Estimation of Nonparametric Models with Simultaneity

科学家们经常遇到这样的情况：他们想要了解黑盒子里的工作原理。通常，他们可以观察黑箱的输出，这些输出是由黑箱的可观察输入生成的。当观测到的输出幅度仅取决于观测到的输入幅度，并且黑箱内的变换是线性的时，简单的数学方程可以帮助唯一地确定这种变换。然而，当转换是非线性的，并且输出的大小取决于不可观察输入的大小以及可观察输入的大小时，问题变得更加困难。作为一个简单的例子，假设黑盒是一个产生某些输出的典型个体。我们可以观察到个体工作的小时数和产出的数量。然而，产出数量不仅取决于工作时数，还取决于不可观察的努力和不可观察的能力。而且，这种关系可以预期是非线性的。举一个稍微复杂一点的例子，考虑一个黑盒，其中产品的总需求和总供给决定了观察到的价格和特定产品的总销售量。我们可以观察到生产投入的价格和消费者的收入水平，这在一定程度上决定了观察到的价格和数量。然而，对产品的需求不仅取决于消费者的收入，还取决于消费者对产品的品味。生产者将收取的价格不仅取决于生产投入的成本，还取决于不可观察的生产率。价格和销售数量是由需求和供给的交集决定的，每一个都取决于可观察和不可观察的变量。这个项目的目标是开发方法来估计黑箱的工作，当黑箱的输出是由几个未知的非线性函数的交集决定的，这些非线性函数依赖于不可观察的和可观察的输入。在需求和供给的例子中，当需求函数取决于消费者的收入和不可观察的品味，供给函数取决于生产投入价格和不可观察的生产率时，这将意味着估计需求和供给函数。这些方法是非参数的。换句话说，这些方法不需要为需求函数或供给函数指定线性或非线性形式。这些方法也不需要为不可观测变量指定特定的分布函数。尽管所考虑的模型比具有可加性不可观察变量的线性模型需要的假设少得多，但新估计器的计算和统计性质与那些更严格的模型相似。在进行第一阶段非参数估计后，通过矩阵反演和乘法计算最终估计量。新估计量的渐近分布是正态分布，允许使用标准程序来计算置信区间。新的估计方法将应用于几种经验情况，例如估计不同支出分配的家庭偏好分布，以及估计产品特征偏好的分布。

Estimation of Nonparametric Models with SimultaneityRosa L. MatzkinScientists are often confronted with a situation where they want to learn the workings inside a black box. Often, they can observe outputs from the black box that are generated from observable inputs into the black box. When the observed magnitude of the outputs depend only on the observed magnitude of the inputs, and the transformation within the black box is linear, simple mathematical equations can help to uniquely determine such transformation. However, when the transformation is nonlinear and the magnitude of the outputs depend on the magnitudes of unobservable inputs as well as on the magnitude of observable inputs, the problem becomes much more difficult. As a simple example, suppose that the black box is a typical individual producing some output. We can observe the quantity of hours the individual works and the output quantity. However, the output quantity will be determined not just by the quantity of hours worked but also by unobservable effort and unobservable ability. Moreover, the relationship can be expected to be nonlinear. As a slightly more complex example, consider a black box where aggregate demand and aggregate supply for a product determine the observed price and aggregate quantity sold of the particular product. We can observe the prices of the production inputs and the income level of the consumers, which partly determine the observed price and quantity. However, the demand for the product will depend not only on consumers' incomes but also on consumers' taste for the product. The price producers will charge will depend not only on the cost of the production inputs but also on unobservable productivity. Prices and quantities sold are determined by the intersection of demand and supply, each of which depend on observable and unobservable variables.The objective of this project is to develop methods to estimate the workings of the black box, when outputs of the black box are determined by the intersection of several unknown nonlinear functions, and these nonlinear functions depend on unobservable as well as observable inputs. In the demand and supply example, this would mean estimating the demand and the supply functions, when the demand function depends on consumer's income and unobservable tastes and the supply function depend on production input prices and unobservable productivity. The methods are nonparametric. In other words, the methods do not require specifying either a linear or a nonlinear form for either the demand or the supply functions. The methods also do not require specifying a particular distribution function for the unobservable variables. Although the models considered require making far less assumptions than linear models with additive unobservable variables, the computation and statistical properties of the new estimators is similar to those of the much more restrictive models. After some first stage nonparametric estimation, the final estimators are calculated by matrix inversion and multiplication. The asymptotic distribution of the new estimators is normal, allowing one to use standard procedures for calculating confidence intervals.The new estimators will be applied to several empirical situations, such as estimating the distribution of preferences across households for different expenditure allocations, and estimating the distribution of preferences for products characteristics.