Estimation of Nonparametric Models with Simultaneity

使用SimultaneityRosa L.Matzkin估计非参数模型科学家们经常面临这样一种情况，即他们想要了解黑匣子中的工作原理。通常，他们可以观察到黑盒的输出，这些输出是从黑盒中的可观察输入生成的。当观测到的输出大小仅取决于观测到的输入大小，并且黑盒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，当变换是非线性的，并且输出的大小取决于不可观测输入的大小和可观测输入的大小时，问题变得困难得多。举个简单的例子，假设黑匣子是一个生产某些输出的典型个体。我们可以观察到个体工作的时数和产出量。然而，产出量不仅取决于工作时数，还取决于不可观测的努力和不可观测的能力。此外，可以预期这种关系是非线性的。作为一个稍微复杂一点的例子，考虑一个黑箱，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，还取决于不可观察到的生产率。价格和销售量是由需求和供给的交集决定的，每个需求和供给都依赖于可观测和不可观测的变量，本项目的目标是发展估计黑匣子工作的方法，当黑匣子的输出由几个未知的非线性函数的交集决定时，这些非线性函数既依赖于不可观测的输入，也依赖于不可观测的输入。在需求和供给的例子中，这将意味着估计需求和供给函数，当需求函数取决于消费者的收入和不可观测的品味，而供给函数取决于生产投入价格和不可观测的生产率时。这些方法都是非参数的。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。该方法也不需要为不可观测变量指定特定的分布函数。虽然所考虑的模型比含有附加不可观测变量的线性模型所作的假设要少得多，但新估计量的计算和统计性质与限制性大得多的模型相似。在一些第一阶段的非参数估计之后，通过矩阵求逆和乘法来计算最终的估计。新估计器的渐近分布是正态的，允许人们使用标准程序来计算可信区间。新估计器将应用于几种经验情况，例如估计不同支出分配下家庭的偏好分布，以及估计产品特征的偏好分布。

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.