Essays on empirical econometrics
Keywords: maximum entropy
partial identification
empirical processes
econometrics

The thesis consists of three main chapters.
Chapter 1 studies methods of identification, model selection, and heterogeneity analysis in discrete choice settings. The chapter considers a general class of revealed preference models and shows that, without further structure imposed on the model, inequality restrictions from revealed prefence allow only for the identification of a compact set of parameters. I show how the principle of maximum entropy can be applied in this setting to identify a unique vector of parameters for each agent, by choosing the solution which is maximally agnostic about missing information. For the purposes of model selection, I apply Selten's measure of predictive success to this setting and I suggest an alternative model selection method based on the Kullback-Leibler divergence. I propose multiple methods for analysing parameter heterogeneity.
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Chapter 2 focuses on the practical and computational aspects of implementing partial identification methods in practice. The aim of the chapter is to bridge the gap between econometric theory and its applications. The chapter focuses on binary-outcome models with random coefficients and endogeneity, and a set-identifying approach to this model developed by Chesher and Rosen (2014). To facilitate practical implementations of the model, I illustrate the mathematical results of Chesher and Rosen (2014) with simple graphical examples and I contrast their model with point-identifying alternatives. To show how the method can be implemented, I provide step-by step pseudo-code and Matlab code. I analyse the increases in the computational complexity of the model arising from increasing the cardinality of the instruments, the endogenous variables, and exogenous explanatory variables; and I discuss computational and practical issues arising from the inclusion of continuous variables. I analyse the empirical practicability of the method by applying it to data from the British Household Panel Survey in the context of analysing female labour supply decisions. The empirical application shows that insufficient structure imposed on the model may lead to unbounded sets, while small cell sizes may result in empty sets. Empirical results derived for a fixed value of the variance of the coefficient on the endogenous variable show that, under this restriction, the set of identified parameters appears bounded and that it has a non-trivial shape. The results also show that the volume of the set is reduced significantly when more instruments are used.
In Chapter 3, the focus shifts from partial identification, analysed in the previous two chapters, to the econometric theory of non-linear regressions. I derive asymptotic results for marked and weighted empirical processes of residuals from non-linear regressions with outlier detection. I show that, under a number of conditions imposed on the regressors, marks, weights, the non-linear function, and the distribution of the errors, the existing asymptotic theory developed for marked and weighted functions of residuals from linear models with outlier detection can be extended to non-linear models with outlier detection. This allows me to apply the theory developed for empirical processes of residuals from linear models, to non-linear models; and to show the asymptotic equivalence of the processes analysed in the chapter and marked and weighted empirical processes based on the true innovations, without estimation errors. My results do not depend on unverifiable assumptions. To illustrate this, I show through an example how each of the assumptions can be checked. I suggest several applications of the theory developed in the chapter.