This thesis is comprised of two parts: Part I It is common in empirical economic applications that use micro-data to exhibit a natural ordering into groups. Angrist (1991) use dummy variables based on such grouping to form instruments for consistent estimation. Khatoon et al. (2014) and Andrews et al. (2016) extend the GEL class of estimators to the case where moment conditions are specified on a group-by-group basis and refer to the resulting estimator as group-GEL. A natural consequence of basing instruments or moment conditions on groups is the degree of over-identification can increase significantly. Following Bekker (1994) it is recognized that inference based on conventional standard errors is incorrect in the presence of many instruments. Furthermore, when using many moment conditions, two-stage GMM is biased. Although the bias of Generalized empirical likelihood (GEL) is robust to the number of instruments, Newey and Windmeijer (2009) show that the conventional standard errors are too small. They propose an alternative variance estimator for GEL that is consistent under conventional and many-weak moment asymptotics. In this part of the thesis I demonstrate that for a particular specification of moment conditions, the group-GEL estimator is more efficient than GEL. I also extend the Newey and Windmeijer (2009) many-moment asymptotic framework to group-GEL. Simulation results demonstrate that group-GEL is robust to many moment conditions, and t-statistic rejection frequencies using the alternative variance estimator are much improved compared to using conventional standard errors. Part II Following the seminal paper of Abowd et al. (1999), Linked Employer-Employee datasets are commonly used in studies decomposing sources of wage variation into unobservable worker and firm effects. If it is assumed that the correlation between these worker and firm effects can be interpreted as a measure of sorting in labour markets, then an efficient matching process between workers and firms would result in a positive correlation. However, empirical evidence has failed to support this ascertain. As a possible answer to this apparent paradox, Andrews et al. (2008) show the estimation of the correlation is biased as a function of the amount of movement of workers between firms, so-called Limited Mobility Bias (LMB); furthermore they provide formula to correct this bias. However, due to computational restrictions, application of these corrections is infeasible, given the size of datasets typically used. In this part of the thesis I introduce an estimation technique to make the bias-correction estimators of Andrews et al. (2008) feasible. Monte Carlo experiments using the bias-corrected estimators demonstrate that LMB can be eliminated from datasets of comparable size to real data. Finally, I apply the bias-correction techniques to a linear model based on the Danish IDA, and find that correcting the correlation between the worker and firm effects due to LMB provides insufficient evidence to resolve the above paradox.