A bivariate simultaneous discrete response model which is a stochastic representation of equilibria in a two-person discrete game is studied. The presence of multiple equilibria in the underlying discrete game maps into a region for the exogenous variables where the model predicts a nonunique outcome. This is an example of an incomplete econometric structure. Economists using this model have made simplifying assumptions to avoid multiplicity. I make a distinction between incoherent models and incomplete models, and then analyse the model in the presence of multiple equilibria, showing that the model contains enough information to identify the parameters of interest and to obtain a well defined semiparametric estimator. I also show that the latter is consistent and √n normal. Moreover, by exploiting the presence of multiplicity, one is able to obtain a more efficient estimator than the existing methods.

In weighted moment condition models, we show a subtle link between identification and estimability that limits the practical usefulness of estimators based on these models. In particular, if it is necessary for (point) identification that the weights take arbitrarily large values, then the parameter of interest, though point identified, cannot be estimated at the regular (parametric) rate and is said to be irregularly identified. This rate depends on relative tail conditions and can be as slow in some examples as n−1/4. This nonstandard rate of convergence can lead to numerical instability and/or large standard errors. We examine two weighted model examples: (i) the binary response model under mean restriction introduced by Lewbel (1997) and further generalized to cover endogeneity and selection, where the estimator in this class of models is weighted by the density of a special regressor, and (ii) the treatment effect model under exogenous selection (Rosenbaum and Rubin (1983)), where the resulting estimator of the average treatment effect is one that is weighted by a variant of the propensity score. Without strong relative support conditions, these models, similar to well known “identified at infinity” models, lead to estimators that converge at slower than parametric rate, since essentially, to ensure point identification, one requires some variables to take values on sets with arbitrarily small probabilities, or thin sets. For the two models above, we derive some rates of convergence and propose that one conducts inference using rate adaptive procedures that are analogous to Andrews and Schafgans (1998) for the sample selection model.

Identification of dynamic nonlinear panel data models is an important and delicate problem in econometrics. In this paper we provide insights that shed light on the identification of parameters of some commonly used models. Using these insights, we are able to show through simple calculations that point identification often fails in these models. On the other hand, these calculations also suggest that the model restricts the parameter to lie in a region that is very small in many cases, and the failure of point identification may, therefore, be of little practical importance in those cases. Although the emphasis is on identification, our techniques are constructive in that they can easily form the basis for consistent estimates of the identified sets.

While English auctions are the most common in practice, their rules typically lack sufficient structure to yield a tractable theoretical model without significant abstractions. Rather than relying on one stylized model to provide an exact interpretation of the data, we explore an incomplete model based on two simple assumptions: bidders neither bid more than their valuations nor let an opponent win at a price they would be willing to beat. Focusing on the symmetric independent private values paradigm, we show that this limited structure enables construction of informative bounds on the distribution function characterizing bidder demand, on the optimal reserve price, and on the effects of observable covariates on bidder valuations. If the standard theoretical model happens to be the true model, our bounds collapse to the true features of interest. In contrast, when the true data†generating process deviates in seemingly small ways from that implied by equilibrium in the standard theoretical model, existing methods can yield misleading results that need not even lie within our bounds. We report results from Monte Carlo experiments illustrating the performance of our approach and comparing it to others. We apply our approach to U.S. Forest Service timber auctions to evaluate reserve price policy.

We provide a practical method to estimate the payoff functions of players in complete information, static, discrete games. With respect to the empirical literature on entry games originated by Bresnahan and Reiss (1990) and Berry (1992), the main novelty of our framework is to allow for general forms of heterogeneity across players without making equilibrium selection assumptions. We allow the effects that the entry of each individual airline has on the profits of its competitors, its “competitive effects,” to differ across airlines. The identified features of the model are sets of parameters (partial identification) such that the choice probabilities predicted by the econometric model are consistent with the empirical choice probabilities estimated from the data. We apply this methodology to investigate the empirical importance of firm heterogeneity as a determinant of market structure in the U.S. airline industry. We find evidence of heterogeneity across airlines in their profit functions. The competitive effects of large airlines (American, Delta, United) are different from those of low cost carriers and Southwest. Also, the competitive effect of an airline is increasing in its airport presence, which is an important measure of observable heterogeneity in the airline industry. Then we develop a policy experiment to estimate the effect of repealing the Wright Amendment on competition in markets out of the Dallas airports. We find that repealing the Wright Amendment would increase the number of markets served out of Dallas Love.

