Stochastic Revealed Preferences with Measurement Error

Stochastic Revealed Preferences with Measurement Error

Introduction

One well-known feature of consumer panel data sets---whether they are based on surveys, experiments, or scanners---is measurement error in prices or consumption.\footnote{See mathiowetz2002measurement, echenique2011money, carroll2014introduction, and gillen2015experimenting.} This is of significant concern because measurement error is responsible for one important limitation of the standard revealed-preference (RP) framework. Specifically, RP tests tend to overreject the utility maximization theory (UMT). To the best of our knowledge, this paper proposes the first fully nonparametric statistical RP framework for consumer panel data sets in the presence of measurement error. We show that taking measurement error into account can significantly change the conclusions about the validity of the UMT in a given context. In the two applications we develop, we cannot reject the validity of the UMT, a finding which contradicts the conclusions of the deterministic RP framework. Measurement error is the difference between the unobserved but true value of the variable of interest and its observed but mismeasured counterpart. If the UMT is valid, the corresponding RP conditions must be satisfied by the true prices and consumption. However, there is no reason to believe that either the mismeasured consumption or mismeasured prices that we usually observe are consistent with the RP conditions. A key concern for RP practitioners is that, in the presence of measurement error, a deterministic RP test may overreject the null hypothesis that the UMT is valid. We provide Monte Carlo evidence that this concern may be relevant in practice. Measurement error in consumption may arise in survey data due to misreporting, in experimental data due to trembling-hand errors, and in scanner data due to recording errors.\footnote{Trembling-hand errors are nonsystematic mistakes incurred by a subject when trying to implement a decision because she has difficulties with the interface of an experiment.} Measurement error in prices may arise in experimental data due to misperception errors and in scanner data due to unobserved coupons.\footnote{Misperception errors are nonsystematic mistakes incurred by a subject because she misperceives information due to the experimental design.} Our methodology covers as special cases the static UMT, and the classic dynamic UMT with exponential discounting.\footnote{We also cover firm-cost minimization varian1984nonparametric, dynamic rationalizability with quasi-hyperbolic discounting blownonparametric2017, homothetic rationalizability varian1985non, quasilinear rationalizability brownquasilinear, expected utility maximization diewert2012afriat, and static utility maximization with nonlinear budget constraints forges2009afriats.} When we apply our methodology to a consumer panel survey of Spanish households and allow for measurement error in consumption, we find that the dynamic UMT with exponential discounting cannot be rejected for single-individual households. This first finding contradicts the conclusions of the deterministic RP test of browning1989anonparametric. When we apply our framework to a widely used experimental data set ahn2014estimating and allow for measurement error in prices that arises due to misperception, we find that the static UMT cannot be rejected. This second finding is the opposite of the conclusions that must be drawn when one applies the deterministic RP test of afriat1967construction and varian1982nonparametric to the same data set. Taken together, these findings suggest that the negative conclusions about the validity of the UMT drawn from the deterministic RP framework may not be robust to measurement error.

