Moment Inequalities Research Articles

Breaking the curse of dimensionality in conditional moment inequalities for discrete choice models

This paper studies inference of preference parameters in semiparametric discrete choice models when these parameters are not point-identified and the identified set is characterized by a class of conditional moment inequalities. Exploring the semiparametric modeling restrictions, we show that the identified set can be equivalently formulated by moment inequalities conditional on only two continuous indexing variables. Such formulation holds regardless of the covariate dimension, thereby breaking the curse of dimensionality for nonparametric inference based on the underlying conditional moment inequalities. We further apply this dimension reducing characterization approach to the monotone single index model and to a variety of semiparametric models under which the sign of conditional expectation of a certain transformation of the outcome is the same as that of the indexing variable.

Instrumental Variables and the Sign of the Average Treatment Effect

This paper considers identification and inference about the sign of the average effect of a binary endogenous regressor (or treatment) on a binary outcome of interest when a binary instrument is available. In this setting, the average effect of the endogenous regressor on the outcome is sometimes referred to as the average treatment effect (ATE). We consider four different sets of assumptions: instrument exogeneity, instrument exogeneity and monotonicity on the outcome equation, instrument exogeneity and monotonicity on the equation for the endogenous regressor, or instrument exogeneity and monotonicity on both the outcome equation and the equation for the endogenous regressor. For each of these sets of conditions, we characterize when (i) the distribution of the observed data is inconsistent with the assumptions and (ii) the distribution of the observed data is consistent with the assumptions and the sign of ATE is identified. A distinguishing feature of our results is that they are stated in terms of a reduced form parameter from the population regression of the outcome on the instrument. In particular, we find that the reduced form parameter being far enough, but not too far, from zero, implies that the distribution of the observed data is consistent with our assumptions and the sign of ATE is identified, while the reduced form parameter being too far from zero implies that the distribution of the observed data is inconsistent with our assumptions. For each set of restrictions, we also develop methods for simultaneous inference about the consistency of the distribution of the observed data with our restrictions and the sign of the ATE when the distribution of the observed data is consistent with our restrictions. We show that our inference procedures are valid uniformly over a large class of possible distributions for the observed data that include distributions where the instrument is arbitrarily “weak.” A novel feature of the methodology is that the null hypotheses involve unions of moment inequalities.

Extended Gravity

AbstractExporting firms often enter foreign markets that are similar to their previous export destinations. We develop a dynamic model in which a firm’s exports in a market may depend on how similar the market is to the firm’s home country (gravity) and to its previous export destinations (extended gravity). Given the large number of export paths from which forward-looking firms may choose, we use a moment inequality approach to estimate our model. Our estimates indicate that sharing similarities with a prior export destination in terms of geographic location, language, and income per capita jointly reduces the cost of foreign market entry by 69–90%. Reductions due to geographic location (25–38%) and language (29–36%) have the largest effect. Extended gravity thus has a large impact on export entry costs.

Estimating Demand for Differentiated Products with Zeroes in Market Share Data

In this paper we introduce a new approach to estimating differentiated product demand systems that allows for products with zero sales in the data. Zeroes in demand are a common problem in product differentiated markets, but fall outside the scope of existing demand estimation techniques. Our solution to the zeroes problem is based on constructing bounds for the conditional expectation of the inverse demand. These bounds can be translated into moment inequalities that are shown to yield consistent and asymptotically normal point estimator for demand parameters under natural conditions for differentiated product markets. In Monte Carlo simulations, we demonstrate that the new approach works well even when the fraction of zeroes is as high as 95%. We apply our estimator to supermarket scanner data and find that correcting the bias caused by zeroes has important empirical implications, e.g., price elasticities become on the order of twice as large when zeroes are properly controlled.

