We introduce a model of dynamic matching with transferable utility, extending the static model of Shapley and Shubik (1971). Forward-looking agents have individual states that evolve with current matches. Each period, a matching market with market-clearing prices takes place. We prove the existence of an equilibrium with time-varying distributions of agent types and show it is the solution to a social planner's problem. We also prove that a stationary equilibrium exists. We introduce econometric shocks to account for unobserved heterogeneity in match formation. We propose two algorithms to compute a stationary equilibrium. We adapt both algorithms for estimation. We estimate a model of accumulation of job-specific human capital using data on Swedish engineers.

Agents in two-sided matching games vary in characteristics that are unobservable in typical data on matching markets. We investigate the identification of the distribution of unobserved characteristics using data on who matches with whom. In full generality, we consider many-to-many matching and matching with trades. The distribution of match-specific unobservables cannot be fully recovered without information on unmatched agents, but the distribution of a combination of unobservables, which we call unobserved complementarities, can be identified. Using data on unmatched agents restores identification.

I explore the estimation of transferable utility matching games, encompassing many-to-many matching, marriage, and matching with trading networks (trades). Computational issues are paramount. I introduce a matching maximum score estimator that does not suffer from a computational curse of dimensionality in the number of agents in a matching market. I apply the estimator to data on the car parts supplied by automotive suppliers to estimate the valuations from different portfolios of parts to suppliers and automotive assemblers.

A matching model takes a set of payoffs or outputs for all possible matches and produces a set of matches where no couple would prefer to deviate and become matched, instead of their assigned matches. Matching models are increasingly being estimated in empirical work in industrial organization, labour economics, public economics, and other fields. This article surveys methods for and applications of structural estimation for two-sided matching games.

We explore multitasking as an explanation for the use of more than one performance measure in the incentive compensation of CEOs. We use detailed data on compensation contracts to describe how incentive pay for multiple performance measures varies across firms. We then use the compensation data to specify and estimate an equilibrium matching game between chief executives and firms. In the matching game, firms and executives sign linear incentive contracts based on both equity and profit performance under a model of hidden executive effort with multitasking. The estimated structural model is used to explore the counterfactual of requiring firms to base incentive pay on only one performance measure in order to quantify the role of current contract forms in ameliorating the efficiency losses from multitasking. *Corresponding Author.

Structural estimation of matching games with transferable utility, including matching games of trading networks and many-to-many matching, is increasingly popular in empirical work. I explore several modeling decisions that need to be made when specifying a structural model for a matching game. One decision is the choice of a game theoretic solution concept to impose in the structural model. I discuss pairwise stability, competitive equilibrium, and noncooperative games such as auctions. Another decision is whether to work with a continuum of agents or a finite number of agents. I explore other issues as well.

We study nonparametric identification of single-agent discrete choice models for bundles and binary games of complete information. We provide conditions under which we can recover both the interaction effects and the distributions of potentially correlated unobservables across goods in single-agent models and across players in games. We establish similarities in identification between these two models. Strengthening our assumptions for games, we provide an equivalence relation between discrete choice models for bundles and binary games that relies on the theory of potential games. Potential games are particularly useful for games of three or more players.

We show how to nonparametrically identify the distribution of unobservables, such as random coefficients, that characterizes the heterogeneity among consumers in multinomial choice models. We provide general identification conditions for a class of nonlinear models and then verify these conditions using the primitives of the multinomial choice model. We require that the distribution of unobservables lie in the class of all distributions with finite support, which under our most general assumptions, resembles a product space where some of the product members are function spaces. We show how identification leads to the consistency of a nonparametric estimator.

I explore the estimation of transferable utility matching games, encompassing many-to-many matching, marriage and matching with trading networks (trades). I introduce a matching maximum score estimator that does not suffer from a computational curse of dimensionality in the number of agents in a matching market. I apply the estimator to data on the car parts supplied by automotive suppliers to estimate the returns from different portfolios of parts to suppliers and automotive assemblers. Jeremy T. Fox Rice University Department of Economics MS22 Baker Hall P.O. Box 1892 Houston, TX 77251-1892 and NBER jeremyfox@gmail.com

Applications of the Price equation partition to the empirical studies of Wardle et al., Belnap et al., and Spehn et al., and to the theoretical model of Ives and Cardinale.

In our Monte Carlo study, we apply our fixed grid estimators to a capital replacement model in the spirit of Rust (1987). A dynamic program must be solved for each grid point or simulation draw, so working with a dynamic program showcases our method’s advantage of requiring computation of choice probabilities only before optimization commences. We vary the number of grid points, the sample sizes and the true distributions of the random coefficients. We compare the least squares and likelihood criteria on a fixed grid (fixed grid estimators) with an alternative likelihood criterion on a flexible grid (flexible grid estimator). For the fixed grid, we optimize over only weights. For the flexible grid, we optimize over both the grid points and the weights, using a much smaller number of weights because of the computational challenges of both optimization and the need to solve the dynamic program many times. The two likelihood estimators are maximized using the EM algorithm. Our results suggest that the two fixed grid estimators are much faster than the flexible grid estimator at the cost of some statistical accuracy.