The leading solution for dealing with measurement error in the RP framework usually consists of perturbing (minimally) any observed individual consumption streams in order to satisfy the conditions of an RP test adams2014consume. However, this approach does not allow for standard statistical hypothesis testing. In particular, one cannot control the probability of erroneously rejecting a particular model when such a rejection could be an artifact of noisy measurements. Other works on RP with measurement error, such as the seminal contribution of varian1985non, allow for statistical hypothesis testing but require knowledge of the distribution of measurement error; this may be impractical because this does not align with the nonparametric nature of the RP framework. In contrast, our procedure does not suffer from these issues. Our main result is the formulation of a statistical test for the null hypothesis that a random data set of mismeasured prices and consumption is consistent with any given model that can be characterized by first-order conditions. Based on this test we provide and implement a general methodology to make out-of-sample predictions or counterfactual analyses with minimal assumptions (e.g., sharp bounds for average or quantile demand). Our approach takes advantage of the work of schennach2014entropic on Entropic Latent Variable Integration via Simulation (ELVIS) to provide a practical implementation of our test. Our RP methodology is fully nonparametric and admits unrestricted heterogeneity in preferences. In addition, we require only a centering condition on the unobserved measurement error. The centering condition captures the application-specific knowledge we have about measurement error. Moreover, our framework is general enough to allow for (i) nonclassical measurement errors in consumption in survey environments, (ii) trembling-hand errors in experimental setups, (iii) misperception of prices due to experimental designs (price measurement error), and (iv) different forms of measurement error in prices in scanner environments. In our first application in particular, we require that consumers be accurate, on average, in recalling and reporting their total expenditures. This assumption is compatible with systematic misreporting of consumption in surveys. For our second application, to an experimental data set, we require that errors in consumption or prices be centered around zero, which is compatible with trembling-hand and misperception errors in subjects' behavior. Measurement error in experimental data sets may arise because the experimental design may fail to elicit the intended choices of consumers. In such a case, classical measurement-error assumptions that allow for nonsystematic mistakes must be taken into account to ensure the external validity of the conclusions drawn from applying any RP test to this type of data set. Our empirical contribution is to apply our methodology to a well-known consumer panel survey of single-individual and couples' households in Spain\footnote{This data set has been used in beatty2011howdemanding,adams2014consume, and blownonparametric2017.} in order to test for the dynamic UMT, and to reexamine the static UMT in a widely used experimental data set ahn2014estimating; in doing this reexamination, we allow (separately) for trembling-hand errors and misperceived prices. For the first application, we note that under the exponential discounting model, the consumer's time preferences are captured by a time-invariant discount factor and a time-invariant instantaneous utility. The main feature of this model is the time-consistency of the exponential discounting consumer. In other words, if the consumer prefers consumption bundle $c$ at time $t$ to $x$ at time $t+k$, then she will always prefer $c$ at time $\tau$ to $x$ at time $\tau+k$. The exponential discounting model remains the workhorse of a large body of applied work in economics. However, several authors, such as browning1989anonparametric, dellavigna2006payingnot, and blownonparametric2017, have provided suggestive evidence against the validity of this model. Our methodology addresses, in a nonparametric fashion, the presence of measurement error in survey data, in order to examine the robustness of these findings. We find support for exponential discounting behavior for single-individual households. This contrasts with the results of applying the deterministic methodology of browning1989anonparametric to the same sample. At the same time, in line with the findings of blownonparametric2017 (who also use the deterministic methodology of browning1989anonparametric), we reject the null hypothesis of exponential discounting for the case of couples. When compared with the single-household evidence, these results suggest that time inconsistencies in consumer behavior in the couples' case arise due to preference aggregation. In our second application, we apply our methodology to test for the validity of the classical static UMT. Since the work of afriat1967construction and varian1982nonparametric, researchers have used the deterministic RP framework to examine the validity of the UMT in experimental data sets. The experimental design proposed by ahn2014estimating is particularly useful for this task since it provides a controlled environment with substantial price variation, variation that guarantees that the UMT has empirical bite. When we apply the deterministic RP test to this experimental data set, we conclude that the UMT is rejected for most subjects. Nonetheless, the external validity of the conclusions drawn from the deterministic RP tests applied to these experimental data sets may be limited. One key reason for this is that the elicitation of consumer behavior may have been subject to measurement error. gillen2015experimenting argue that experimental elicitations of choices are subject to random variation in participants' perception and focus. Moreover, RP practitioners since afriat1967construction have recognized that the deterministic RP test for the static UMT may be too demanding in the presence of imperfect devices for the elicitation of choices. Many researchers have studied how to allow for optimization mistakes in the RP framework and how to measure the intensity of any departure from rationality.\footnote{See afriat1967construction, varian1990goodness, and echenique2011money.} However, none of the existing approaches designed to introduce the possibility of mistakes in the RP framework has allowed for a fully nonparametric approach to doing standard statistical hypothesis testing. In our application, we allow for the possibility of nonsystematic mistakes by requiring that the measurement error in consumption or prices be mean zero. We cannot reject the null hypothesis of the validity of the static UMT with misperception of prices. However, when we allow only for trembling-hand errors in consumption, we must strongly reject the static UMT. Our findings call into question the robustness of the deterministic RP test that is due to afriat1967construction and varian1982nonparametric to measurement error in prices.

