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Nonparametric Testability of Slutsky Symmetry

Florian Gunsilius, Lonjezo Sithole

TL;DR

This paper tackles the problem of testing Slutsky symmetry nonparametrically in consumption models with heterogeneity and endogeneity. It develops a bivariate extension of identification results and derives a testable condition expressed in terms of conditional and marginal quantiles and their derivatives, including nonlinear correction terms $C_{ij}$ and $D_{ij}$ that arise in multigood settings. The approach shows that, unlike the two-good case, symmetry imposes observable restrictions that involve these correction factors, enabling empirical validation of Slutsky symmetry and enabling nonparametric welfare analysis under general heterogeneity. The results thus provide a practical, quantile-based framework for verifying rationality conditions in empirical demand systems beyond two goods, with implications for welfare analysis and the treatment of endogeneity.

Abstract

Economic theory implies strong limitations on what types of consumption behavior are considered rational. Rationality implies that the Slutsky matrix, which captures the substitution effects of compensated price changes on demand for different goods, is symmetric and negative semi-definite. While empirically informed versions of negative semi-definiteness have been shown to be nonparametrically testable, the analogous question for Slutsky symmetry has remained open. Recently, it has even been shown that the symmetry condition is not testable via the average Slutsky matrix, prompting conjectures about its non-testability. We settle this question by deriving nonparametric conditional quantile restrictions on observable data that constitute a testable implication of Slutsky symmetry in an empirical setting with individual heterogeneity and endogeneity. The theoretical contribution is a multivariate generalization of identification results for partial effects in nonseparable models without monotonicity, which is of independent interest. This result has implications for different areas in econometric theory, including nonparametric welfare analysis with individual heterogeneity for which, in the case of more than two goods, the symmetry condition introduces nonlinear correction factors.

Nonparametric Testability of Slutsky Symmetry

TL;DR

This paper tackles the problem of testing Slutsky symmetry nonparametrically in consumption models with heterogeneity and endogeneity. It develops a bivariate extension of identification results and derives a testable condition expressed in terms of conditional and marginal quantiles and their derivatives, including nonlinear correction terms and that arise in multigood settings. The approach shows that, unlike the two-good case, symmetry imposes observable restrictions that involve these correction factors, enabling empirical validation of Slutsky symmetry and enabling nonparametric welfare analysis under general heterogeneity. The results thus provide a practical, quantile-based framework for verifying rationality conditions in empirical demand systems beyond two goods, with implications for welfare analysis and the treatment of endogeneity.

Abstract

Economic theory implies strong limitations on what types of consumption behavior are considered rational. Rationality implies that the Slutsky matrix, which captures the substitution effects of compensated price changes on demand for different goods, is symmetric and negative semi-definite. While empirically informed versions of negative semi-definiteness have been shown to be nonparametrically testable, the analogous question for Slutsky symmetry has remained open. Recently, it has even been shown that the symmetry condition is not testable via the average Slutsky matrix, prompting conjectures about its non-testability. We settle this question by deriving nonparametric conditional quantile restrictions on observable data that constitute a testable implication of Slutsky symmetry in an empirical setting with individual heterogeneity and endogeneity. The theoretical contribution is a multivariate generalization of identification results for partial effects in nonseparable models without monotonicity, which is of independent interest. This result has implications for different areas in econometric theory, including nonparametric welfare analysis with individual heterogeneity for which, in the case of more than two goods, the symmetry condition introduces nonlinear correction factors.
Paper Structure (14 sections, 2 theorems, 23 equations)

This paper contains 14 sections, 2 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. A nonparametric Slutsky symmetry condition
  3. Setup and assumptions
  4. A generalization of Hoderlein2007
  5. A Slutsky symmetry condition on conditional and marginal quantiles
  6. Slutsky Symmetry and Nonparametric Welfare Analysis
  7. Empirical content of the condition
  8. The $C_{ij}$ terms.
  9. The $D_{ij}$ terms.
  10. Conclusion
  11. Proofs
  12. Regularity Assumptions
  13. Proof of Lemma \ref{['lem:hod_mam']}
  14. Proof of Theorem \ref{['thm:symmetry']}

Key Result

Lemma 2.1

Under Assumption ass:mod and Assumption ass:ind along with the regularity conditions in Assumption ass:reg, fix a point $(y_i^*, y_j^*)$ in the interior of the support of $(Y_i, Y_j)$ given $W = w^*$ and define so that $k_{\alpha_j,j}(w^*) = y_j^*$ and $k_{\gamma_{i|j},i}(w^*, k_{\alpha_j,j}(w^*)) = y_i^*$. Then for $L \geq 3$, the following holds for $s = 1, \ldots, L$.

Theorems & Definitions (4)

  • Lemma 2.1
  • Theorem 2.1: Slutsky symmetry
  • proof
  • proof