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This paper characterizes the minimax linear estimator of the value of an unknown function at a boundary point of its domain in a Gaussian white noise model under the restriction that the first-order derivative of the unknown function is Lipschitz continuous. The estimator is obtained through solving a corresponding single-class modulus problem, which involves optimally configuring the first-order derivative of the least favorable function at the boundary point, bringing an additional complexity relative to the interior-point case. The result is then applied to construct minimax optimal estimators for the regression discontinuity design model, where the parameter of interest involves function values at boundary points.
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This paper develops exact finite sample and asymptotic distributions for structural equation tests based on partially restricted reduced form estimates. Particular attention is given to models with large numbers of instruments, wherein the use of partially restricted reduced form estimates is shown to be especially advantageous in statistical testing even in cases of uniformly weak instruments. Comparisons are made with methods based on unrestricted reduced forms, and numerical computations showing finite sample performance of the tests are reported. Some new results are obtained on inequalities between noncentral chi-squared distributions with different degrees of freedom that assist in analytic power comparisons.
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This paper develops a model of risk sharing in which each individual’s income shock is locally shared ex-post given an ex-ante strategically formed network. Emphasizing the informational constraint of the network such that transfers can only be contingent on local information, the model provides characterizations of the ex-ante efficient network and the pairwise stable networks under the local equal sharing rule. We characterize the efficient and pairwise stable networks: while it is no surprise that the unique efficient network is the complete graph, it is interesting that any pairwise stable network features low average degree and almost 2-regular structures, even under individual risk heterogeneity. This suggests that, in real-world networks with average degrees often much larger than 2, risk-sharing considerations tend to generate negative incentives for network linkage. Moreover, we find that pairwise stable networks are likely to exhibit positive assortativity in terms of risk variances: people of similar income volatility are more likely to be connected in equilibrium.
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This paper proposes a simple yet robust method for semiparametric identification and estimation in panel multinomial choice models, where we allow for infinite dimensional fixed effects in the presence of additive nonseparability, thus incorporating rich forms of unobserved heterogeneity.
Our identification strategy exploits the standard notion of multivariate monotonicity in its contrapositive form, which provides powerful leverage for converting observable events into identifying restrictions on unknown parameters. Specifically, we show how certain configurations of conditional choice probabilities preserve weak monotonicity in an index vector, despite the presence of infinite-dimensional nuisance parameters. Then, by taking the logical contraposition of an intertemporal inequality on conditional choice probabilities from two time periods, we obtain an identifying restriction on the index values. Based on our identification result, we construct consistent set (or point) estimators, together with a computational algorithm adapted to the challenges of this framework. The first step of our two-stage procedure nonparametrically estimates a collection of inequalities concerning intertemporal differences in conditional choice probabilities, where we adopt a machine learning algorithm using artificial neural networks. In the second stage, we compute the final estimator as the minimizers of our sample criterion function. Here, we adopt a spherical-coordinate reparameterization to exploit a combination of topological, geometric and computational advantages. The estimated model is then shown to be further utilizable for counterfactual analysis, such as predicting the effect of a promotional campaign on product sales. We conduct a simulation study to analyze the finite-sample performance of our method and the adequacy of our computational procedure for practical implementation. We then apply our procedure to the Nielsen data on popcorn sales to explore the effects of marketing promotion effects. In our model, we permit rich unobserved heterogeneity in factors such as brand loyalty or responsiveness to subtle flavor and packaging designs, which may affect choices in complex ways. The results show that our procedure produces estimates that conform well with economic intuition. For example, we find that special in-store displays boost sales not only through a direct promotion effect but also through the attenuation of consumers’ price sensitivity |
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This paper considers the effect of contracting limitations in risk-sharing networks that arise, for example, from local information constraints. We derive necessary and sufficient conditions for Pareto efficiency under these constraints in a general setting and provide an explicit characterization of the optimal risk-sharing arrangements under CARA utilities and normally distributed endowments. In our model, individuals with higher centralities become quasi-insurance providers to more peripheral individuals. We show that network centrality is (asymptotically) positively correlated with consumption volatility in dense and moderately sparse random graphs, and empirically corroborate this prediction using consumption and network data on rural villages in Thailand. We also provide a discussion about the implications of network heterogeneity on empirical tests of risk-sharing efficiency.
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The focus of this paper is the identification of an index model of dyadic link formation with nonparametric homophily effects and unobserved degree heterogeneity. The paper derives sharp nonparametric identification results for these unknown elements. The key to the identification strategy is a novel form of scale normalization that controls an arbitrary inter-quantile range of the error distribution and provides a convenient linkage between observable conditional choice probabilities and the unknown index values. Under this normalization we deploy a new recursive in-fill and out-expansion algorithm to establish the main identification results. Extensions of the results are explored to accommodate other relevant features such as additive nonseparability and network sparsity. As a byproduct of the analysis a notion of modeling equivalence is proposed as a refinement of the traditional concept of observational equivalence. The relevance of this notion in econometric modeling is illustrated in a formal discussion about normalization, identification and their interplay with counterfactual analysis.
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This paper considers a semiparametric model of dyadic network formation under nontransferable utilities. Such dyadic links arise frequently in real-world social interactions that require bilateral consent but by their nature induce additive nonseparability. The formation of friendship among U.S. high-school students, which naturally requires mutual acceptance, is one particularly relevant example of considerable academic and policy interest. In our model we show how two-way fixed effects (corresponding to unobserved individual heterogeneity in sociability) can be canceled out without requiring additivity. The approach uses a new method we call logical differencing. The key idea is to construct an observable event involving the intersection of two mutually exclusive restrictions on the fixed effects, while these restrictions are obtained by taking the logical contraposition of multivariate monotonicity. Based on this identification strategy we provide consistent estimates of the network formation model. Finite-sample performance is analyzed in a simulation study.
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This paper considers an umbrella theoretical framework under which the problem of network formation and the problem of network bargaining (i.e., the division of surplus obtained in specific network structures) are solved jointly in a consistent way. A network formation solution and a network bargaining solution (together with a disagreement protocol) are defined to be consistent with each other if (1) the networks reached after a counterfactual disagreement are equilibrium networks defined by the network formation solution, and (2) the network allocation rule at each post-disagreement network is determined by further application of the network bargaining solution. Assuming that each disagreement is irreversible, we inductively construct a family of network bargaining solutions that are consistent with pairwise stability and satisfy a fairness condition with respect to endogenously generated outside options. We provide examples under which the proposed solution concept induces novel and realistic theoretical features, such as a Cournot type of “local market power”.
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