Econometrics, Quantitative Economics, Data Science

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Vector quantiles


In dimension one, the quantile is the inverse of the cumulative distribution function. Quantiles are extremely useful objects because they fully characterize the distribution of an outcome, and they allow to provide directly a number of statistics of interest such as the median, the extremes, the deciles, etc. They allow to express maximal dependence between two random variables (comonotonicity) and are used in decision theory (rank-dependent expected utility); in finance (value-at-risk and Tail VaR); in microeconomic theory (efficient risksharing); in macroeconomics (inequality); in biometric (growth charts) among other disciplines. Quantile regression, pioneered by Roger Koenker, allows to model the dependence of a outcome with respect to a set of explanatory variables in a very flexible way, and has become extremely popular in econometrics.
The classical definition of quantiles based on the cumulative distribution function, however, does not lend itself well to a multivariate extension, and given the number of applications of the notion of quantiles, many authors have suggested various proposals.
We have proposed a novel definition of multivariate quantiles called “Vector quantiles” based on optimal transport. The idea is that instead of viewing the quantile map as the inverse of the cumulative distribution function, it is more fruitful to view it as the map that rescales a distribution of interest to the uniform distribution over [0,1] in the least possible distortive way, in the sense that the average squared distance between an outcome and its preimage by the map should be minimized. While equivalent to the classical definition in the univariate case, this definition lends itself to a natural multivariate generalization using Monge-Kantorovich theory. We thus define the multivariate quantile map as the one that attains the L2-Wasserstein distance, which is known in the optimal transport literature as Brenier’s map. With Carlier and Santambrogio, we gave a precise connection between vector quantiles and a celebrated earlier proposal by Rosenblatt. We have shown that many of the desirable properties and uses of univariate quantiles extend to the multivariate case if using our definition. With Ekeland and Henry, we have shown that this notion is a multivariate analog of the notion of Tail VaR used in finance. With Henry, we have shown that it allows to construct an extension of Yaari’s rank-dependent utility in decision theory, and have provided an axiomatization for it. With Carlier and Dana, we have shown that this notion extends Landsberger and Meilijson’s celebrated characterization of efficient risksharing arrangements. With Charpentier and Henry, we have shown the connection with Machina’s theory of local utility. With Carlier and Chernozhukov, we have shown that Koenker’s quantile regression can be naturally extended to the case when the dependent variable is multivariate if one adopts our definition of multivariate quantile. With Chernozhukov, Hallin and Henry we have defined empirical vector quantiles and have studied their consistency.
A number of challenges remains. Among these, an empirical process theory for multivariate quantiles (extending the univariate theory of the empirical quantile process) is the obvious next step, although this is highly non-trivial. Invariance issues are also interesting and challenging questions. Finally, while the link with other multivariate quantiles (such as Rosenblatt’s) is by now well understood, the link with other related concepts, such as Tukey’s halfspace depth, remains to be explored.
A brief description of vector quantiles and vector quantile regression can be found in my book, Optimal transport methods in economics, chap. 9.4-9.5.


My co-authors:

Guillaume Carlier, Arthur Charpentier, Victor Chernozhukov, Rose-Anne Dana, Ivar Ekeland, Marc Hallin, Marc Henry, and Filippo Santambrogio.


Presentation slides:

A presentation on vector quantile regression can be found here.



The code for vector quantile regression can be found in the following Github repository.



Guillaume Carlier, Alfred Galichon, and Filippo Santambrogio (2010). From Knothe’s transport to Brenier’s map. SIAM Journal on Mathematical Analysis 41, Issue 6, pp. 2554-2576. Available here.
Ivar Ekeland, Alfred Galichon, and Marc Henry (2012). Comonotonic measures of multivariate risks. Mathematical Finance 22 (1), pp. 109-132. Available here.
Alfred Galichon and Marc Henry (2012). Dual theory of choice with multivariate risks. Journal of Economic Theory 147(4), pp. 1501–1516. Available here.
Guillaume Carlier, Rose-Anne Dana, and Alfred Galichon (2012). Pareto efficiency for the concave order and multivariate comonotonicity. Journal of Economic Theory 147(1), pp. 207–229. Available here.
Arthur Charpentier, Alfred Galichon, and Marc Henry (2016). Local utility and risk aversion. Mathematics of Operations Research 41(2), pp. 466—476.
Guillaume Carlier, Victor Chernozhukov, and Alfred Galichon (2016). Vector quantile regression: an optimal transport approach.Annals of Statistics 44 (3), pp. 1165–1192. Available here. Software available here.
Victor Chernozhukov, Alfred Galichon, Marc Hallin, and Marc Henry (2016). Monge-Kantorovich Depth, Quantiles, Ranks and Signs. Annals of Statistics. Available here.
Guillaume Carlier, Victor Chernozhukov, and Alfred Galichon (2017). Vector quantile regression beyond correct specification. Journal of Multivariate Analysis. Available here.


