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Nonparametric Testability of Slutsky Symmetry footnote-1
Rationality is the foundational assumption of consumer demand theory. A smooth demand function is rationalizable, that is, consistent with utility maximization, if and only if the associated Slutsky matrix of compensated price effects is symmetric and negative semidefinite samuelson1948consumption, houthakker1950revealed, hurwicz1971integrability. Whether these conditions hold in data is therefore a question of first-order importance for applied welfare analysis.
Testing the Slutsky conditions nonparametrically, i.e., without committing to a functional form for demand, has proven difficult. For negative semidefiniteness, DETTE2016129 provide a nonparametric test in an empirical model with individual heterogeneity and endogeneity. For symmetry, no analogous result has been available to date. Parametric tests exist barten1967evidence, laitinen1978demand, MEISNER1979231, BERA1981101, bewley1983tests, TAYLOR1985327, silver1989testing, but conflate violations of symmetry with misspecification. Nonparametric approaches such as lewbel1995consistent, HAAG200933, and hoderlein2011many rely on additional restrictions that are either unverifiable or impose substantial structure on the demand system. Recently, kono2025untestability shows that symmetry cannot be tested from the average Slutsky matrix alone, and maes2024beyond conjecture that in the general setting of DETTE2016129, nonparametric testability of Slutsky symmetry is impossible.
We show that this conjecture is false in the general setting. We derive a nonparametric restriction on observable conditional and marginal quantiles that is necessary for Slutsky symmetry, providing a testable characterization of the symmetry condition in the empirical setting of DETTE2016129. The key theoretical contribution is a bivariate extension of the identification result of Hoderlein2007 for average partial effects in nonseparable models without monotonicity. This extension, which is of independent interest, reveals that in the multivariate setting the identification result acquires correction terms that are absent in the univariate case. These terms have concrete economic content: they capture how the composition of preference types at a given demand level shifts with prices, and they govern the gap between conditional quantile demands and true Hicksian demands in multi-good welfare analysis.
Our approach is based on continuous demands in a nonseparable model with unrestricted heterogeneity. hausman2016individual provide the foundational framework for nonparametric welfare analysis in this setting, but their analysis is restricted to two goods where symmetry holds by construction. We generalize their approach to multiple goods, where symmetry has empirical content and must be verified.
This section contains the main identification result for the Slutsky symmetry condition, which leads to a testable nonparametric restriction. To state it, we start with the required assumptions on the data and an extension of the ideas in Hoderlein2007 to conditional quantiles.
The key idea is based on the connection between quantile functions of the data-generating process and properties of the Slutsky matrix in an empirical model with individual heterogeneity and endogeneity. DETTE2016129 posit an IV model for a nonparametric demand system with a potentially endogenous covariate (for example, income or total expenditures). The endogeneity of total expenditure is, in part, because categories of individual expenditures are often mismeasured, and hence the aggregate expenditure measure is mismeasured. Moreover, the authors model expenditure while allowing for unobserved individual heterogeneity. In the sequel, we carry over the notation and model from DETTE2016129 to offer a unified treatment:
Here $Y$, a vector of quantities of goods demanded, takes values in $\mathbb{R}^{L-1}$; and $X$ is an endogenous variable, for instance income. We assume that there exists exactly one exogenous shifter $S$ for the endogenous variable. This is done for ease of exposition, analogous to the result in DETTE2016129, and can be generalized to higher dimensions gunsilius2023condition. $V$ captures the unobservable variation in this first stage regression. $V$ can be solved for and used as residuals in a control function approach. For standard identification arguments imbens2009identification, therefore, we assume $\mu$ is invertible in its last argument. $P \in \mathbb{R}^{L-1}$ is the vector of prices for the goods; and $U$ is an unrestricted unobservable describing the preferences. Making $U$ as general as possible is crucial to account for any form of preference function. By writing $U=\nu(Q,A)$, we assume that the preference function is a function of observable characteristics $Q$, like age for instance, and unobservable characteristics $A$. $A$ can therefore be interpreted as residual unobserved heterogeneity after controlling for the observable part of individual heterogeneity. Putting everything together, we have the same assumption on the model as DETTE2016129, to which we refer for further detail.
