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Quantifier Elimination for Deduction in Econometrics -- by Casey B. MulliganWhen combined with the logical notion of partially interpreted functions, many nonparametric results in econometrics and statistics can be understood as statements about semi-algebraic sets. Tarski's quantifier elimination (QE) theorem therefore guarantees that a universal algorithm exists for deducing such results from their assumptions. This paper presents the general framework and then applies QE algorithms to Jensen's inequality, omitted variable bias, partial identification of the classical measurement error model, point identification in discrete choice models, and comparative statics in the nonparametric Roy model. This paper also discusses the computational complexity of real QE and its implementation in software used for program verification, logic, and computer algebra. I expect that automation will become as routine for abstract econometric reasoning as it already is for numerical matrix inversion.
http://papers.nber.org/papers/w24601#fromrss
http://papers.nber.org/papers/w24601#fromrssNon-linear Real Arithmetic Benchmarks derived from Automated Reasoning in Economics -- by Casey B. Mulligan, Russell Bradford, James H. Davenport, Matthew England, Zak TonksWe consider problems originating in economics that may be solved automatically using mathematical software. We present and make freely available a new benchmark set of such problems. The problems have been shown to fall within the framework of non-linear real arithmetic, and so are in theory soluble via Quantifier Elimination (QE) technology as usually implemented in computer algebra systems. Further, they all can be phrased in prenex normal form with only existential quantifiers and so are also admissible to those Satisfiability Module Theory (SMT) solvers that support the QF_NRA logic. There is a great body of work considering QE and SMT application in science and engineering, but we demonstrate here that there is potential for this technology also in the social sciences.
http://papers.nber.org/papers/w24602#fromrss
http://papers.nber.org/papers/w24602#fromrss