The mass transport approach to demand inversion in multinomial choice models



Multinomial choice models constitute a fundamental toolbox of microeconomic analysis. Although this classification is a bit arbitrary, they usually divide into discrete choice models, in which the choice set is finite (e.g. a consuming choosing a model of car), and hedonic models, in which the choice set is continuous (e.g. a consumer choosing the quality of a wine). An important problem in these models is the problem of demand inversion, namely how to recover the payoffs associated with each alternative based on the corresponding market shares. We have developed a methodology called the “mass transport approach” to perform demand inversion in choice models using matching theory.
Multinomial choice models are usually thought of as conceptually distinct from matching models. The traditional wisdom is that matching models are “two-sided” (on the labor market, workers and firms choose each other), while demand models are “one-sided” (consumers choose yoghurts, but yoghurts don’t choose consumers). In work with Bonnet, O’Hara and Shum, we build on the findings of earlier papers with Salanié and with Chiong and Shum to show that this distinction has no bite, and that in fact, a model where consumers choose yoghurts is observationally equivalent to a (hypothetical) dual model where yoghurts choose consumers, or to a model where consumers “match” with yoghurts. At the heart of the “mass transport” approach to demand inversion lies our equivalence theorem: identifying the systematic payoffs in a multinomial choice model is equivalent to the determining a stable pair in a matching model. We use this reformulation to make use of matching theory in order to provide new theoretical results and new computational techniques in demand models. This finding gives rise to a novel class of efficient computational algorithms to invert multinomial choice models, that are based on matching algorithms. In ongoing work with Chernozhukov, Henry and Pass, we extend these methods to the case when the alternative are continuous, i.e. hedonic models.
See a brief description in my book, Optimal transport methods in economics, chap. 9.2.


My co-authors:

Odran Bonnet, Khai Chiong, Victor Chernozhukov, Marc Henry, Keith O’Hara, Brendan Pass, and Bernard Salanié.


Presentation slides:

Available here.



See arum routines of the TraME library.



Alfred Galichon, and Bernard Salanié (2012). Cupid’s Invisible Hand: Social Surplus and Identification in Matching Models. Revision requested (2nd round), Review of Economic Studies. Available here.
Khai Chiong, Alfred Galichon, and Matt Shum (2016). Duality in dynamic discrete choice models. Quantitative Economics 7(1), pp. 83—115. Available here.
Odran Bonnet, Alfred Galichon, Keith O’Hara, and Matt Shum (2017). Yogurts choose consumers? Identification of Random Utility Models via Two-Sided Matching. Available here.
Victor Chernozhukov, Alfred Galichon, Marc Henry, and Brendan Pass (2017). Single market nonparametric identification of multi-attribute hedonic equilibrium models. Available here.


Lecture series

Economic applications of optimal transport

Lake Como School of Advanced Studies from May 7-11, 2018 (6h)

Course material

The lecture slides will be available before each lecture on this Github link.

Description of the Course

These lectures will deal with economic applications of optimal transport.


Galichon, A. (2016). Optimal transport methods in economics. Princeton.

Monday 5/7, 2:30pm-4:30pm
Tuesday 5/8, 11am-12:30pm
Thursday 5/20, 9am-10:30am
Thursday 5/20, 11am-12:30pm

1. Multinomial choice models and their inversion (1h30)
Based on:
– Chiong, Galichon, Shum (2016). Duality in dynamic discrete choice models. Quantitative Economics.
– Bonnet, Galichon, Shum (2017). Yogurts choose consumers? Identification of Random Utility Models via Two-Sided Matching. Preprint.

2. Separable matching models with heterogeneity (1h30)
Based on:
– Galichon, Salanié (2010) Matching with Trade-offs: Revealed Preferences over Competing Characteristics. Technical report.
– Galichon, Salanié (2017) Cupid’s Invisible Hand: Social Surplus and Identification in Matching Models. Preprint.