Isoperimetric Inequalities for Moments of Inertia and Stability of Stationary Motions of a Flexible Thread

Isoperimetric Inequalities for Moments of Inertia and Stability of Stationary Motions of a Flexible Thread

Inference in Moment Inequality Models That Is Robust to Spurious Precision under Model Misspecification

Standard tests and confidence sets in the moment inequality literature are not robust to model misspecification in the sense that they exhibit spurious precision when the identified set is empty. This paper introduces tests and confidence sets that provide correct asymptotic inference for a pseudo-true parameter in such scenarios, and hence, do not suffer from spurious precision.

Inference for Linear Conditional Moment Inequalities

Inference for Linear Conditional Moment Inequalities

Inference in Moment Inequality Models That Is Robust to Spurious Precision under Model Misspecification

Standard tests and confidence sets in the moment inequality literature are not robust to model misspecification in the sense that they exhibit spurious precision when the identified set is empty. This paper introduces tests and confidence sets that provide correct asymptotic inference for a pseudo-true parameter in such scenarios, and hence, do not suffer from spurious precision.

Partial Identification and Inference in Duration Models with Endogenous Censoring

This paper studies identification and inference in transformation models with endogenous censoring. Many kinds of duration models, such as the accelerated failure time model, proportional hazard model, and mixed proportional hazard model, can be viewed as transformation models. I allow the censoring of duration outcome to be arbitrarily correlated with observed covariates and unobserved heterogeneity. I impose no parametric restrictions on the transformation function or the distribution function of the unobserved heterogeneity. In this setting, I partially identify the regression parameters and the transformation function, which are characterized by conditional moment inequalities of U-statistics. I provide an inference method for them by constructing an inference approach for the conditional moment inequality models of U-statistics. As an empirical illustration, I apply the proposed inference method to evaluate the effect of heart transplants on patients' survival time using data from the Stanford Heart Transplant Study.

Moment inequalities for matrix-valued U-statistics of order 2

We present Rosenthal-type moment inequalities for matrix-valued U-statistics of order 2. As a corollary, we obtain new matrix concentration inequalities for U-statistics. One of our main technical tools, a version of the non-commutative Khintchine inequality for the spectral norm of the Rademacher chaos, could be of independent interest.

Economies of scope in price setting: A moment inequalities estimation

Using weekly data on prices, costs and units sold by a supermarket chain, a moment inequalities approach is used to estimate a discrete-choice dynamic model of a multiproduct firm facing menu costs. This empirical methodology allows to estimate two types of fixed costs of price adjustment: costs that are independent of the number of items that change prices and costs that are incurred at each item’s price change. I find that both types of menu costs exist and are substantial. The total cost of changing prices is estimated to be bounded between 0.3% and 1.3% of revenues and between 17% and 65% of net margins. The first type of fixed cost accounts for between 23% and 99% of this expense, pointing to substantial economies of scope in price setting.

Convergence rates in the central limit theorem for weighted sums of Bernoulli random fields

We prove moment inequalities for a class of functionals of i.i.d. random fields. We then derive rates in the central limit theorem for weighted sums of such randoms fields via an approximation by $m$-dependent random fields.

On moment inequalities for branching processes

On moment inequalities for branching processes

Applying Multiple Stochastic iterated integral to moment inequalities for Wiener Functional

Applying Multiple Stochastic iterated integral to moment inequalities for Wiener Functional

Inference on Causal and Structural Parameters using Many Moment Inequalities

Abstract This article considers the problem of testing many moment inequalities where the number of moment inequalities, denoted by $p$, is possibly much larger than the sample size $n$. There is a variety of economic applications where solving this problem allows to carry out inference on causal and structural parameters; a notable example is the market structure model of Ciliberto and Tamer (2009) where $p=2^{m+1}$ with $m$ being the number of firms that could possibly enter the market. We consider the test statistic given by the maximum of $p$ Studentized (or $t$-type) inequality-specific statistics, and analyse various ways to compute critical values for the test statistic. Specifically, we consider critical values based upon (1) the union bound combined with a moderate deviation inequality for self-normalized sums, (2) the multiplier and empirical bootstraps, and (3) two-step and three-step variants of (1) and (2) by incorporating the selection of uninformative inequalities that are far from being binding and a novel selection of weakly informative inequalities that are potentially binding but do not provide first-order information. We prove validity of these methods, showing that under mild conditions, they lead to tests with the error in size decreasing polynomially in $n$ while allowing for $p$ being much larger than $n$; indeed $p$ can be of order $\exp (n^{c})$ for some $c > 0$. Importantly, all these results hold without any restriction on the correlation structure between $p$ Studentized statistics, and also hold uniformly with respect to suitably large classes of underlying distributions. Moreover, in the online supplement, we show validity of a test based on the block multiplier bootstrap in the case of dependent data under some general mixing conditions.

