There is a substantial literature estimating the responsiveness of charitable donations to tax incentives for giving in the USA. One approach estimates the price elasticity of giving based on tax return data of individuals who itemize their deductions, a group substantially wealthier than the average taxpayer. Another estimates the price elasticity for the average taxpayer based on general population survey data. Broadly, results from both arms of the literature present a counterintuitive conclusion: the price elasticity of donations of the average taxpayer is larger than that of the average, wealthier, itemizer. We provide theoretical and empirical evidence that this conclusion results from a heretofore unrecognized downward bias in the estimator of the price elasticity of giving when non-itemizers are included in the estimation sample (generally with survey data). An intuitive modification to the standard model used in the literature is shown to yield a consistent and more efficient estimator of the price elasticity for the average taxpayer under a testable restriction. Strong empirical support is found for this restriction, and we estimate a bias in the price elasticity around − 1, suggesting the existing literature significantly over-estimates (in absolute value) the price elasticity of giving. Our results provide evidence of an inelastic price elasticity for the average taxpayer, with a statistically significant and elastic price response found only for households in the top decile of income.
This paper studies the properties of generalised empirical likelihood (GEL) methods for the estimation of and inference on partially identi ed parameters in models speci ed by unconditional moment inequality constraints. The central result is, as in moment equality condition models, a large sample equivalence between the scaled optimised GEL objective function and that for generalised method of moments (GMM) with weight matrix equal to the inverse of the e cient GMM metric for moment equality restrictions. Consequently, the paper provides a generalisation of results in the extant literature for GMM for the non-diagonal GMM weight matrix setting. The paper demonstrates that GMM in such circumstances delivers a consistent estimator of the identi ed set, i.e., those parameter values that satisfy the moment inequalities, and derives the corresponding rate of convergence. Based on these results the consistency of and rate of convergence for the GEL estimator of the identi ed set are obtained. A number of alternative equivalent GEL criteria are also considered and discussed. The paper proposes simple conservative consistent con dence regions for the identi ed set and the true parameter vector based on both GMM with a non-diagonal weight matrix and GEL. A simulation study examines the e cacy of the non-diagonal GMM and GEL procedures proposed in the paper and compares them with the standard diagonal GMM method.
The particular concern of this paper is the construction of a confidence region with pointwise asymptotically correct size for the true value of a parameter of interest based on the generalized Anderson-Rubin (GAR) statistic when the moment variance matrix is singular. The large sample behaviour of the GAR statistic is analysed using a Laurent series expansion around the points of moment variance singularity. Under a condition termed first order moment singularity the GAR statistic is shown to possess a limiting chi-square distribution on parameter sequences converging to the true parameter value. Violation, however, of this condition renders the GAR statistic unbounded asymptotically. The paper details an appropriate discretisation of the parameter space to implement a feasible GAR-based confidence region that contains the true parameter value with pointwise asymptotically correct size. Simulation evidence is provided that demonstrates the efficacy of the GAR-based approach to moment-based inference described in this paper.
The particular concern of this paper is the construction of a confidence region with pointwise asymptotically correct size for the true value of a parameter of interest based on the generalized Anderson-Rubin (GAR) statistic when the moment variance matrix is singular. The large sample behaviour of the GAR statistic is analysed using a Laurent series expansion around the points of moment variance singularity. Under a condition termed first order moment singularity the GAR statistic is shown to possess a limiting chi-square distribution on parameter sequences converging to the true parameter value. Violation, however, of this condition renders the GAR statistic unbounded asymptotically. The paper details an appropriate discretisation of the parameter space to implement a feasible GAR-based confidence region that contains the true parameter value with pointwise asymptotically correct size. Simulation evidence is provided that demonstrates the efficacy of the GAR-based approach to moment-based inference described in this paper. 2 Notation p →, d → denote convergence in probability and distribution, respectively, 'w.p.(a.)1' is 'with probability (approaching) 1' and 'i.i.d.' is 'independent and identically distributed'. o p (a) and O p (a) respectively indicate a variate that, after division by a, converges to zero w.p.a.1 and to a variate that is bounded w.p.a.1 by a bounded non-stochastic sequence; similar definitions apply for their deterministic counterparts o(a), O(a). For an arbitrary random variable x, a.s.(x) denotes 'almost surely' x. E P 0 [·] and Var P 0 [·] denote expectation and variance taken with respect to the true population probability law (P 0 ) of z. For arbitrary random variables y and x, E P 0 [y|x] is the conditional expectation of y given x. χ 2 k denotes a central chi-square distributed random variable with k degrees of freedom. rk(A) and N (A) denote the rank and (right) null space, respectively, of a matrix A whereas tr(A) and det(A) are the trace and determinant, respectively, of a square matrix A. For integer k > 0, I k denotes a k × k identity matrix. For a full column rank k × p matrix A and k × k nonsingular matrix K, P A (K) denotes the oblique projection matrix A(A ′ K −1 A) −1 A ′ K −1 and M A (K) = I k − P A (K) its orthogonal counterpart. We abbreviate this notation to P A and M A if K = I k and, if p = 0, set M A = I k . For square matrices A and B, diag(A, B) denotes a block-diagonal matrix with diagonal blocks A and B. ∥A∥ = tr(A ′ A) 1/2 denotes the Euclidean norm of the matrix A and d(x, y) = ∥x − y∥ the Euclidean distance between vectors x, y ∈ R d for integer d ≥ 1. The Hausdorff distance between sets A and B is defined as d H (A, B) = max{sup a∈A d(a, B), sup b∈B d(A, b)}, where d(a, B) = inf b∈B ∥b − a∥, and d H (A, B) = ∞ if either A or B is the null set ∅. The expectations of the moment vector, Jacobian matrix and moment function second moment matrix are defined as g(θ) = E P 0 [g(z, θ)], G(θ) = E P 0 [∂g(z, θ)/∂θ ′ ] and Ω(θ) = E P 0 [g(z, θ)g(z, θ) ′ ] respectively, θ...
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