We study moment inequality estimation of a general class of Random Utility Models where the unobserved components enter additively in payoffs but they are completely unrestricted in terms of their correlation across choice alternatives and individuals, and in the probability distribution. Under mild regularity conditions, the expectation of the unobserved payoff associated to the optimal choice is a positive and finite constant. We introduce an incidental parameter K>0 that is an upper bound to the value of this expectation, and we obtain moment inequalities that include the structural payoff parameters and the constant K. We provide conditions for point identification and for set identification of the structural parameters. Using an estimator within the class of Chernozhukov, Hong, and Tamer (2007), we propose a Cross-Validation method for the choice of the incidental parameter. We implement several Monte Carlo experiments to illustrate the trade-offs in the choice of K and the performance of our method. In all the experiments, the cross-validation function is convex in the parameter K: absolute bias declines and variance increases monotonically with K; for low values of K, the bias-reduction-effect of increasing this parameter is stronger than variance-increase-effect, and the opposite occurs for high values of K. We show that the estimation under the restriction that the expectation of the unobserved (optimal) payoff is not zero (i.e., estimation with K=0 ignoring selection term) implies substantial biases in the estimates of structural parameters. Furthermore, our cross-validation method implies substantially smaller mean square errors than the estimation that ignores the selection term. |