A Distribution-Free Test of Monotonicity with an Application to Auctions

This study develops a simple distribution-free test of monotonicity of conditional expectations. The test is based solely on ordinary least squares (OLS) and exploits the property between conditional expectation and projection; we prove that the monotonicity of a conditional expectation function restricts the sign of a corresponding projection coefficient. The estimated projection coefficient is used for a one-tailed t-test. The test -- which is notably simpler than other monotonicity tests -- is applied to bidding data from Japanese construction procurement auctions to empirically test first-price sealed bid auction models with independent private values (IPV), assuming the data are generated from a symmetric Bayesian Nash equilibrium. We regress the bid level on the number of bidders and use the estimated projection coefficient for testing. We find that the test results depend on public work categories.