Abstract: In econometrics, many parameters of interest can be written as ratios of expectations. The main approach to construct confidence intervals for such parameters is the delta method. However, this asymptotic procedure yields intervals that may not be relevant for small sample sizes or, more generally, in a sequence-of-model framework that allows the expectation in the denominator to decrease to 0 with the sample size. In this setting, we prove a generalization of the delta method for ratios of expectations and the consistency of the nonparametric percentile bootstrap. We also investigate finite-sample inference and show a partial impossibility result: nonasymptotic uniform confidence intervals can be built for ratios of expectations but not at every level. Based on this, we propose an easy-to-compute index to appraise the reliability of the intervals based on the delta method. Simulations and an application illustrate our results and the practical usefulness of our rule of thumb.

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In the introductory chapter, we compare views on estimation and inference in the econometric and statistical learning disciplines.

In the second chapter, our interest lies in a generic class of nonparametric instrumental models. We extend the estimation procedure in Otsu (2011) by adding a regularisation term to it. We prove the consistency of our estimator under Lebesgue’s L2 norm.

In the third chapter, we show that when observations are jointly exchangeable rather than independent and identically distributed (i.i.d), a modified version of the empirical process converges weakly towards a Gaussian process under the same conditions as in the i.i.d case. We obtain a similar result for a modified version of the bootstrapped empirical process. We apply our results to get the asymptotic normality of several nonlinear estimators and the validity of bootstrap-based inference. Finally, we revisit the empirical work of Santos Silva and Tenreyro (2006).

In the fourth chapter, we address the issue of conducting inference on ratios of expectations. We find that when the denominator tends to zero slowly enough when the number of observations n increases, bootstrap-based inference is asymptotically valid. Secondly, we complement an impossibility result of Dufour (1997) by showing that whenever n is finite it is possible to construct confidence intervals which are not pathological under some conditions on the denominator.

In the fifth chapter, we present a Stata command which implements estimators proposed in de Chaisemartin and D’Haultfoeuille (2018) to measure several types of treatment effects widely studied in practice.

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