We explore least squares and likelihood nonparametric mixtures estimators of the joint distribution of random coefficients in structural models. The estimators fix a grid of heterogeneous parameters and estimate only the weights on the grid points, an approach that is computationally attractive compared to alternative nonparametric estimators. We provide conditions under which the estimated distribution function converges to the true distribution in the weak topology on the space of distributions. We verify most of the consistency conditions for three discrete choice models. We also derive the convergence rates of the least squares nonparametric mixtures estimator under additional restrictions. We perform a Monte Carlo study on a dynamic programming model.

With advances in telecommunications, an increasing number of services rely on high data rate spectrum access. These critical services include banking, telemedicine, and exchange of technical information. As a result, spectrum resources are in ever-greater demand and the radio spectrum has become overly crowded. For efficient usage of spectrum, smart or cognitive radios are sought after. However, current wireless phones can only select a few specific bands. In this paper, we discuss the advantages of reconfigurable radios in not only increasing the efficiency of spectrum usage but also in potentially reducing the cost of wireless handsets and the barriers for new wireless service providers to enter the market. We review available technologies that make the implementation of reconfigurable radios possible and discuss technical challenges that need to be overcome before multistandard reconfigurable radios are put into practice. We also evaluate the ability of reconfigurable radios in reducing entry costs for new competitors in wireless service.

FCC spectrum auctions sell licenses to provide mobile phone service in designated geographic territories.We propose a method to structurally estimate the deterministic component of bidder valuations and apply it to the 1995-1996 C-block auction.We base our estimation of bidder values on a pairwise stability condition, which implies that two bidders cannot exchange licenses in a way that increases total surplus.Pairwise stability holds in many theoretical models of simultaneous ascending auctions, including some models of intimidatory collusion and demand reduction.Pairwise stability is also approximately satisfied in data that we examine from economic experiments.The lack of post-auction resale also suggests pairwise stability.Using our estimates of deterministic valuations, we measure the allocative efficiency of the C-block outcome.

The widely used estimator of Berry, Levinsohn, and Pakes (1995 )p roduces estimates of consumer preferences from a discrete-choice demand model with random coefficients, market-level demand shocks, and endogenous prices. We derive numerical theory results characterizing the properties of the nested fixed point algorithm used to evaluate the objective function of BLP’s estimator. We discuss problems with typical implementations, including cases that can lead to incorrect parameter estimates. As a solution, we recast estimation as a mathematical program with equilibrium constraints, which can be faster and which avoids the numerical issues associated with nested inner loops. The advantages are even more pronounced for forward-looking demand models where the Bellman equation must also be solved repeatedly. Several Monte Carlo and real-data experiments support our numerical concerns about the nested fixed point approach and the advantages of constrained optimization. For static BLP, the constrained optimization approach can be as much as ten to forty times faster for large-dimensional problems with many markets.

Agents in two-sided matching games vary in characteristics that are unobservable in typical data on matching markets.We investigate the identification of the distribution of unobserved characteristics using data on who matches with whom.In full generality, we consider many-tomany matching and matching with trades.The distribution of match-specific unobservables cannot be fully recovered without information on unmatched agents, but the distribution of a combination of unobservables, which we call unobserved complementarities, can be identified.Using data on unmatched agents restores identification.We estimate the contribution of observables and unobservable complementarities to match production in venture capital investments in biotechnology and medical firms.

The widely-used estimator of Berry, Levinsohn and Pakes (1995) produces estimates of consumer preferences from a discrete-choice demand model with random coefficients, market-level demand shocks and endogenous prices.We derive numerical theory results characterizing the properties of the nested fixed point algorithm used to evaluate the objective function of BLP's estimator.We discuss problems with typical implementations, including cases that can lead to incorrect parameter estimates.As a solution, we recast estimation as a mathematical program with equilibrium constraints, which can be faster and which avoids the numerical issues associated with nested inner loops.The advantages are even more pronounced for forward-looking demand models where Bellman's equation must also be solved repeatedly.Several Monte Carlo and real-data experiments support our numerical concerns about the nested fixed point approach and the advantages of constrained optimization.

The random coefficients multinomial choice logit model, also known as the mixed logit, has been widely used in empirical choice analysis for the last thirty years. We prove that the distribution of random coefficients in the multinomial logit model is nonparametrically identified. Our approach requires variation in product characteristics only locally and does not rely on the special regressors with large supports used in related papers. One of our two identification arguments is constructive. Both approaches may be applied to other choice models with random coefficients.