3. Affinity estimation: a framework for statistical inference in matching models (1h30)
Based on:
– Dupuy, Galichon (2014) Personality traits and the marriage market. Journal of Political Economy.
– Dupuy, Galichon, Shum (2017) Estimating matching affinity matrix under low-rank constraints. Preprint.

4. Equilibrium transport: incorporating taxes in matching models (1h30)
Based on:
– Galichon, Kominers, Weber (2017) Costly Concessions: An Empirical Framework for Matching with Imperfectly Transferable Utility. Preprint.
– Dupuy, Galichon, Jaffe, Kominers (2017) Taxation in matching markets. Preprint.


ECON-GA 3503

‘math+econ+code’ in Paris: A masterclass on optimal transport, choice and matching models

NYU Paris (57 bd St-Germain, 75005 Paris), June 17-21, 2019 (30 hours)

Instructor: A. Galichon (NYU Econ+Math). Email:

** Note: Interested students must contact the instructor ahead of time.**


This intensive course, part of the ‘math+econ+code’ series, is focused on models of demand, matching models, and optimal transport methods, with various applications pertaining to labor markets, economics of marriage, industrial organization, matching platforms, networks, and international trade, from the crossed perspectives of theory, empirics and computation. It will introduce tools from economic theory, mathematics, econometrics and computing, on a needs basis, without any particular prerequisite other than the equivalent of a first year graduate sequence in econ or in applied math.
Because it aims at providing a bridge between theory and practice, the teaching format is somewhat unusual: each teaching “block” will be made of 50 minutes of theory followed by 1 hour of coding, based on an empirical application related to the theory just seen. Students are expected to write their own code, and we will ensure that it is operational at the end of each block. This course is therefore closer to cooking lessons than to traditional lectures.
The course is open to graduate students in the fields of economics and applied mathematics, but also in other quantitative disciplines. Students need to bring a laptop with them to the lectures. The knowledge of a particular programming language is not required; students are however expected to have some experience with programming. The course can be taken for credit or as a registered auditor.
The lecturer is Alfred Galichon (professor of economics and of mathematics at NYU). James Nesbit and Octavia Ghelfi (graduate Economics student at NYU) helped prepare the course material. The course is partly based on Galichon’s book, Optimal Transport Methods in Economics.

Support from NSF grant DMS-1716489 is acknowledged.

Course material

Available on Github here.

Practical information

• Schedule: Monday to Friday, 8am-12noon and 2pm-4pm. Location: NYU Paris, 57 boulevard Saint-Germain, 75005 Paris, Room TBA.
• Credits: 2, assessed through a take-home exam or a short final paper, at the student’s option.
• A syllabus is available at
• NYU Students need to register on Albert (code ECON-GA 3503). Non-NYU student can be allowed to audit the course. All students need to contact the instructor ( ahead of time.


• Monday: linear programming, dynamic programming, network flows
• Tuesday: optimal transport toolbox 1 (discrete, one-dimensional, semi-discrete cases)
• Wednesday: optimal transport toolbox 2 (continuous transport, convex analysis, entropic regularization)
• Thursday: static and dynamic multinomial choice
• Friday: statistical estimation of models of matching with transfers


Part I: Tools
Day 1: linear programming (Monday)
Block 1. Basics of linear programming (morning 1st half)
• Theory: linear programming duality; complementary slackness; minimax formulation
• Coding: How to eat optimally? Dataset: Stigler’s original diet data (1945).
Block 2. Network flow problems (morning 2nd half)
• Theory: directed graphs and min-cost flow problem
• Coding: How to find the shortest path through a network? Dataset: Paris subway; New York City street network.
Block 3. Dynamic programming as linear programming (afternoon)
• Theory: Bellman’s equation; interpretation of duality; forward induction, backward induction
• Coding: When to repair mechanical engines? Dataset: Rust’s bus maintenance data (1994).

Day 2: optimal transport I (Tuesday)
Block 4. Discrete matching (morning 1st half)
• Theory: Shapley-Shubik duality; stability; decentralized equilibrium
• Coding: How to solve it? Dataset from Dupuy and Galichon (JPE 2014).
Block 5. Positive assortative matching (morning 2nd half)
• Theory: Becker’s model; compensating differentials; comonotonicity
• Coding: What is a CEO worth? Dataset: Gabaix-Landier’s (QJE 2008) CEO pay data.