The Slutsky matrix $\mathcal{S}(p, x, u)$ takes the form \[ \mathcal{S}=D_{p} \psi(p, x, u)+\partial_{x} \psi(p, x, u) \psi(p, x, u), \] where $D_{p}$ is the Jacobian of Marshallian demands with respect to prices and $\partial_x$ is a vector of partial derivatives of Marshallian demands with respect to income.\\ Following DETTE2016129, we denote $W=(P, X, Q, V)$ and introduce $k(\alpha, b \mid w)$ as the $\alpha$-quantile defined as
\[ P\left(b^{\prime} Y \leq k(\alpha, b \mid w) \mid W=w\right)=\alpha\] for any nonzero $b \in \mathbb{R}^{L-1}$. To derive a condition on the observable quantiles that captures the symmetry condition of the Slutsky matrix, only considering univariate marginal quantiles is not enough, as one has to compare off-diagonal terms. However, we can focus on only two elements at a time to capture the symmetry condition. Still, only considering bivariate joint quantiles does not provide identification of the relevant conditions, because the result in Hoderlein2007 cannot be generalized to the multivariate setting directly Hoderlein2009.
We show that symmetry becomes testable in this setting by considering a combination of conditional- and marginal quantiles, for which we provide a generalization of the result in Hoderlein2007 in section (ref). For symmetry, we only need to consider the case where $b$ is a unit vector, that is, we only need to consider elements of the form $e_{i}^{\prime} Y$ for $e_{i} \in \mathbb{R}^{L-1}$ the $i$-th unit vector in $\mathbb{R}^{L-1}$. In this case the above defined $\alpha$-quantile reduces to the $\alpha$-quantile of the marginal distribution of $Y_{i}=e_{i}^{\prime} Y$. For notational purposes, we define the marginal $\alpha$-quantile as
In the same vein, we define the conditional $\gamma$-quantile of the element $Y_{i} \equiv e_{i}^{\prime} Y$ given that the element $Y_{j} \equiv e_{j}^{\prime} Y$ lies at the $\alpha$ marginal level by
This conditional quantile depends not only on $W$ but also on the marginal quantile $k_{\alpha, j}(w)$, which in turn also depends on $W$. In fact, it will be the interplay of the conditional and the marginal quantile which will enable us to derive conditions on the observable data that captures the Slutsky symmetry. We also need to make use of the joint bivariate $\alpha$-quantile $K_{i j}(\alpha \mid w)$ for elements $\left(Y_{i}, Y_{j}\right) \equiv\left(e_{i}^{\prime} Y, e_{j}^{\prime} Y\right)$ of the vector $Y \in \mathbb{R}^{L-1}$, which is defined as
Note that $K_{i j}$ is not unique without further assumption: it is not a point but an isoquant in $\mathbb{R}^{2}$, which is the main complication for testing symmetry nonparametrically.
Since we generalize the identification result in Hoderlein2007 to conditional and marginal quantiles, we aim to keep the same notation and assumptions. These assumptions are also stated in the same notation as the ones in DETTE2016129 to keep the two results unified. We set $Z:=(Q, V)$.
The independence assumption requires that after controlling for observable preference characteristics and unobservable first-stage, the residual unobserved heterogeneity represented by $A$ is independent of prices and the endogenous variable. This conditional independence assumption is key to identification; it maps the space of nonseparable demand functions, which is inherently unobservable, to something we can observe: regression quantiles. It is this feature that allows us to derive a testable restriction on the quantiles of the data generating process, develop a test based on the restriction and infer Slutsky symmetry for the unobserved nonseparable demand functions.
In the following, we denote by $\partial_{w_{1} s} k_{\alpha, j}(w^*)$ the partial derivative of the quantile $k_{\alpha, j}$ with respect to the price of good s (the $s^{th}$ element in the price vector, which in turn is the first element of $w$) at $w^{*}$. For the conditional quantile $k_{\gamma, i}\left(w^{*}, k_{\alpha, j}\left(w^{*}\right)\right)$, we define by $\partial_{1, w_{1 s}} k_{\gamma, i}\left(w^{*}, k_{\alpha, j}\left(w^{*}\right)\right)$ the partial derivative with respect to price of good $s$ (the $s^{th}$ element of the price vector, which in turn is the first argument of $w$) at $w^{*}$ and by $\partial_{2} k_{\gamma, i}\left(w^{*}, k_{\alpha, j}\left(w^{*}\right)\right)$ the partial derivative with respect to the second argument, i.e. $k_{\alpha, j}\left(w^{*}\right)$.
