# Christophe Gaillac

I am a PhD candidate in Economics at the Toulouse School of Economics under the supervision of Eric Gautier.

I am also a research fellow at CREST (Center for Research in Economics and Statistics). I was coordinator of statistical and mathematical teachings at ENSAE in 2017-2019.

My research focuses on **Econometrics, Statistics**, and **Machine Learning**.

I have also interests in **Labor Economics** and **Political Science**.

You can find my **CV** here.

You can also find the handout of the course *Machine learning for Econometrics *here.

**Contact: **christophe.gaillac@tse-fr.eu ; Phone: (+33)6 69 04 93 30

Toulouse School of Economics, 1, Esplanade de l'université**,** 31000 Toulouse, France

# Research

**WORKING PAPERS**

**Rationalizing Rational Expectations? Characterization and Tests**, with Xavier D’Haultfoeuille (CREST) and Arnaud Maurel (Duke university),

**Keywords:** Rational expectations, Test, Subjective expectations, Data combination.

**Summary:** In this paper, we build a new test of rational expectations based on the marginal distributions of realizations and subjective beliefs. This test is widely applicable, including in the common situation where realizations and beliefs are observed in two different datasets that cannot be matched. We show that whether one can rationalize rational expectations is equivalent to the distribution of realizations being a mean-preserving spread of the distribution of beliefs. The null hypothesis can then be rewritten as a system of many moment inequality and equality constraints, for which tests have been recently developed in the literature. The test is robust to measurement errors under some restrictions and can be extended to account for aggregate shocks. Finally, we apply our methodology to test for rational expectations about future earnings. While individuals tend to be right on average about their future earnings, our test strongly rejects rational expectations.

**R Package:** **RationalExp**** **and vignette, with Xavier D’Haultfoeuille (CREST) and Arnaud Maurel (Duke university). This package implements a test of the rational expectations hypothesis based on the marginal distributions of realizations and subjective beliefs. The package also computes the estimator of the minimal deviations from rational expectations than can be rationalized by the data. **R** and the package **RationalExp** are open-source software projects and can be freely downloaded from CRAN: http://cran.r-project.org

**Adaptive estimation in the linear random coefficients model when regressors have limited variation,**with Eric Gautier (TSE).

**Keywords**: Adaptation, Ill-posed Inverse Problem, Minimax, Random Coefficients.

**Summary: **We consider a linear model where the coefficients - intercept and slopes - are random with a distribution in a nonparametric class and independent from the regressors. The main drawback of this model is that identification usually requires the regressors to have a support which is the whole space. This is rarely satisfied in practice. Rather, in this paper, the regressors can have a support which is a proper subset. This is possible by assuming that the slopes do not have heavy tails. Lower bounds on the supremum risk for the estimation of the joint density of the random coefficients density are derived for this model and a related white noise model. We present an estimator, its rates of convergence, and a data-driven rule which delivers adaptive estimators.

** R Package:** **RandomCoefficients** and vignette, with Eric Gautier (TSE). This package implements the estimator proposed in Gaillac and Gautier (2019), which is based on Prolate Spheroidal Wave functions which are computed efficiently in RandomCoefficients based on Osipov, Rokhlin, and Xiao (2013). This package also provides a parallel implementation of the estimator.

**Estimates for the SVD of the truncated Fourier transform on L2(exp(b|×|)) and stable analytic continuation****,**with Eric Gautier (TSE).

**Keywords: **Analytic continuation, Nonbandlimited functions, Heavy tails, Uniform estimates, Extrapolation, Singular value decomposition, Truncated Fourier transform, Singular Sturm Liouville Equations, Superresolution.

**Summary:** The Fourier transform truncated on [-c,c] is usually analyzed when acting on L2(-1/b,1/b) and its right-singular vectors are the prolate spheroidal wave functions. This paper considers the operator acting on the larger space L2(exp(b|.|)) on which it remains injective. We give nonasymptotic upper and lower bounds on the singular values with similar qualitative behavior in m (the index), b, and c. The lower bounds are used to obtain rates of convergence for stable analytic continuation of possibly nonbandlimited functions whose Fourier transform belongs to L2(exp(b|.|)). We also derive bounds on the sup-norm of the singular functions. Finally, we propose a numerical method to compute the SVD and apply it to stable analytic continuation when the function is observed with error on an interval.

# Teaching

**Machine Learning for Econometrics, ENSAE Paris and Institut Polytechnique de Paris **(previously "High-Dimensional Econometrics")

*joint with **Jérémy L'hour** *(INSEE, CREST) and Bruno Crépon (CREST)

**You can now download a copy of the handout: **link

This course covers recent applications of high-dimensional statistics and machine learning to econometrics, including variable selection, inference with high-dimensional nuisance parameters in different settings, heterogeneity, networks and text data. The focus will be on policy evaluation problems. Recent advances in causal inference such as the synthetic controls method will be reviewed.

The goal of the course is to give insights about these new methods, their benefits and their limitations. It will mostly benefit students who are highly curious about recent advances in econometrics, whether they want to study theory or use them in applied work. Students are expected to be familiar with Econometrics 2 (2A) and Statistical Learning (3A).

In 2020, the outline was:

Introduction

High-Dimension, Variable Selection and Post-Selection Inference

Methodology: Using Machine Learning Tools in Econometrics

High-Dimension and Endogeneity

The Synthetic Control Method

Machine Learning Methods for Heterogeneous Treatment Effects

Prediction Policy Problems

Fairness and optimal treatment allocation

**Courses:**

Machine Learning for Econometrics (2020), ENSAE Paris, Master in Economics of Institut Polytechnique de Paris, with Jérémy L’Hour (CREST) and Bruno Crépon (CREST)

Machine Learning for Econometrics (2018, 2019), ENSAE Paris, Master in Economics of Institut Polytechnique de Paris, with Jérémy L’Hour (CREST)

Mathematics for Economists (Analysis and Optimisation) (2018), Master in Economics, Paris-Saclay university, Phd track

Mathematics for Economists (2017, 2018), Sciences-Po Paris, Phd track

Mathematics for Economists (2018), ENSAE Paris, Specialised Master

Algebra and Python (2018), HEC Paris and ENSAE Paris, Undergraduate.

**TA sessions:**

Statistics 1 (2017-2018), ENSAE Paris, Nicolas Chopin

Numerical Analysis (2016-2018), ENSAE Paris, Cristina Butucea

Econometrics 2, (2017-2018), ENSAE Paris, Xavier D’Haultfoeuille

Simulations and Monte-Carlo (2018), ENSAE Paris, Nicolas Chopin

Time Series analysis (2015-2017), ENSAE Paris, Christian Franck