Outline

The paper proceeds as follows. Section (ref) presents the first-order conditions approach to the deterministic RP methodology. Section (ref) contains our statistical test. Section (ref) presents a framework for recoverability and counterfactual analysis on the basis of our testing methodology. Section (ref) provides an econometric framework for our methodology. Section (ref) provides a guide specifying the centering condition in different environments. Section (ref) implements our empirical test for the case of the dynamic UMT in a consumer panel-survey data set. Section (ref) implements our methodology for the case of the static UMT in an experimental data set. Section (ref) presents a brief discussion of related literature. Finally, we conclude in Section (ref). All proofs can be found in Appendix (ref).

The Revealed-Preference Methodology and the First-Order Conditions Approach

The main objective of this section is to provide a brief summary in a united fashion of two very important deterministic consumer models and their RP characterization. In particular, we study the static UMT or rational model ($\ensuremath{\mathrm{R}}$), and the dynamic UMT with exponential discounting ($\ensuremath{\mathrm{ED}}$). These models are at the center of many applied and theoretical works. We show that they can be completely characterized by their first-order conditions in an RP fashion. All quantities used here are assumed to be measured precisely. Let the consumption space be ${\mathds{R}}_{+}^{L}\setminus\{0\}$, where $L\in{\mathds{N}}$ is the number of commodities.\footnote{We use ${\mathds{N}}$ to denote the set of natural numbers. The expression ${\mathds{R}}_{+}^L$ denotes the set of componentwise nonnegative elements of the $L$-dimensional Euclidean space ${\mathds{R}}^L$, and ${\mathds{R}}_{+}^L\setminus\{0\}$ denotes the set of vectors $v\in{\mathds{R}}_{+}^L$ that are distinct from zero ($v\neq0$). Similarly, ${\mathds{R}}^L_{++}$ denotes the set of componentwise positive elements of ${\mathds{R}}^L_{+}$. The inner product of two vectors $v_1,v_2\in{\mathds{R}}^L$ is denoted by $v_1^{\prime} v_2$.} Consider a consumer who is endowed with a utility function $u:{\mathds{R}}_{+}^{L}\to{\mathds{R}}$ that is assumed to be concave, locally nonsatiated, and continuous. The consumer faces a sequence of decision problems indexed by $t\in \mathcal{T}$, where $\mathcal{T}=\{0,\cdots,T\}$, with a known and finite $T\in{\mathds{N}}$. At each decision problem $t\in \mathcal{T}$, the consumer faces the price vector $p_{t}\in{\mathds{R}}_{++}^{L}$.

Next we focus on the dynamic UMT. We assume that an individual consumer has preferences over a stream of dated consumption bundles $(c_{t})_{t\in\mathcal{T}}$, where $\mathcal{T}=\{0,\cdots,T\}$, $T\in{\mathds{N}}$, and $c_{t}\in{\mathds{R}}_{+}^{L}\setminus\{0\}$. (The number of goods, $L$, is kept the same across the time interval.) At time $\tau$, the consumer chooses how much $c_{\tau}$ she will consume by maximizing \[ V_{\tau}(c)=u(c_{\tau})+\sum_{j=1}^{T-\tau}d^{j}u(c_{\tau+j}), \] subject to the linear budget or flow constraints shown here: \[ p_{t}^{\prime} c_{t}-y_{t}+s_{t}-a_{t}=0,\quad t=\tau,\dots,T, \] where $d\in(0,1]$ is the discount factor; $p_{t}\in{\mathds{R}}_{++}^{L}$ is the price vector as before; $y_{t}\in{\mathds{R}}_{++}$ is income received by the individual at time $t$; $s_{t}$ is the amount of savings held by the consumer at the end of time $t$; and $a_{t}$ is the volume of assets held at the start of time $t$. The consumer invests all her savings. Moreover, the assets evolve according to the following law of motion: \[ a_{t}=(1+r_{t})s_{t-1}, \] where $r_{t+1}>-1$ is the interest rate that is accessible for the consumer. The holdings of assets in the last period ($t=T$) are set to be zero. The intertemporal value function, $V_{t}:{\mathds{R}}_{+}^{L\times(T-t+1)}\to{\mathds{R}}_{++}$, represents the consumer preferences at a given time $t$. The components of this representation are the parameters of the model. First, $d\in(0,1]$ is a scalar number that measures the degree of discount that the consumer gives to the future. Second, $u:{\mathds{R}}_{+}^{L}\to{\mathds{R}}_{++}$ is an instantaneous utility function that is assumed to be concave, locally nonsatiated, and continuous. The exponential discounting consumer is time-consistent, that is, she will solve the dynamic problem above the same way at any point of the time window.

