Tang, Xun (2011), "Bounds on revenue distributions in counterfactual auctions with reserve prices." The RAND Journal of Economics, 42: 175–203.

Measuring auction revenues under counterfactual reserve prices or formats requires knowledge of distributions of bidders' values and private signals. This poses a challenge when bids are observed from first-price, common-value auctions. I bound counterfactual revenue distributions without imposing parametric restrictions on the model structure. Using data from U.S. municipal bond auctions, I find first-price and second-price auctions under optimal reserve prices lead to little improvement in revenues over existing first-price formats. The number of bidders has a more significant impact on revenues in optimal auctions. I also find invoking an incorrect assumption of private values in counterfactual analyses results in small errors in predicting revenues from optimal second-price auctions.