Furthermore, we write the conditional density $f_{e_{i}^{\prime} Y \mid W, e_{j}^{\prime} Y}\left(y_{i}\right)$ at a point $\left(w^{*}, k_{\alpha, j}\left(w^{*}\right)\right)$ as $f_{e_{i}^{\prime} Y \mid W=w^{*}, e_{j}^{\prime} Y=k_{\alpha, j}\left(w^{*}\right)}\left(y_{i}\right)$. Similarly, for the conditional distribution function $F_{Y_i \mid W, Y_j = y_j}(y_i)$, we write $\nabla_{p,1} F_{Y_i \mid W=w^*, Y_j = y_j}(y_i)$ for the gradient with respect to the price components of $W$ holding the conditioning value $y_j$ fixed, $\partial_{x,1} F_{Y_i \mid W=w^*, Y_j = y_j}(y_i)$ for the partial derivative with respect to income holding $y_j$ fixed, and $\partial_2 F_{Y_i \mid W=w^*, Y_j = y_j}(y_i)$ for the partial derivative with respect to the conditioning value $y_j$. We impose regularity conditions on the conditional distribution function, the conditional density, the conditional quantile and the demand function, which we have relegated to the appendix.
We can now state a multivariate extension of the identification result in Hoderlein2007:
This result is analogous to the univariate version of Hoderlein2007. The difference is that the conditional quantile is affected through several paths when changing $w$, instead of just the marginal $k_{\alpha, j}\left(w\right)$. This results in the additional complex correction terms in the expression. The underlying idea of the proof lies in the fact that the inherently $2$-dimensional quantile function $K_{ij}(\alpha, w)$ can be represented in two different ways by the two respective marginal- and the corresponding conditional quantiles. This representation is unique for regular quantiles.
Lemma (ref) can be straightforwardly extended to more than two dimensions by repeated conditioning. In this case, the expression becomes even more complicated with additional terms; this stems from the fact that for $d$-dimensions, one needs to iteratively condition on $d-1$ quantiles. Since we only care about the Slutsky symmetry case, which is bivariate, we omit the d-dimensional version. Note that this result does not hinge on the generality of Assumption (ref): in applications without endogeneity, this result still applies with a \enquote{reduced form} nonparametric demand model (the second stage model in Assumption (ref)).
Lemma (ref) now enables us to derive the condition on the quantile functions which captures the symmetry condition of the Slutsky matrix. Based on Lemma (ref), the intuition is as follows: the joint quantile $K_{ij}(\alpha|w)$ can be represented uniquely in two ways, corresponding to two different combinations of marginal- and conditional quantiles, which captures the “off-diagonal” terms in the Slutsky matrix via these marginal and corresponding conditional quantile functions. This leads to the following representation:
Equation (ref) gives a necessary condition for Slutsky Symmetry in the empirical setting described by Model (ref) and Assumptions (ref) and (ref). If this condition is not satisfied for any $i,j$ pair of goods under the model assumptions, then the Slutsky symmetry is not satisfied for that set of goods in this model. The complication arises from the correction terms $C_{ij}$, $C_{ji}$, $D_{ij}$, $D_{ji}$, $D_{ij}^{(x)}$, and $D_{ji}^{(x)}$, which do not appear in the negative semidefinite case.
Our results yield new insights for nonparametric welfare analysis with individual heterogeneity in multi-good settings. hausman2016individual show that in a two-good setting with a numeraire, quantile demands coincide with true demands, using the univariate identification result of Hoderlein2007 and the negative semidefiniteness condition of DETTE2016129. In that setting, Slutsky symmetry holds by construction. With more than two goods, symmetry has empirical content, and the univariate identification argument no longer suffices. Existing approaches either assume symmetry outright or impose restrictive conditions such as monotonicity in scalar unobserved heterogeneity.
