A Winner's Curse for Econometric Models: On the Joint Distribution of In-Sample Fit and Out-of-Sample Fit and its Implications for Model Selection

Abstract

We consider the case where a parameter, �; is estimated by maximizing a criterion function, Q(X;�). The estimate, ^ = ^(X); is then used to evaluate the criterion function with the same data, X, as well as with an independent data set, Y. The insample …t and out-of-sample …t relative to that of the true, or quasi-true, parameter, � � ; are de…ned by � = Q(X; ^) Q(X;� � ) and ~ = Q(Y; ^) Q(Y;� � ), respectively. We derive the joint limit distribution of (�;~) for a broad class of criterion functions and the