Nonparametric Testability of Slutsky Symmetry

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.

💡 Research Summary

The paper tackles a long‑standing gap in the non‑parametric testing of Slutsky symmetry, a cornerstone condition of consumer rationality alongside negative semi‑definiteness. While recent work (Dette et al., 2016) has shown that the negative semi‑definiteness of the Slutsky matrix can be tested non‑parametrically even with individual heterogeneity and endogeneity, the symmetry condition was believed to be non‑testable after Kono (2025) proved that the average Slutsky matrix does not contain enough information. Maes and Malhotra (2024) further conjectured that in the general setting of Dette et al. (2016) symmetry could not be identified.

The authors refute this conjecture by deriving a set of conditional and marginal quantile restrictions that are necessary for Slutsky symmetry in the same general non‑parametric IV framework. Their approach rests on a structural demand model
(Y = \psi(P,X,U) = \phi(P,X,Q,A))
with endogenous income (or total expenditure) (X) generated by a first‑stage equation (X = \mu(P,Q,S,V)). The unobserved preference heterogeneity (U) is allowed to be an arbitrary function of observable characteristics (Q) and an unrestricted residual (A). A key independence assumption states that ((P,X)) is independent of (A) conditional on ((Q,V)). This assumption maps the unobservable non‑separable demand function into observable conditional quantiles, making identification possible.

The technical heart of the paper is a multivariate extension of the Höderlein‑Mammen (2007) identification result, which originally dealt with a single‑dimensional outcome. Lemma 2.1 shows that for any pair of goods (i) and (j) the derivative of a conditional quantile with respect to prices can be decomposed into five terms: (1) the direct price derivative of the conditional quantile, (2) the price derivative of the marginal quantile that enters the conditional quantile, (3) the derivative of the conditional quantile with respect to its second argument (the marginal quantile), (4) terms involving the conditional density and the price‑ and income‑derivatives of the conditional distribution, and (5) analogous income‑derivative terms. These additional “correction terms” (denoted (C_{ij}, D_{ij}, D_{ij}^{(x)}) etc.) are absent in the univariate case and capture how the composition of preference types changes with prices.

Theorem 2.1 translates this decomposition into a concrete symmetry condition: for every pair ((i,j)) the following equality must hold at any interior point of the support of ((Y_i,Y_j)): \