Our bivariate extension of Hoderlein2007 in Lemma (ref), together with the symmetry condition in (ref), restores the quantile-based approach to welfare analysis under general heterogeneity. Theorem (ref) shows that with vector-valued demands, true demands are identified by conditional and marginal quantiles satisfying (ref). Unlike in hausman2016individual, where marginal quantile demands coincide with true demands directly, conditional quantile demands for each good coincide with true demands only up to the correction factors $C_{ij}(w^*)$, $D_{ij}(w^*)$, and $D_{ij}^{(x)}(w^*)$ induced by conditioning on the demand level of the other good.
\paragraph{Empirical content of the condition} The symmetry condition in Theorem (ref) involves two types of correction terms beyond the leading conditional and marginal quantile derivatives: the terms $C_{ij}(w^*)$ and $D_{ij}(w^*)$ (along with their income counterparts $D_{ij}^{(x)}(w^*)$). These terms have distinct economic content that is worth unpacking, as they govern the gap between conditional quantile demands and true Hicksian demands in multi-good settings.
\paragraph{The $C_{ij}$ terms.} The term $C_{ij}(w^*)$ captures how the conditional quantile of $Y_i$ responds to shifts in the level of $Y_j$ at which we condition. It combines a direct effect ($\partial_2 k_{\gamma_{i|j},i}$, the mechanical response of the conditional quantile to a change in the conditioning value) with an indirect effect through the conditional distribution. Economically, $C_{ij}(w^*)$ is small when the demand for good $i$ at a given quantile is approximately insensitive to the demand level of good $j$. This is plausible when the two goods are weakly related in preferences (for instance, under approximate weak separability) or when each good individually constitutes a small fraction of total expenditure, so that the income reallocation channel linking their demands is muted.
\paragraph{The $D_{ij}$ terms.} The terms $D_{ij}(w^*)$ and $D_{ij}^{(x)}(w^*)$ capture a fundamentally different phenomenon: how the conditional distribution of $Y_i$ given $Y_j = y_j^*$ shifts when prices or income change, holding the conditioning level $y_j^*$ fixed. This effect is absent in the univariate setting of Hoderlein2007, where the independence assumption $A \perp\!\!\!\perp (P,X) \mid Z$ suffices to eliminate any such dependence. In the bivariate setting, the conditional distribution $F_{Y_i | W, Y_j = y_j^*}$ conditions on the level set $\{a : e_j'\phi(p, x, z^*, a) = y_j^*\}$, which shifts with $(p,x)$ through the demand function $\phi$, even though the marginal distribution of the preference heterogeneity $A$ given $Z$ does not depend on prices or income.
Economically, $D_{ij}(w^*) = 0$ when the composition of preference types demanding a fixed quantity $y_j^*$ of good $j$ is stable with respect to price changes. This is a natural condition in settings where the mapping from preferences to demand for good $j$ is approximately one-to-one (so that the level set $\{a : e_j'\phi(\cdot, a) = y_j^*\}$ shifts rigidly rather than changing its composition), or more generally when the heterogeneity in demand for good $i$ among consumers who demand the same quantity of good $j$ is not systematically related to the price sensitivity of demand for good $j$. A sufficient condition for $D_{ij}(w^*) = 0$ for all $w^*$ is that $\phi$ is monotone in the component of $A$ that drives $Y_j$, reducing the bivariate problem to the univariate case of Hoderlein2007. The results of this paper, when integrated with existing negative-semidefiniteness tests, now allow researchers to comprehensively verify Slutsky conditions prior to welfare analysis with unobserved heterogeneity.
We provide a fully nonparametric analysis of the testability of the symmetry of the Slutsky matrix via empirical quantiles, complementing existing work that provides a test for the negative semidefiniteness. This work, therefore, closes the gap on testing an empirically informed version of rationality, providing a testable symmetry condition relevant for nonparametric welfare analysis with individual heterogeneity. Incidentally, the results show that the symmetry condition is testable using conditional quantile restrictions. This shows that a recent conjecture maes2024beyond is false and opens the door to testing a empirically informed version of rationality with heterogeneity. The symmetry condition provides empirical restrictions in empirical welfare analysis. Since it involves only conditional and marginal quantiles and their derivatives, all of its components are estimable from data via conditional quantile inference methods. In the multi-good case, the condition provides nonlinear correction factors that govern the gap between conditional quantile demands and true Hicksian demands, which must be accounted for in any nonparametric welfare analysis with individual heterogeneity beyond the two-good setting.
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