$\ensuremath{\mathrm{ED}}$-rationalizability implicitly assumes perfect foresight (e.g., individuals know their future income) and homogeneity of consumers within a household (e.g., household members have the same discount factors). In Appendix (ref) we show that our methodology covers two extensions of this model: the dynamic UMT with income uncertainty, and the collective model of adams2014consume.

The First-Order Conditions Approach

Now we establish that any consumer model $\ensuremath{\mathrm{m}}\in\{\ensuremath{\mathrm{R}},\ensuremath{\mathrm{ED}}\}$ can be completely characterized in terms of its first-order conditions with respect to (i) a concave, locally nonsatiated, and continuous utility function $u:{\mathds{R}}_{+}^{L}\to{\mathds{R}}$, (ii) the effective (or transformed) prices $\rho_t^\ensuremath{\mathrm{m}}\in{\mathds{R}}_{++}^{L}$, and (iii) restrictions on some constants $\lambda_t^\ensuremath{\mathrm{m}} \in {\mathds{R}}_{++}$ and $\delta^{\ensuremath{\mathrm{m}}}_t\in(0,1]$, interpreted as the marginal utility of income and the discount rate, respectively. We call this the first-order conditions approach. Observe that the utility function is model-independent, but the effective prices, the marginal utility of income, and the discount rate are not. We define the effective prices in Table (ref).

The following lemma summarizes the results in browning1989anonparametric for the exponential discounting case, and it is trivial for the static rationalizability case. Let $\nabla u(c_t)$ denote a supergradient of $u$ at the point $c_t$.\footnote{The supergradient is $\nabla u(c_t)=\{\xi\in{\mathds{R}}^{L}\::\:u(c)-u(c_t)\leq \xi^{\prime}(c-c_t),\:\forall\:c\in{\mathds{R}}_{+}^{L}\setminus\{0\}\}$. Under differentiability, $\nabla u(c_t)$ is a gradient.}

We want to highlight that while we focus on these two models for expositional and motivational purposes, our methodology is applicable to any model that can be characterized using the first-order conditions approach.

The Elimination of a Latent Infinite-Dimensional Parameter

Since our objective is not to estimate but to test $\ensuremath{\mathrm{m}}$-rationalizability, we will eliminate the utility function $u$ from its characterization. We follow the theorists of RP to eliminate the latent infinite-dimensional parameters by exploiting their shape restrictions. In particular, we follow afriat1967construction, varian1985non, browning1989anonparametric, and rockafellar1970convexanalysis to formulate a result that eliminates the utility function $u$ (an infinite dimensional parameter) from the first-order conditions. The cost of doing this is that we have to replace the first-order conditions by a set of inequalities that require only the concavity of $u$. As a result, the inequalities are exact and do not involve any form of approximation; this is an advantage compared to other nonparametric methods (e.g., sieves, kernel estimators) or the parametric approach used in many applied papers. To formulate our result, we first recall the definition of the concavity of $u$.

The nonparametric characterization of the $\ensuremath{\mathrm{m}}$-rationalizability of observed consumption and prices without measurement error is captured by the following result.

Theorem (ref) summarizes known results from the RP literature.\footnote{The proof is a consequence from the results in afriat1967construction, varian1985non, and browning1989anonparametric taken together.} Observe that Theorem (ref) has transformed the first-order conditions that depend on the infinite-dimensional $u$ to a set of inequalities that depend only on a deterministic finite-dimensional array $(v_{t},\delta_t^{\ensuremath{\mathrm{m}}},\lambda^{\ensuremath{\mathrm{m}}}_{t})_{t\in\mathcal{T}}$. Nonetheless, this set of conditions is satisfied if and only if we can find a utility function that satisfies the conditions in Lemma (ref).\footnote{In contrast to Afriat's theorem for the static UMT, the assumption of concavity of the utility function is necessary in our framework. The reason is that concavity is testable in some cases that are different from static utility maximization (e.g., expected utility maximization, polisson2020revealed). Concavity guarantees that first-order conditions are necessary and sufficient in a wide variety of models beyond the static UMT. For the additional generality our result requires this additional constraint.} Checking the set of inequalities is a parametric problem and it tells us whether a consumption stream is $\ensuremath{\mathrm{m}}$-rationalizable. This methodology is traditionally applied at the individual level in panel data sets, assuming that the data contains no measurement error. In the next section we extend the RP framework to a noisy or stochastic environment.