This paper analyzes the linear regression model y = x + with a conditional median assumption Med( j z) = 0 where z is a vector of exogenous random variables. Added complication arise due to the censoring of the outcome y. We treat the censored model as a model with interval-observed outcomes thus obtaining an incomplete model with inequality restrictions on conditional median regressions. This allows us to use the estimator introduced by Manski and Tamer (2000) to analyze the information contained in these inequality restrictions. We give identication conditions in the absence of censoring and introduce a p N-consistent estimator based on the minimum distance method. We then give suÆcient conditions for global identication of with censored y and endogenous x. In the case of interval data on y and endogenous x, we provide a set-consistent estimator that is based on a modied minimum distance method. In the case where we have point identication, we show that the estimator is p N-normal and derive its asymptotic distribution with a feasible asymptotic variance. A Montecarlo analysis illustrates our estimator. We thank Bo Honore for comments and the Econometrics Research Program at Princeton for support.

This paper examines inference on regressions when interval data are available on one variable, the other variables being measured precisely. Let a population be characterized by a distribution P(y, x, v, v0, v1), where y∈R1, x∈Rk, and the real variables (v, v0, v1) satisfy v0≤v≤v1. Let a random sample be drawn from P and the realizations of (y, x, v0, v1) be observed, but not those of v. The problem of interest may be to infer E(y|x, v) or E(v|x). This analysis maintains Interval (I), Monotonicity (M), and Mean Independence (MI) assumptions: (I) P(v0≤v≤v1)=1; (M) E(y|x, v) is monotone in v; (MI) E(y|x, v, v0, v1)=E(y|x, v). No restrictions are imposed on the distribution of the unobserved values of v within the observed intervals [v0, v1]. It is found that the IMMI Assumptions alone imply simple nonparametric bounds on E(y|x, v) and E(v|x). These assumptions invoked when y is binary and combined with a semiparametric binary regression model yield an identification region for the parameters that may be estimated consistently by a modified maximum score (MMS) method. The IMMI assumptions combined with a parametric model for E(y|x, v) or E(v|x) yield an identification region that may be estimated consistently by a modified minimum-distance (MMD) method. Monte Carlo methods are used to characterize the finite-sample performance of these estimators. Empirical case studies are performed using interval wealth data in the Health and Retirement Study and interval income data in the Current Population Survey.

We study the problem of parameter inference in (possibly non-linear and non-smooth) econometric models when the data are measured with error. We allow for arbitrary correlation between the true variables and the measurement errors. To solve the identification problem, we require the existence of an auxiliary data-set that contains information about the conditional distribution of the true variables given the mismeasured variables. Our main assumption requires that the conditional distribution of the true variables given the mismeasured variables is the same in the primary and auxiliary data. Our methods allow the auxiliary data to be a validation sample, where the primary and validation data are from the same distribution, and more importantly, a stratified sample where the auxiliary data-set is not from the same distribution as the primary data. We also show how to combine the two data-sets to obtain a more efficient estimator of the parameter of interest. We establish the large sample properties of the sieve based estimators under verifiable conditions. In particular, we allow for the mismeasured variables to have unbounded supports without employing the tedious trimming scheme typically used in kernel based methods. We illustrate our methods by estimating a returns to schooling censored quantile regression using the CPS/SSR 1978 exact match files where the dependent variable is measured with error of arbitrary kind.

We provide an econometric framework for estimating a game of simultaneous entry and pricing decisions while allowing for correlations between unobserved cost and demand shocks. We use our framework to account for selection in the pricing stage. We estimate the model using data from the US airline industry and find that not accounting for endogenous entry leads to biased estimation of demand elasticities. We simulate a merger between American and US Airways and find that product repositioning and postmerger outcomes depend on how we model the characteristics of the merged firm as a function of the premerger firms’ characteristics.