The thesis comprises four papers in theoretical econometrics: two papers on the subject of robust estimation and inference in moment condition models, one on semiparametric estimation in the presence of missing data, and one on survival analysis with competing risks data.

The first chapter considers estimation of moment condition models when some data are missing. The inverse probability tilting (IPT) estimator of Graham et al. [2] re-weights fully observed data to account appropriately for missingness. This paper considers a generalisation of the IPT estimator that allows for more flexible nuisance parameter estimation. It is shown that an IPT estimator with nonparametrically-estimated generated regressors retains some key asymptotic efficiency and robustness properties. A simulation study illustrates that these robustness properties allow IPT estimators to be insensitive to the choice of tuning parameter.

The second chapter concerns semiparametric moment condition models where the parameter of interest is described by one set of moment restrictions, while nuisance functions are identified from another set of moment restrictions. A two-step GEL-weighted estimator, a generalisation of Hellerstein and Imbens [3] and Bravo [1] to the semiparametric setting with estimated nuisance functions, is proposed that guarantees an efficiency gain arising from exploiting auxiliary moment restrictions that may involve nonparametric components. It is shown that in order to achieve this, moment restrictions generally need to be adjusted to account for first-stage nuisance estimation of nonparametric components. The theory is applied to a semiparametric missing data model where it is shown that the two-step GEL-weighted estimator possesses good efficiency and robustness properties when nuisance models are misspecified.

The third chapter represents my contribution on a project led by James Wason and Chien-Ju Lin of the MRC Biostatistics Unit. The paper focuses on time-to-event studies with several possible causes of event for each individual. When these competing risks are mutually dependent and only information on the time-to-first-event is available, marginal survival functions for each risk cannot be identified. Copula-Graphic estimators (Zheng and Klein [5]) exploit information on the dependence structure between risks to return consistent estimators. The paper derives asymptotic results for a class of parametric Copula-Graphic estimators, allowing for the construction of asymptotic confidence intervals for marginal survival functions. The performance of these confidence intervals is investigated in a simulation study.

The final chapter (co-authored with my supervisor, Richard Smith) considers methods to investigate the validity of over-identified moment restrictions when violations may occur only in small subgroups of the population. Hansen's J-test and likelihood-based variants aim to have non-trivial power against a wide range of alternatives, whereas power against particular forms of heterogeneity or parameter instability are often of concern. The paper addresses this issue by providing concentration inequalities designed to detect patterns of model misspecification. The associated bounds can be used to identify subsets of individual characteristics that are not consistent with the moment restrictions. These results are applied to show the consistency of goodness-of-fit statistics (Ramalho and Smith [4]) with data-dependent partitions.

References: [1] Bravo, Francesco, "Efficient M-estimators with auxiliary information", Journal of Statistical Planning and Inference 140, 11 (2010), pp. 3326--3342. [2] Graham, Bryan S. and Campos De Xavier Pinto, Cristine and Egel, Daniel, "Inverse probability tilting for moment condition models with missing data", Review of Economic Studies 79, 3 (2012), pp. 1053--1079. [3] Hellerstein, Judith K. and Imbens, Guido W., "Imposing Moment Restrictions from Auxiliary Data by Weighting", The Review of Economics and Statistics 81, 1 (1999), pp. 1--14. [4] Ramalho, Joaquim J S and Smith, Richard J., "Goodness of Fit Tests for Moment Condition Models" (2006). [5] Zheng, Ming and Klein, John P., "Estimates of marginal survival for dependent competing risks based on an assumed copula", Biometrika 82, 1 (1995), pp. 127--138.