The Revealed-Preference Approach with Measurement Error

In this section, we introduce a new statistical notion of $\ensuremath{\mathrm{m}}$-rationalizability (henceforth, $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability) with mismeasured consumption or prices, and provide a result similar to Theorem (ref) in the presence of measurement error. From here on, we use boldface font to denote random objects and regular font for deterministic ones.

Statistical Rationalizability

We are interested in testing a statistical model of consumption such that each individual is an independent, identically distributed (i.i.d.) draw from some stochastic consumption rule. Note that by Lemma (ref) the choice of a particular model $\ensuremath{\mathrm{m}}$ only affects the definition of the effective price, and the restrictions on the marginal utility of income and the discount rate. Henceforth, we fix some model such that the effective prices, and the restrictions on the marginal utility of income and the discount rate are known, and we omit the superscript $\ensuremath{\mathrm{m}}$ from the notation. Using Lemma (ref) as motivation, we directly define $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability as follows. Let $\boldsymbol{\rho}^*_t\in P^*_t\subseteq\mathbb{R}^L_{++}$ and $\mathbf{c}^*_t\in C_t^*\subseteq\mathbb{R}^L_+ \setminus{\{0\}}$ denote random vectors of true effective prices and true consumption at time $t$, respectively.\footnote{For short, we use $\ensuremath{\mathrm{a.s.}}$ instead of “almost surely.” We denote (i) the probability of an event $A$ by the expression $\mathds{P}\left(A\right)$; (ii) the indicator function by $\mathds{1}\left(\,A\,\right)=1$ when the statement $A$ is true, otherwise it is zero; (iii) the mathematical expectation of any random vector $\mathbf{z}$ by the expression $\mathds{E}\left[\mathbf{z}\right]$; (iv) the cardinality of a set $\mathcal{A}$ is given by the expression $|\mathcal{A}|$; and (v) the norm of a vector $v$ is given by $\left\lVertv\right\rVert$.}

This definition means that for a given realization of (i) the utility function, (ii) the marginal utility of income, and (iii) the discount rate, the realized effective prices and the realized true consumption should fulfill the inequality $\delta_t\nabla u(c_{t}^{*})\leq\lambda_t \rho_{t}^*$. This is a special case of the dynamic random utility model in which the preferences (captured by $\rand u$), the random discount factor (captured by $(\boldsymbol{\delta}_t)_{t\in \mathcal{T}})$, and the distribution of the marginal utility of income (captured by $(\boldsymbol{\lambda}_t)_{t\in\mathcal{T}})$ are drawn at some initial time for each consumer, and then are kept fixed over time. Several consumer models can be characterized by their first-order conditions and by restrictions on the marginal utility of income, as we observed in Section (ref). For instance, we define the statistical version of $\ensuremath{\mathrm{R}}$-rationalizability or $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{R}}$-rationalizability by requiring that the support of the marginal utility of income be strictly positive (i.e., $\Lambda={\mathds{R}}_{++}^{|\mathcal{T}|}$), and the discount rate to be one (i.e., $\Delta=\{1\}^{|\mathcal{T}|}$). Similarly, we define $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{ED}}$-rationalizability by imposing that $\Lambda=\{1\}^{|\mathcal{T}|}$, and the support $\Delta$ be given by the restriction $\boldsymbol{\delta}_t=\mathbf{d}^{t}$, where $\mathbf{d}$ is a random variable supported on $(0,1]$. The effective prices in each case are to be defined according to Table (ref). Given the definition of $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability, we can now formulate the stochastic version of Theorem (ref).

Lemma (ref) allows us to statistically test the $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability of $(\boldsymbol{\rho}^*_{t},\rand c_{t}^{*})_{t\in\mathcal{T}}$. However, as the following example demonstrates, any test based on this notion of rationalizability cannot differentiate between “almost” $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability and exact $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizabilty (an issue first identified by Galichon2013db).

Example (ref) presents a random choice rule that is not $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizable, but is arbitrarily close to being $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizable. In other words, there may exist sequences of random arrays that are not $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizable that converge to random arrays that are $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizable. That is why we need to extend the notion of the consistency of a data set that is characterized by $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability.