Block 6. Hotelling’s characteristics model (afternoon)
• Theory: power diagrams, Aurenhammer’s method
• Coding: How to infer the unobservable quality of a car model? Dataset: Feenstra-Levinsohn (Restud 1994) car data.

Day 3: optimal transport II (Wednesday)
Block 7. Continuous multivariate matching (morning 1st half)
• Theory: Knott-Smith criterion; Brenier’s map; McCann’s theorem
• Applications: exercises.
Block 8. A short tutorial on convex analysis (morning 2nd half)
• Theory: convex duality; Fenchel’s inequality; subdifferentials and their inverses
• Application: exercises.
Block 9. Regularized optimal transport (afternoon)
• Theory: optimal transport with entropic regularization, and with other regularizations.
• Coding: coordinate descent and the IPFP algorithm.

Part II. Models
Day 4: models of static and dynamic multinomial choice (Thursday)
Block 10. Basics of static discrete choice (morning 1st half)
• Theory: Dary-Zachary-Williams theorem, generalized entropy of choice, the inversion theorem
• Coding: How to solve it? simulation methods; AR, SARS, and GHK. Dataset: Greene and Hensher (1997) data on choice of travel mode.
Block 11. Demand models, old and new (morning 2nd half)
• Theory: the GEV model; the random coefficient logit model and the pure characteristics models
• Coding: How to estimate demand for automobiles? Dataset: BLP.
Block 12. Dynamic discrete choice methods (afternoon)
• Theory: Rust’s model; estimation; normalization issues
• Coding: maintenance choice.

Day 5: empirical matching models, the quasilinear case (Friday)
Block 13. Separable models of matching (morning 1st half)
• Theory: matching with unobservable heterogeneity
• Coding: Did Roe vs. Wade decrease the value of marriage? Dataset: Choo and Siow (JPE 2006).
Block 14. The gravity equation (morning 2nd half)
• Theory: optimal transport and the gravity equation; generalized linear models and pseudo-Poisson maximum likelihood estimation
• Coding: How to forecast international trade flows? estimating the gravity equation based on WTO international trade data.
Block 15. High-dimensional matching models (afternoon)
• Theory: estimation of rank-constrained models
• Application: Does physical appearance have a price? matching on socioeconomic and anthropomorphic characteristics. Dataset: Chiappori, Oreffice and Quintana-Domeque’s (JPE 2012).


Short course

Optimal Transport Methods in Economics

Toulouse School of Economics, Fall 2017 (15h)

Course material

The lecture slides will be available before each lecture.

Description of the course

These lectures will introduce the theory of optimal transport, and various applications to economics and finance.


These lectures will be based on my monograph, Optimal Transport Methods in Economics, Princeton, 2016.


Monday, Oct 9, 3:30pm–6:30pm
Monday, Oct 16, 3:30pm–6:30pm
Monday, Nov 6, 3:30pm–6:30pm
Monday, Nov 13, 3:30pm–6:30pm
Monday, Nov 20, 3:30pm–6:30pm


Older Lecture series

(Recent and upcoming lecture series can be found here.)


Older Talks

(Recent and upcoming talks can be found here.)

  • June 12, 2015, CORE seminar, Louvain-la-Neuve
  • April 30, 2015, Econometrics lunch, MIT
  • April 16, 2015, Econometrics workshop, University of Chicago
  • April 15, 2015, Econometrics seminar, University of Iowa
  • April 10, 2015, Econometrics seminar, Boston College
  • April 9, 2015, Empirical Microeconomics workshop, University of Pennsylvannia
  • April 2, 2015, Microeconomics seminar, University of Zurich
  • March 31, 2015, Econometrics Journal special invited session, Royal Economic Society Conference 2015, Manchester. Video here.
  • March 6, 2015, “Big Data Finance” Conference, Courant Institute, New York University
  • December 16, 2014, Cemmap seminar, University College London
  • November 21, 2014, Stochastics and statistics seminar, MIT Sloan School of Management
  • November 18, 2014, Theory lunch, MIT
  • November 14, 2014, Econometrics seminar, Boston University
  • November 12, 2014, Applied Mathematics seminar, Courant Institute for Mathematical Sciences, NYU
  • October 30, 2014, Econometrics lunch, MIT
  • October 16, 2104, Econometrics seminar, University of California at Berkeley
  • October 15, 2014, Econometrics seminar, Stanford University
  • October 9, 2014, Econometrics workshop, Harvard-MIT
  • October 7, 2014, Industrial Organization seminar, UCLA
  • October 6, 2014, Econometrics and Applied Microeconomics seminar, CalTech
  • September 30, 2014, Joint Econometrics and Applied Microeconomics seminar, New York University
  • September 25, 2014, Economic Theory seminar, Carnegie Mellon University
  • September 23, 2014, Econometrics seminar, Princeton University
  • September 18, 2014, Labor/Public Economic Workshop, Yale University
  • September 17, 2014, Invited lectures, Conference on Optimization, Transportation and Equilibrium in Economics, Fields Institute, Toronto
  • September 11, 2014, Econometrics seminar, Columbia University
  • September 9, 2014, Labor lunch, MIT