A Combinatorial Approach to Small Ball Inequalities for Sums and Differences

Small ball inequalities have been extensively studied in the setting of Gaussian processes and associated Banach or Hilbert spaces. In this paper, we focus on studying small ball probabilities for sums or differences of independent, identically distributed random elements taking values in very general sets. Depending on the setting – abelian or non-abelian groups, or vector spaces, or Banach spaces – we provide a collection of inequalities relating different small ball probabilities that are sharp in many cases of interest. We prove these distribution-free probabilistic inequalities by showing that underlying them are inequalities of extremal combinatorial nature, related among other things to classical packing problems such as the kissing number problem. Applications are given to moment inequalities.

Nonparametric Tests for Superior Predictive Ability

Abstract A nonparametric method for comparing multiple forecast models is developed and implemented. The hypothesis of Optimal Predictive Ability generalizes the Superior Predictive Ability hypothesis from a single given loss function to an entire class of loss functions. Distinction is drawn between General Loss functions, Convex Loss functions and Symmetric Convex Loss functions. The research hypothesis is formulated in terms of moment inequality conditions. The empirical moment conditions are reduced to an exact and finite system of linear inequalities based on piecewise-linear loss functions. The hypothesis can be tested in a statistically consistent way using a blockwise Empirical Likelihood Ratio test statistic. A computationally feasible test procedure computes the test statistic using Convex Optimization methods, and estimates conservative, data-dependent critical values using a majorizing chi-square limit distribution and a moment selection method. An empirical application to inflation forecasting reveals that a very large majority of thousands of forecast models are redundant, leaving predominantly Phillips Curve type models, when convexity and symmetry are assumed.

TESTING THE QUANTAL RESPONSE HYPOTHESIS

AbstractWe develop a nonparametric test for consistency of player behavior with the quantal response equilibrium (QRE). The test exploits a characterization of the equilibrium choice probabilities in any structural QRE as the gradient of a convex function; thereby, QRE‐consistent choices satisfy the cyclic monotonicity inequalities. Our testing procedure utilizes recent econometric results for moment inequality models. We assess our test using lab experimental data from a series of generalized matching pennies games. We reject the QRE hypothesis in the pooled data but cannot reject individual‐level quantal response behavior for over half of the subjects.

Inequalities for Functions of the Sum of the Indicators of Events

We obtain moment inequalities for the sum of the indicators of events and an upper estimate for a convex function of such a sum. Our results generalize inequalities that were obtained earlier for moment characteristics of the sojourn time of a random walk on a half-axis.

Identifying Treatment Effects in the Presence of Confounded Types

In this paper, I consider identification of treatment effects when the treatment is endogenous. The use of instrumental variables is a popular solution to deal with endogeneity, but this may give misleading answers when the instrument is invalid. I show that when an (unobserved) instrument is invalid due to correlation with the first stage unobserved heterogeneity, a proxy for the instrument helps partially identify not only the local average treatment effect, but also the entire potential outcomes distributions for compliers. I exploit the fact that the distribution of the observed outcome in each group defined by the treatment and the instrument is a mixture of the distributions of interest. I write the identified set in the form of conditional moment inequalities, and provide an easily implementable inference procedure. Under some tail restrictions, the potential outcomes distributions are point-identified for compliers. Finally, I illustrate my methodology on data from the National Longitudinal Survey of Young Men to estimate returns to college using college proximity as a proxy for the instrument low college cost. I find that a college degree increases the average hourly wage of the compliers by 15-30%.