Introducing Measurement Error

Theorem (ref) and Lemma (ref) provide testable implications of $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability. These implications depend solely on the distribution of $\boldsymbol{\lambda}=(\boldsymbol{\lambda}_t)_{t\in\mathcal{T}}$, $\boldsymbol{\delta}=(\boldsymbol{\delta}_t)_{t\in\mathcal{T}}$ and $\rand v=(\rand v_{t})_{t\in\mathcal{T}}$. The usual approach to testing $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability would amount to solving a (non)linear programming problem corresponding to Theorem (ref) at the level of individual consumers. However, this common practice does not work any more in the presence of measurement error. When true consumption or true prices are measured erroneously, we observe not $\rand c_{t}^{*}$ or $\boldsymbol{\rho}^*_t$ but rather perturbed versions of them. (See Section (ref) for a discussion of the reasons measurement error in consumption and prices arises in survey, experimental, and scanner data sets.) Define the measurement error $\mathbf{w}=(\rand w_{t})_{t\in\mathcal{T}}\in W$ as the difference between reported consumption and prices, $\mathbf{c}=(\rand c_t)_{t\in\mathcal{T}}$ and $\boldsymbol{\rho}=(\boldsymbol{\rho}_t)_{t\in\mathcal{T}}$; and true consumption and prices, $(\rand c_t^*)_{t\in\mathcal{T}}$ and $(\boldsymbol{\rho}_t^*)_{t\in\mathcal{T}}$. That is, \[ \rand w_t=\left(

\right), \] where $\rand w_{t}^{c}=\rand c_{t}-\rand c_{t}^{*}$ and $\rand w_{t}^{p}=\rands \rho_{t}-\rands \rho_{t}^{*}$ for all $t\in\mathcal{T}$. It is important to note that we define the measurement error. We do not make any assumptions about how the difference between observed and true quantities arises (i.e., we allow for measurement error to be multiplicative or additive).\footnote{Formally, this makes the support $W$ depend on the support of both the observed and the true quantities. For simplicity we omit this dependency from the notation.} Moreover, we do not need to assume that measurement error is independent of other variables, independent within time periods, or independent across goods. By Lemma (ref) we can immediately conclude that the observed $\mathbf{x}=(\boldsymbol{\rho}_{t},\rand c_{t})_{t\in\mathcal{T}}$ can be $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalized if and only if there exist $(\boldsymbol{\lambda}_t, \boldsymbol{\delta}_t,\rand v_t,\mathbf{w}_t)_{t\in\mathcal{T}}$, with $(\boldsymbol{\lambda}_t)_{t\in\mathcal{T}}$ supported on or inside $\Lambda$, and $(\boldsymbol{\delta}_t)_{t\in\mathcal{T}}$ supported on or inside $\Delta$ such that \[ \mathbf{v}_t-\mathbf{v}_s\geq\dfrac{\boldsymbol{\lambda}_t}{\boldsymbol{\delta}_t}(\boldsymbol{\rho}_{t}-\rand w^p_t)^{\prime}(\mathbf{c}_{t}-\mathbf{c}_{s}+\mathbf{w}^c_{s}-\mathbf{w}^c_{t})\quad\ensuremath{\mathrm{a.s.}},\forall s,t\in\mathcal{T}. \] However, we know that without restrictions on the distribution of measurement error, RP tests have no power. That is, there always exists a measurement error $\rand w$ such that the observed $\rand x$ is consistent with $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability. Hence, we require access to additional validation information about measurement error. The source of measurement error is different in different applications. That is why in this section we formulate a general restriction on the measurement error distribution that can be tailored for a given empirical application. Recall that $\mathbf{x}\in X$ denotes observed quantities. Let $\mathbf{e}=(\rands\lambda^{\prime},\boldsymbol{\delta}^{\prime},\rand v^{\prime},\rand w^{\prime})^{\prime}\in E|X$ denote the vector of latent random variables, supported on or inside the conditional support $E|X$. We say that a mapping $g_{M}:X\times E|X\to{\mathds{R}}^{d_{M}}$ is a measurement error moment. We only require the following condition on measurement error.

The choice of function $g_M$ depends on the application and the assumptions the researcher is willing to make on the basis of the available knowledge about the nature of measurement error. In Section (ref) we provide examples of moment conditions in data sets that are often used in the RP literature. The objects of interest for us are measurement error in consumption, expenditure, and prices.