  • July 10, 2014, Economics seminar, EIEF, Rome
  • June 19, 2014, Conference in honor of Ivar Ekeland’s 70th birthday, Université Paris Dauphine
  • June 16-29, 2014, Guest Lectures on the Econometrics of Matching Markets, Toulouse School of Economics
  • June 5, 2014, Workshop on Econometrics Methods, Sciences Po
  • June 4, 2014, Economics seminar, Aalto University, Helsinki
  • May 19, 2014, NERA / STICERD Industrial Organization seminar, London School of Economics
  • April 10, 2014, Malinvaud Seminar, CREST, Paris
  • March 17, 2014, Roy Seminar, Paris
  • March 4, 2014, EPFL, Lausanne
  • February 14, 2014, Séminaire Léon Brillouin, IRCAM, Paris. Video here.
  • November 22, 2013, Research seminar, Austrian Central Bank, Wien
  • October 19, 2013, Harvard-MIT econometrics seminar
  • September 13, 2013, lunch seminar, Sciences Po Department of Economics


  • June 11, 2013, workshop “Advances in Mechanism Design”, Paris School of Economics
  • June 6, 2013, workshop on Economic Theory, University of Manchester
  • June 5, 2013,  Finance & Stochastics seminar, Imperial College London
  • May 16, 2013, Econometrics and Statistics Seminar, Ecares, Université Libre de Bruxelles
  • May 10, 2013,  joint Econometrics-Family economics workshop, University of Chicago
  • April 26, 2013, Econometrics Seminar, Università della Svizzera italiana, Lugano
  • April 22-26, 2013, Workshop “Partial Identification”, Oberwolfach
  • April 8, 2013, Lunch seminar, Economics Department, Ecole Polytechnique
  • March 5, 2013, brown bag seminar, CalTech
  • February 19, 2013, Conference in honor of Rose-Anne Dana, Dauphine
  • December 14, 2012, Groupe de Travail Humaniste, Université Pierre-et-Marie-Curie, Paris
  • November 26-30, 2012, Workshop “Frontiers in Quantile Regression”, Oberwolfach
  • November 22, 2012, Econometrics seminar, Universite de Montreal
  • November 12, 2012, Econometrics seminar, Queen Mary University, London
  • November 5, 2012, Economics Research Seminar, ETH, Zurich


  • June 21, 2012, “OTtO” workshop, Orsay
  • June 8, 2012, FiME workshop, IHP, Paris
  • May 21-25, 2012, Guest lecture, French Statistical Society, Brussels
  • May 16, 2012, Economics Seminar, Stanford GSB
  • May 4, 2012, Economics Seminar, SciencesPo, Paris
  • May 2, 2012, Economics Seminar, Paris School of Economics
  • April 11, 2012, Economics Seminar, HEC, Paris
  • March 29, 2012, Econometrics Seminar, Columbia University
  • March 15, 2012, Malinvaud Seminar, CREST, Paris
  • March 5, 2012, Economics Seminar, Queen Mary University, London
  • March 2, 2012, Economics Seminar, University of Alicante
  • December 14, 2011, Econometrics and Statistics seminar, Tilburg University
  • December 7, 2011, ESRC Seminar on testability in game theory, Warwick
  • November 25, 2011, Plenary speaker, Conference on Optimization & Practices in Industry, Clamart


