbrot distribution led to what is actually a poorly supported conclusion that these data have infinite variance. This in turn led to the dominant paradigm of the movie business that ‘nobody knows anything’ and hence that box-office revenues cannot be predicted. Using the Box–Cox power exponential distribution within the generalized additive models for location, scale and shape framework, we are able to model box-office revenues and develop probabilistic statements about revenues.     

Empirical Likelihood-Based Inference for Poverty Measures with Relative Poverty Lines

In this article, we propose an empirical likelihood-based method of inference for decomposable poverty measures utilizing poverty lines which are some fraction of the median of the underlying income distribution. Specifically, we focus on making poverty comparisons between two subgroups of the population which share the same poverty line. Our proposed method is assessed using a Monte Carlo simulation and is applied to some Canadian household income data.     

Distributional properties of a model for the spread or drug abuse

We examine the properties of the distribution of the number of drug abusers in a previously free community assuming that initiators enter the community during a ‘latent’ period in which they randomly infect other members of the community, and that during the subsequent ‘control’ period the spread of abuse follows a linear birth and death process. The form of distribution is shown to be unaltered by a series of steady changes in the birth and death parameters. The distribution can be regarded as the convolution of three distributions:pseudo-binomial or binomial, negative binomial, and Polya-Aeppli. Special cases include the Laguerre series distribution.     

公共预算转移支付反贫困瞄准：以低保为例的ROC方法分析

本文使用中国家庭动态跟踪调查（CFPS）数据，基于代理工具测试模型（Proxy means testing,PMT）并结合ROC曲线方法（Receiver Operating Characteristics）研究低保的反贫困瞄准问题。结果显示：城市单个贫困指数以卫生间类型、电脑拥有情况来衡量较为有效，农村单个贫困指数则以户主年龄、冰箱拥有情况来衡量较为有效。更改贫困概率门槛值会影响公共预算转移支付贫困瞄准结果，对于一个较低的贫困概率门槛值，其对应的覆盖率（公共预算转移支付覆盖贫困人口）和漏损率（非贫困人口被纳入公共预算转移支付）都比较高。当政策制定者把对覆盖贫困人口和排除非贫困人口的目标赋予同等权重时，城乡贫困概率门槛值约为0.5左右。当贫困率较低且使用与贫困率相同的瞄准率时，基于PMT模型的贫困瞄准较差。在贫困率给定的条件下，随着受益比率（包含率）的增加，贫困瞄准的精确性在提高。使用全覆盖所需预算的百分比下降时，覆盖率和漏损率也都呈现下降态势，贫困率逐渐上升，预算中做覆盖之用的比例上升，而预算中漏损部分的比例下降。贫困线的变动会影响覆盖率、漏损率。     

最低工资政策的城镇家庭收入分配效应研究

家庭在福利分析中具有重要地位，但家庭收入如何受最低工资政策影响却很少得到关注。本文采用2005-2015年中国综合社会调查（CGSS）数据，利用多时期双重差分和再中心化影响函数回归方法实证考察了最低工资政策对我国城镇家庭收入及其分布的影响。研究发现：①最低工资标准上涨显著提升了处于贫困标准1~2倍家庭的收入水平，对贫困标准以下或高于贫困标准2倍以上的家庭收入没有影响；②最低工资标准提升可有效减缓城镇家庭收入不平等，尤其对家庭收入分布较低分位部分的不平等程度有明显改善；③最低工资政策对家庭收入的最终影响取决于工资溢出效应和就业挤出效应的比较，且其对不同收入等级家庭中成员的收入和就业影响存在明显的异质性。政府在制定最低工资政策时，应针对低收入女性劳动力出台相应的配套措施，以保证各类低收入家庭均能从最低工资标准提升中受益。     

Statistical inference for the quintile share ratio

In recent years, the Quintile Share Ratio (or QSR) has become a very popular measure of inequality. In 2001, the European Council decided that income inequality in European Union member states should be described using two indicators: the Gini Index and the QSR. The QSR is generally defined as the ratio of the total income earned by the richest 20% of the population relative to that earned by the poorest 20%. Thus, it can be expressed using quantile shares, where a quantile share is the share of total income earned by all of the units up to a given quantile. The aim of this paper is to propose an improved methodology for the estimation and variance estimation of the QSR in a complex sampling design framework. Because the QSR is a non-linear function of interest, the estimation of its sampling variance requires advanced methodology. Moreover, a non-trivial obstacle in the estimation of quantile shares in finite populations is the non-unique definition of a quantile. Thus, two different conceptions of the quantile share are presented in the paper, leading us to two different estimators of the QSR. Regarding variance estimation, [Osier, 2006] and [Osier, 2009] proposed a variance estimator based on linearization techniques. However, his method involves Gaussian kernel smoothing of cumulative distribution functions. Our approach, also based on linearization, shows that no smoothing is needed. The construction of confidence intervals is discussed and a proposition is made to account for the skewness of the sampling distribution of the QSR. Finally, simulation studies are run to assess the relevance of our theoretical results.     

劳动份额“ 型”演进规律

 要素分配是收入分配研究的基础和逻辑起点,具有重要的理论意义与政策价值。劳动份额演进特征,集中反映收入分配随经济发展的内在关系,是要素分配研究的核心议题。文章基于国际与历史比较视角,挖掘工业革命以来主要发达国家要素分配数据,发现劳动份额具有“ 型”演进规律,“水平型”和“U型”规律均可视为其阶段性特例。以此为经验标准,审视我国要素分配结构变化态势,剖析其对一般演进规律的偏离,并探讨其政策涵义。     

Outlier detection for multivariate skew-normal data: a comparative study

A general way of detecting multivariate outliers involves using robust depth functions, or, equivalently, the corresponding ‘outlyingness’ functions; the more outlying an observation, the more extreme (less deep) it is in the data cloud and thus potentially an outlier. Most outlier detection studies in the literature assume that the underlying distribution is multivariate normal. This paper deals with the case of multivariate skewed data, specifically when the data follow the multivariate skew-normal [1] distribution. We compare the outlier detection capabilities of four robust outlier detection methods through their outlyingness functions in a simulation study. Two scenarios are considered for the occurrence of outliers: ‘the cluster’ and ‘the radial’. Conclusions and recommendations are offered for each scenario.     

Conditional interval estimation of the mean following rejection of a two sided test

The effect of rejecting a two-sided preliminary test of significance for the mean of a normal distribution upon subsequent interval estimation of the mean is examined. For the case where the variance is known, conditional confidence intervals may be shorter than unconditional intervals, in contrast to the one-sided preliminary test case examined by Meeks and D’Agostino (1983, The American Statistician, 7, 134-136) . For the case where the variance is unknown and must be estimated by the sample variance, it is shown that customary intervals do not offer uniformly greater or lesser coverage than the nominal level.     

Simulating Media Exposure Distributions

We examine three media exposure distribution (e.d.) simulation methods. The first is based on the maximum likelihood estimate of an individual's exposure, the second on ‘personal probability’ (Greene 1970) and the third on a dependent Bernoulli trials model (Klotz 1973). The last method uses population exposure probabilities rather than individual exposure probabilities, thereby markedly reducing computation time. Magazine exposure data are used to compare the accuracy and computation times of the simulation methods with a log–linear e.d. model (Danaher 1988b) and the popular Metheringham (1964) model based on the beta–binomial distribution (BBD). The results show that the simulation methods are not as accurate as the log– linear model but are more accurate than Metheringham's model, However, all the simulation methods take less computation time than the log–linear model for schedules with more than six magazines, making them viable competitors for large schedule sizes