Recoverability and Counterfactuals

varian1982nonparametric,varian1984nonparametric exploits the connections between empirical content and counterfactuals. In particular, varian1982nonparametric seems to be the first to think of nonparametric counterfactual analysis as specification testing.\footnote{Recent work building on these connections includes blundell2003nonparametric and allen2019identification in demand analysis, and norets2014semiparametric in the analysis of dynamic binary choice models.} Following these ideas, our formulation of rationalizability allows us to answer questions about the recoverability of, and counterfactual predictions for, different objects of interest. In Section (ref) we show how to recover different quantities of interest (e.g., average true consumption at a given $t=\tau$) from the $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizable data set. In Section (ref) we demonstrate how to make out-of-sample predictions for expected consumption in a way that is analogous to blundell2014bounding. In the presence of measurement error, distributional information about the primitives of the model of interest is inevitably lost. Hence, we cannot apply the traditional approach proposed by varian1982nonparametric to recover preferences and to do counterfactual analysis on an individual basis. Instead, we use this section to pose questions about the primitives of the model at the level of the population.

Recoverability

Assume that $\mathbf{x}=(\boldsymbol{\rho}_{t},\rand c_{t})_{t\in\mathcal{T}}$ can be $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalized and Assumption (ref) holds. Suppose that there is a finite-dimensional parameter of interest $\theta_0\in\Theta$, where $\Theta$ is a compact subset of the Euclidean space. The parameter of interest is related to the model via the user-specified moment condition \[ \mathds{E}\left[g_{R}(\mathbf{x},\mathbf{e};\theta_0)\right]=0\in{\mathds{R}}^{d_{R}}. \] The function $g_{R}$ can take different forms depending on the different questions the user wants to answer. We provide some examples here.

The user may also be interested in testing the joint null hypothesis that (i) the consumer is $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{ED}}$-rationalizable and (ii) the random discount factor distribution has certain properties.

In addition, our framework allows us to have, as a special case, latent random variables with flexible support.

Counterfactual Out-of-Sample Predictions

We consider a counterfactual situation in which the user is given an out-of-sample effective random price vector $\boldsymbol{\rho}^*_{T+1}$ (supported in ${\mathds{R}}_{++}^L$), a data set $\mathbf{x}=(\boldsymbol{\rho}_{t},\rand c_{t})_{t\in\mathcal{T}}$ such that Assumption (ref) holds, and she then asks two related questions. First, the user wants to know if there exists a counterfactual random consumption vector $\rand c^*_{T+1}$ such that the augmented random array $\{(\boldsymbol{\rho}^*_{t},\mathbf{c}^*_{t})_{t\in\mathcal{T}}, (\boldsymbol{\rho}^*_{T+1},\mathbf{c}^*_{T+1})\}$ is approximately $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizable, where $\mathbf{c}^*_t=\mathbf{c}_t-\mathbf{w}^c_t$ and $\boldsymbol{\rho}^*_t=\boldsymbol{\rho}_t-\mathbf{w}^p_t$. Second, if the answer to the first question is affirmative, then the user will be interested in constructing confidence sets for some counterfactual finite-dimensional parameter $\theta_0\in\Theta$. The parameter $\theta_0$ satisfies the user-specified moment condition \[ \mathds{E}\left[g_C\left((\boldsymbol{\rho}^*_{t},\rand c^*_{t})_{t\in\mathcal{T}},\mathbf{c}^*_{T+1};\boldsymbol{\rho}^*_{T+1},\theta_0\right)\right]=0\in{\mathds{R}}^{d_{C}}. \] Both questions can be answered simultaneously with our characterization of $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability. Observe that the answer to the first question is negative if the random array $(\boldsymbol{\rho}^*_{t},\rand c^*_{t})_{t\in\mathcal{T}}$ is not $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizable. In contrast, if the random array $(\boldsymbol{\rho}^*_{t},\rand c^*_{t})_{t\in\mathcal{T}}$ is $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizable, then the counterfactual exercise is equivalent to checking that the counterfactual price/consumption distribution is simultaneously compatible with $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability and the user-specified moment condition. Formally, to answer both questions, we define what it means for a random array $(\boldsymbol{\rho}^*_{t},\rand c^*_{t})_{t\in\mathcal{T}}$ to be counterfactually rationalizable ($\ensuremath{\mathrm{C}}/\ensuremath{\mathrm{m}}$-rationalizability) for a given $\boldsymbol{\rho}^*_{T+1}$, $\theta_0$, and $g_C$.

