Identification in Separable Matching with Observed Transfers

Imposing a separability assumption on the joint surplus in transferable utility matching models has proved very useful in empirical work. Yet when only “who matches whom” is observed, the distributions of unobserved heterogeneity cannot be identified separately. This note derives the distribution of equiilibrium transfers and shows that if the distribution of transfers within cells is observed, the distribution of heterogeneity can often be recovered, separability can be tested, and complementarities in surplus inferred

Cupid’s Invisible Hand: Social Surplus and Identification in Matching Models

We investigate a model of one-to-one matching with transferable utility when some of the characteristics of the players are unobservable to the analyst. We allow for a wide class of distributions of unobserved heterogeneity, subject only to a separability assumption that generalizes Choo and Siow (2006). We first show that the stable matching maximizes a social gain function that trades off exploiting complementarities in observable characteristic sand matching on unobserved characteristics. We use this result to derive simple closed-form formulæ that identify the joint surplus in every possible match and the equilibrium utilities of all participants, given any known distribution of unobserved heterogeneity. If transfers are observed, then the pre-transfer utilities of both partners are also identified. We discuss computational issues and provide an algorithm that is extremely efficient in important instances. Finally, we present two estimators of the joint surplus and we revisit Choo and Siow’s empirical application to illustrate the potential of our more general approach

From Aggregate Betting Data to Individual Risk Preferences

As a textbook model of contingent markets, horse races are an attractive environment to study the attitudes towards risk of bettors. We innovate on the literature by explicitly considering heterogeneous bettors and allowing for very general risk preferences, including non-expected utility. We build on a standard single-crossing condition on preferences to derive testable implications; and we show how parimutuel data allow us to uniquely identify the distribution of preferences among the population of bettors. We then estimate the model on data from US races. Within the expected utility class, the most usual specfications (CARA and CRRA) fit the data very badly. Our results show evidence for both heterogeneity and nonlinear probability weighting

Does Fertility Respond to Financial Incentives?

There has been little empirical work evaluating the sensitivity of fertility to financial incentives at the household level. We put forward an identification strategy that relies on the fact that variation of wages induces variation in benefits and tax credits among "comparable" households. We implement this approach by estimating a discrete choice model of female participation and fertility, using individual data from the French Labor Force Survey and a fairly detailed representation of the French tax-benefit system. Our results suggest that financial incentives play a notable role in determining fertility decisions in France, both for the first and for the third child. As an example, an unconditional child benefit with a direct cost of 0.3% of GDP might raise total fertility by about 0.3 point

Identifying Effects of Multivalued Treatments

Multivalued treatment models have only been studied so far under restrictive assumptions: ordered choice, or more recently unordered monotonicity. We show how marginal treatment effects can be identified in a more general class of models. Our results rely on two main assumptions: treatment assignment must be a measurable function of threshold-crossing rules; and enough continuous instruments must be available. On the other hand, we do not require any kind of monotonicity condition. We illustrate our approach on several commonly used models; and we also discuss the identification power of discrete instruments

Modeling Competition and Market Equilibrium in Insurance: Empirical Issues

In the last decade or so, numerous papers have been devoted to empirical investigations based on contract theory. Many contributions use insurance data, and specifically files provided by firms. A typical paper would analyze the relationship between individual characteristics, the contracts chosen and the corresponding “outcome,” as measured by claims. The natural next step in this research agenda is to model empirically market equilibrium on insurance markets. Empirical models of competitive insurance markets are important in many respects. First, such models are an indispensable first step for the empirical analysis of existing markets. The discussion of optimal pricing strategies or the definition of new insurance contract would greatly benefit from such models. From a policy perspective, the design of any regulation requires estimating its likely impact on the market allocation. For instance, while a ban on specific pricing options (based, say, on gender or age) is often advocated on ethical grounds, a precise assessment of its impact on insurance markets is needed before any decision is made; and an empirical model is required to provide such an assessment. From a purely theoretical perspective, any description of insurance markets that aims at a modicum of realism needs to come to terms with a host of complex features (horizontal differentiation of products, unobserved heterogeneity of preferences, frictions of various types), the theoretical analysis of which may be forbiddingly complex. A simple model that can be solved or at least numerically simulated may, in that case, be particularly helpful. Finally, a tractable model of insurance equilibrium can be used to run experiments, which should help us understand individual behavior in such strategic settings as competition under asymmetric information. On the other hand, modeling insurance markets raises several theoretical and empirical issues, starting, of course, with the well-known pitfalls in modeling equilibrium in contracts. The goal of the present paper is to discuss these problems and summarize the knowledge acquired so far. We successively discuss modeling of the demand side, the supply side, and the equilibrium itself

The Econometrics of Matching Models

In October 2012 the Nobel prize was attributed to Al Roth and Lloyd Shapley for their work on matching. Both the seminal Gale-Shapley (1962) paper and most of Roth’s work were concerned with allocation mechanisms when prices or other transfers cannot be used—what we will call non-transferable utility (NTU) in this survey. Gale and Shapley used college admissions, marriage, and roommate assignments as examples; and Roth’s fundamental work in market design has led to major improvements in the National Resident Matching Program (Roth and Peranson 1999) and to the creation of a mechanism for kidney exchange (Roth, Sönmez and Ünver 2004.) The resulting insights have been applied to a host of issues, including the allocation of students to schools, the marriage market with unbalanced gender distributions, the role of marital prospects in human capital investment decisions, the social impact of improved birth control technologies and many others. The econometrics of matching models have recently been reconsidered, from different and equally innovative perspectives. The goal of the present project will be to survey these methodological advances. We shall describe the main difficulties at stake, the various answers provided so far, and the issues that remain open

Partial Identification of Finite Mixtures in Econometric Models

We consider partial identification of finite mixture models in the presence of an observable source of variation in the mixture weights that leaves component distributions unchanged, as is the case in large classes of econometric models. We first show that when the number J of component distributions is known a priori, the family of mixture models compatible with the data is a subset of a J(J1)-dimensional space. When the outcome variable is continuous, this subset is defined by linear constraints, which we characterize exactly. Our identifying assumption has testable implications, which we spell out for J=2. We also extend our results to the case when the analyst does not know the true number of component distributions and to models with discrete outcomes. Keywords. Partial identification, finite mixture models. JEL classification. C24

Identifying Finite Mixtures in Econometric Models

Mixtures of distributions are present in many econometric models, such as models with unobserved heterogeneity. It is thus crucial to have a general approach to identify them nonparametrically. Yet the literature so far only contains isolated examples, applied to specific models. We derive the identifying implications of a conditional independence assumption in finite mixture models. It applies for instance to models with unobserved heterogeneity, regime switching models, and models with mismeasured discrete regressors. Under this assumption, we derive sharp bounds on the mixture weights and components. For models with two mixture components, we show that if in addition the components behave differently in the tails of their distributions, then components and weights are fully nonparametrically identified. We apply our findings to the nonparametric identification and estimation of outcome distributions with a misclassified binary regressor. This provides a simple estimator that does not require instrumental variables, auxiliary data, symmetric error distributions or other shape restrictions