This dissertation consists of three chapters, each of which proposes methods to deal with the "many moments" problem in a different model. Chapter I develops shrinkage methods for solving the "many moments" problem in the context of instrumental variable estimation. The procedure can be understood as a two-step process of shrinking some of the coefficient estimates from the regression of the endogenous variables on the instruments, then using the predicted values of the endogenous variables (based on the shrunk coefficient estimates) as the instruments. The optimal shrinkage parameter, which minimizes the asymptotic mean square error, has a closed form that makes it easy to implement. A Monte Carlo study shows that the shrinkage methods work well. Chapter 2 proposes the shrinkage generalized method of moments (SGMM) estimator for the estimation of conditional moment restriction models, which extends the idea of the shrinkage methods proposed in the previous chapter. The SGMM estimator is obtained as the minimizer of the objective function of the GMM estimator modified by dividing the objective function into two parts, then shrinking the effect of the second part. As in the previous chapter, the optimal shrinkage parameter has a closed form that makes it easy to implement. In the simulation, the SGMM estimator always performs at least as well, and often outperforms, the conventional GMM estimator. Chapter 3 considers the problem of choosing which moment conditions to use in estimating dynamic panel data models. It derives the asymptotic mean squared error of the GMM estimator in an autoregressive model with fixed effects. It shows that additional instruments should be included on the basis of how well they approximate the fixed effects. Using this result, a procedure for choosing the number of instruments and a procedure for shrinking the effects of additional instruments are proposed. Both methods are based on minimization of the asymptotic mean squared error. A Monte Carlo study shows that applying the proposed procedures greatly improves the performance of the GMM estimator.
