Chapter 1: This chapter looks at a dynamic panel data model with fixed effects. Estimating the model with GMM is consistent but suffers from small sample bias. We apply Helmert's transformation to the model, assume that error terms and nuisance parameters are homoskedastic and independent across observations and of one another, and utilize the GMM bias calculation of Newey & Smith (2001). This leads to a closed form expression for the GMM bias applied to AR(1) model. Chapter 2: This chapter develops specification tests for quantile regression under various data types. We consider what happens to the quantile regression estimator under local and global misspecification and design specification tests that handle a wide range of data types. We consider how to carry out such tests in practice and present Monte Carlo results to show the effectiveness of such tests. Chapter 3: Through a Taylor expansion, We compute the bias of a general GMM model where the weighting matrix A of the moment conditions g(z, [beta]) is left unspecified, except for some general conditions. Our bias results are compared to those of Newey and West (2003). An important case of GMM estimation with a general weighting matrix A is when A is a function of a vector of parameters with fixed dimension. Arellano's IVE estimator is an example of this type of estimator--we consider the bias properties of Arellano's IVE estimator in the AR(1) setting and compare them to our results from Chapter 1.