"Simple Inference in First-Price Auctions" (with Cristián Hernández)
Abstract
In this paper, we propose a new estimator for the distribution of valuations in first-price auctions with independent private valuations which applies in parametric and nonparametric settings. A distinguishing feature of our estimator is that its distribution is well-approximated by a sequence of normal distributions with consistently estimable covariance matrices. Consequently, our method provides simple inference for a wide array of objects of interest, which includes—but is not limited to—forming confidence intervals for the density of valuations at a fixed point, the optimal reserve price, the expected revenue of the auction and quantiles of the distribution of valuations as well as providing uniform confidence bands for the density of valuations. In first-price auctions, it is well-known that the support of the bid distribution depends on the parameters of the valuation distribution, which violates the standard regularity conditions for the asymptotic normality of the maximum likelihood estimator. We circumvent this issue by applying a simple modification to a method of moments estimator, which uses moments derived from the likelihood function. Simulations suggest our estimator and inference procedure perform well as our confidence bands provide empirical coverage close to nominal size and the mean squared error of functions of interest using our estimator compare favorably against alternatives in the literature. We demonstrate the usefulness of our approach in an application to timber auctions conducted by the United States Forest Service. To illustrate the flexibility of our inference procedure, we show how policy makers can use our results to form confidence sets for the welfare-maximizing reserve price when non-revenue considerations enter the welfare function.
"Inference under Shape Restrictions" (with Joachim Freyberger)
Abstract
We propose a uniformly valid inference method for an unknown function or parameter vector satisfying certain shape restrictions. The method applies very generally, namely to a wide range of finite dimensional and nonparametric problems, such as regressions or instrumental variable estimation, to both kernel or series estimators, and to many different shape restrictions. A major application of our inference method is to construct uniform confidence bands for an unknown function of interest. Our confidence bands are asymptotically equivalent to standard unrestricted confidence bands if the true function strictly satisfies all shape restrictions, but they can be much smaller if some of the shape restrictions are binding or close to binding. We illustrate these sizable width gains as well as the wide applicability of our method in Monte Carlo simulations and in an empirical application.
"Inference under Shape Restrictions with Partial Identification"