英文标题：
《Orthogonal Polynomials for Seminonparametric Instrumental Variables
  Model》
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作者：
Yevgeniy Kovchegov, Nese Yildiz
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最新提交年份：
2014
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英文摘要：
  We develop an approach that resolves a {\\it polynomial basis problem} for a class of models with discrete endogenous covariate, and for a class of econometric models considered in the work of Newey and Powell (2003), where the endogenous covariate is continuous. Suppose $X$ is a $d$-dimensional endogenous random variable, $Z_1$ and $Z_2$ are the instrumental variables (vectors), and $Z=\\left(\\begin{array}{c}Z_1 \\\\Z_2\\end{array}\\right)$. Now, assume that the conditional distributions of $X$ given $Z$ satisfy the conditions sufficient for solving the identification problem as in Newey and Powell (2003) or as in Proposition 1.1 of the current paper. That is, for a function $\\pi(z)$ in the image space there is a.s. a unique function $g(x,z_1)$ in the domain space such that $$E[g(X,Z_1)~|~Z]=\\pi(Z) \\qquad Z-a.s.$$ In this paper, for a class of conditional distributions $X|Z$, we produce an orthogonal polynomial basis $Q_j(x,z_1)$ such that for a.e. $Z_1=z_1$, and for all $j \\in \\mathbb{Z}_+^d$, and a certain $\\mu(Z)$, $$P_j(\\mu(Z))=E[Q_j(X, Z_1)~|~Z ],$$ where $P_j$ is a polynomial of degree $j$. This is what we call solving the {\\it polynomial basis problem}.   Assuming the knowledge of $X|Z$ and an inference of $\\pi(z)$, our approach provides a natural way of estimating the structural function of interest $g(x,z_1)$. Our polynomial basis approach is naturally extended to Pearson-like and Ord-like families of distributions.
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中文摘要：
对于一类具有离散内生协变量的模型，以及Newey和Powell（2003）工作中考虑的一类经济计量模型，我们发展了一种解决{it多项式基问题}的方法，其中内生协变量是连续的。假设$X$是一个$d$维内生随机变量，$Z_1$和$Z_2$是工具变量（向量），并且$Z=\\left（\\begin{array}{c}Z_1\\\\Z_2\\end{array}\\right）$。现在，假设$X$给定$Z$的条件分布满足Newey和Powell（2003）或本论文命题1.1中所述的足以解决识别问题的条件。也就是说，对于图像空间中的函数$\\pi（z）$有一个a.s.在域空间中有一个唯一的函数$$g（x，z_1）$$E[g（x，z_1）~| ~z]=\\pi（z）qquad z-a.s.$。在本文中，对于一类条件分布$$x | z$，我们产生一个正交多项式基$$Q_j（x，z_1）$，这样对于a.E.$z_1=z_1$，对于所有的条件分布$$x |z{z}和}d$$，$$P_j（\\mu（Z））=E[Q_j（X，Z_1）~|~Z]，$$，其中，$P_j$是一个次为$j$的多项式。这就是我们所说的解决{\\it多项式基问题}。假设知道$X | Z$并推断出$\\pi（Z）$，我们的方法提供了一种估算利息$g（X，Z|1）$结构函数的自然方法。我们的多项式基方法自然地扩展到了类皮尔逊分布族和类Ord分布族。
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分类信息：

一级分类：Mathematics        数学
二级分类：Statistics Theory        统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Quantitative Finance        数量金融学
二级分类：Economics        经济学
分类描述：q-fin.EC is an alias for econ.GN. Economics, including micro and macro economics, international economics, theory of the firm, labor economics, and other economic topics outside finance
q-fin.ec是econ.gn的别名。经济学，包括微观和宏观经济学、国际经济学、企业理论、劳动经济学和其他金融以外的经济专题
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一级分类：Statistics        统计学
二级分类：Applications        应用程序
分类描述：Biology, Education, Epidemiology, Engineering, Environmental Sciences, Medical, Physical Sciences, Quality Control, Social Sciences
生物学，教育学，流行病学，工程学，环境科学，医学，物理科学，质量控制，社会科学
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一级分类：Statistics        统计学
二级分类：Statistics Theory        统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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