Untestability of Average Slutsky Symmetry

TL;DR

This paper investigates the empirical testability of Slutsky symmetry in demand functions, revealing that the average Slutsky matrix's symmetry is untestable, but certain bounds can lead to testable inequalities.

Contribution

It demonstrates the untestability of average Slutsky symmetry and introduces bounds on income elasticity to derive testable inequality constraints.

Findings

Average Slutsky matrix is not identified.

Symmetry of the average Slutsky matrix is untestable.

Bounds on income elasticity lead to testable inequalities.

Abstract

Slutsky symmetry and negative semidefiniteness are necessary and sufficient conditions for the rationality of demand functions. While the empirical implications of Slutsky negative semidefiniteness in repeated cross-sectional demand data are well understood, the empirical content of Slutsky symmetry remains largely unexplored. This paper takes an important first step toward addressing this gap. We show that the average Slutsky matrix is not identified and that its identified set always contains a symmetric matrix, implying that the symmetry of the average Slutsky matrix is untestable and that individual Slutsky symmetry cannot be tested through the average. Nevertheless, we demonstrate that, by imposing bounds on the income elasticity of demand, Slutsky symmetry implies a set of functional inequality constraints that are testable.