  • June 7, 2011, Finance seminar, Imperial College, London
  • May 20, 2011, Econometrics seminar, DEFI, Université de la Méditerranée
  • May 9, 2011, Séminaire Parisien d’Optimisation, Institut Henri Poincare
  • March 2, 2011, Econometrics colloquium, Columbia University
  • Feb 21, 2011, Economic Theory Seminar, Columbia University
  • Jan 8, 2011, Econometric Society Winter Meeting, Denver
  • Dec 13, 2010, 2nd meeting of the French Econometrics Society, Paris
  • Nov 30, 2010, Econometrics seminar, Cemmap, University College London
  • Nov 25, 2010, Workshop “Recent Advances in Revelealed Preferences,” Universite Paris-Dauphine, Paris
  • Oct 27, 2010, Economics department, University of British Columbia, Vancouver
  • Sept 24, 2010, Conference “Partial Identification and Revealed Preferences,” Montreal
  • Sept 1, 2010 Conference OKASE, Toulouse School of Economics
  • Aug 19, 2010, Econometric Society World Congress, Shanghai


  • June 2, 2010, Econometrics workshop, UCLA
  • May 19, 2010, Econometrics seminar, UC Riverside
  • May 18, 2010, Econometrics seminar, UC San Diego
  • May 13, 2010, Labor Economics and Econometrics seminar, Northwestern University
  • May 12, 2010, Econometrics workshop, University of Chicago, Economics Department
  • May 5, 2010, Stochastic Analysis Seminar, Institut Henri Poincaré
  • April 27, 2010, Economic Theory seminar, Vanderbilt University
  • April 15, 2010, Econometrics and Statistics seminar, University of Chicago Booth School of Business
  • April 7, Econometrics and applied microeconomics seminar, CalTech
  • March 16, 2010, Conference “Large portfolio, Concentration and Granularity,” Paris
  • March 8, 2010, Econometrics seminar, Paris School of Economics
  • Feb 4, 2010, Econometrics seminar, Columbia University
  • Jan, 4, 2010, North American Winter meeting of the Econometric Society, Atlanta
  • Dec 17, 2009, Stochastics seminar, University of Freiburg
  • Dec 4, 2009, Bachelier Seminar, Paris
  • Oct 13, 2009, Economic Theory seminar, Toulouse School of Economics


  • July 7, 2009, “Optimization, Transport and Equilibrium” workshop, Paris
  • June 3, 2009, North American Summer Meeting of the Econometric Society, Boston
  • April 20, 2009, Risk Seminar, Department of Statistics, Columbia University
  • March 20, 2009, 2nd International Financial Research Forum, Europlace Institute of Finance, Paris
  • Feb 28, 2009, conference “New Economics of the Family”, Milton Friedman Institute, the University of Chicago
  • Jan 16, 2009, IHPST, Paris
  • Nov 32, 2008, Statistics seminar, LUISS, Rome
  • Oct 24, 2008, Cireq conference on Inference with Incomplete Models, Montreal
  • Oct 17, 2009, Workshop on dynamic and multivariate measures, IHP, Paris
  • Oct 6, 2008, Collegio Carlo Alberto, Torino


  • Jul 18, 2008, “Optimization, Transport and Equilibrium” workshop, University of British Columbia, Vancouver
  • June 26, 2008, “Risk, Decision and Uncertainty” conference, Oxford
  • June 12, 2008, Workshop “Nonsmooth Inference, Analysis and Dependence,” Goteborg
  • June 12, 2008, Finance seminar, Toulouse School of Economics
  • May 22, 2008, Finance seminar, Universite de Geneve
  • May 20, 2008, Workshop on New Directions in Quantitative Finance, Paris
  • May 16, 2008, Bachelier Seminar, Paris
  • March 28, 2008, Conference “Inference in Partially Identified Models and Applications,” UCL, London
  • Jan 5, 2008, North American Winter meeting of the Econometric Society, New Orleans
  • Nov 29, 2007, Malinvaud seminar, CREST, Paris
  • Nov 9, 2007, Workshop “Model Validation, Predictive Ability and Model Risk,” Banque de France, Paris
  • June 25, 2007, “Optimization, Transport and Equilibrium” workshop, Columbia University
  • May 22, 2007, Econometrics Seminar, Northwestern University
  • April 27, 2007, PhD Defense, Harvard University



Past Classes

(Current classes can be found here.)

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