We propose a likelihood-based estimator for random coefficients discrete choice demand models that is applicable in a broad range of data settings. Intuitively, it combines the likelihoods of two mixed logit estimators—one for consumer level data, and one for product level data—with product level exogeneity restrictions. Our estimator is both efficient and conformant: its rates of convergence will be the fastest possible given the variation available in the data. The researcher does not need to pre-test or adjust the estimator and the inference procedure is valid across a wide variety of scenarios. Moreover, it can be tractably applied to large datasets. We illustrate the features of our estimator by comparing it to alternatives in the literature.

We introduce several new estimation methods that leverage shape constraints in auction models to estimate various objects of interest, including the distribution of a bidder’s valuations, the bidder’s ex ante expected surplus, and the seller’s counterfactual revenue. The basic approach applies broadly in that (unlike most of the literature) it works for a wide range of auction formats and allows for asymmetric bidders. Though our approach is not restrictive, we focus our analysis on first–price, sealed–bid auctions with independent private valuations. We highlight two nonparametric estimation strategies, one based on a least squares criterion and the other on a likelihood criterion. We establish several theoretical properties of our methods to guide empirical analysis and inference. In addition to providing the asymptotic distributions of our estimators, we identify ways in which methodological choices should be tailored to the objects of interest. For objects like the bidders’ ex ante surplus and the seller’s counterfactual expected revenue with an additional symmetric bidder, we show that our input–parameter–free estimators achieve the semiparametric efficiency bound. For objects like the bidders’ inverse strategy function, we provide an easily implementable boundary–corrected kernel smoothing and transformation method in order to ensure the squared error is integrable over the entire support of the valuations. An extensive simulation study illustrates our analytical results and demonstrates the respective advantages of our least–squares and maximum likelihood estimators in finite samples. Compared to estimation strategies based on kernel density estimation, the simulations indicate that the smoothed versions of our estimators enjoy a relatively large degree of robustness to the choice of an input parameter.

We estimate the density and its derivatives using a local polynomial approximation to the logarithm of an unknown density function 𝑓 . The estimator is guaranteed to be nonnegative and achieves the same optimal rate of convergence in the interior as on the boundary of the support of 𝑓 . The estimator is therefore well–suited to applications in which nonnegative density estimates are required, such as in semiparametric maximum likelihood estimation. In addition, we show that our estimator compares favorably with other kernel–based methods, both in terms of asymptotic performance and computational ease. Simulation results confirm that our method can perform similarly or better in finite samples compared to these alternative methods when they are used with optimal inputs, i.e. an Epanechnikov kernel and optimally chosen bandwidth sequence. We provide code in several languages.

We consider a situation in which we have data from ascending auctions with symmetric bidders, independent private values, and exogenous entry in which the bidders’ value distribution is partially identified. Focusing on the case in which the seller intends to use a second price auction, we discuss how to determine an optimal reserve price. We justify the use of maximum entropy, explore the properties of the estimand, determine the asymptotic properties of our maximum entropy estimator, evaluate its behavior in a simulation study, and demonstrate its use in a modest application. As an extension, we propose a maxmin decision rule with entropy regularization, which includes Aryal and Kim (2013) and the maximum entropy solution as extreme cases.