PANEL DATA A N D EULER ETHAN LIGON EQUATIONS 1. INTRODUCTION Dynamic rational expectations models featuring agents w i t h addi- tively time-separable u t i l i t y functions typically yield some sort of mar­ tingale restriction which may be used for either testing the model or estimating its parameters. W h i l e different models may yield a variety of these sorts of restrictions, the one most commonly observed is the Euler equation. Martingale restrictions are exceptionally useful i n a t i m e series con­ text, since the martingale property delivers a very useful sort of i n ­ dependence. However, as noted by Chamberlain (1984), this indepen­ dence property does not extend to the analysis of panel data. Most panels have many agents observed over a small number of t i m e peri­ ods. For these data, the most natural asymptotic theory would hold the number of t i m e periods ( T ) fixed, while l e t t i n g the number of agents (JV) approach infinity. However, when working w i t h the Euler equation or similar restrictions, this procedure w i l l yield inconsistent estimators whenever there are i m p o r t a n t aggregate components to the innovations observed by each agent (Pakes 1994). This paper develops a characterization of estimators t h a t rely on N oo, but which hold T fixed. I n particular, we show t h a t l i m i t i n g estimator is a random variable, and show how to calculate its distribu­ t i o n when there are overidentifying restrictions. W h e n the distribution of the l i m i t i n g estimator is nondegenerate, the estimator cannot be consistent, but knowledge of this d i s t r i b u t i o n permits us to engage i n the usual sort of inference and hypothesis testing. 2. M O D E L Suppose t h a t we have a dataset of observations on N agents over the course of T periods. A n economic model implies t h a t Vu = f(x , it b ) + u , it Date: May, 1997.