Scientists 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.