Observe that if a random array $(\boldsymbol{\rho}^*_{t},\rand c^*_{t})_{t\in\mathcal{T}}$ is $\ensuremath{\mathrm{C}}/\ensuremath{\mathrm{m}}$-rationalizable for a given $(\boldsymbol{\rho}^*_{T+1},\mathbf{c}_{T+1}^*)$ and $\theta_0$, then it is also $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizable. However, the opposite is not always true. We can apply Lemma (ref) to Definition (ref) and get an extended system of the RP inequalities coupled with the counterfactual moment conditions $g_C$. Moreover, we can define an identified set for counterfactual parameter values $\Theta_0$. Formally, \[ \Theta_0=\left\{\theta_0\in\Theta\::\: \text{$(\boldsymbol{\rho}^*_{t},\rand c^*_{t})_{t\in\mathcal{T}}$ is approximately $\ensuremath{\mathrm{C}}/\ensuremath{\mathrm{m}}$-rationalizable given $\boldsymbol{\rho}^*_{T+1}$, $\theta_0$, and $g_C$}\right\}. \] We highlight that our framework can accommodate additional support restrictions on the counterfactual objects. A classical support constraint is a target out-of-sample expenditure level (i.e., $\boldsymbol{\rho}^{*\prime}_{T+1}\mathbf{c}^{*}_{T+1}=1\:\ensuremath{\mathrm{a.s.}}$) as in varian1982nonparametric. We omit these constraints from our discussion to simplify exposition.

Some counterfactual questions (e.g., the Average Varian Support Set) lead to convex identified sets $\Theta_0$.

Proposition (ref) imposes two restrictions on $g_C$: (i) additive separability between $\mathbf{c}^*_{T+1}$ and $\theta$; (ii) affinity of the moment condition in $\theta$. In Section (ref) we provide a framework to construct confidence sets for the counterfactual parameters by means of the test inversion. Convexity of $\Theta_0$ can substantially simplify the computation of the confidence sets since one does not need to conduct test inversion at every point of the parameter space.

Econometric Framework

In Sections (ref) and (ref) we showed how testing, recoverability, and counterfactuals in RP models with measurement error can be framed in the form of moment conditions. In this section we recast the empirical content of the RP inequalities in a form amenable to statistical testing. To simplify the exposition we will focus on testing $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability in the presence of measurement error ($g_M$ and $g_I$ only).

Characterization of the Model via Moment Conditions

First, we write a set of moment conditions that will summarize the empirical content of $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability. Recall that $\rand x\in X$ denotes observed quantities and $\rand e=(\rands\lambda^{\prime},\rands \delta^{\prime},\rand v^{\prime},\rand w^{\prime})^{\prime}\in E|X$ denote the vector of latent random variables. The support $E|X$ depends on the fixed supports $\Lambda$ and $\Delta$ that characterizes the particular model of interest. We use $\mathcal{P}_{X}$, $\mathcal{P}_{E,X}$, and $\mathcal{P}_{E|X}$ to denote the set of all probability measures defined over the support of $\rand x$, $(\rand e^{\prime},\rand x^{\prime})^{\prime}$, and $\mathbf{e}|\mathbf{x}$, respectively. (Recall that the boldface font letters denote random objects.) Define the following moment functions:

We have $k=\left\lvert\mathcal T\right\rvert^2-\left\lvert\mathcal T\right\rvert$ and $q=d_M$ moment functions which correspond to inequality conditions ($g_{I}$) and the measurement error centering conditions ($g_{M}$), respectively.\footnote{If in addition, the user includes moments $g_{R}$ or $g_C$, then $q=d_M+d_R$ or $q=d_M+d_C$, respectively.} Define $\mathds{E}_{\mu\times\pi}\left[\,g(\mathbf{x},\mathbf{e})\,\right]=\int_{X}\int_{E|X}g(x,e)d\mu d\pi$, where $\mu\in\mathcal{P}_{E|X}$ and $\pi\in\mathcal{P}_{X}$.

Theorem (ref) establishes the equivalence between (i) $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability with the centered measurement error condition and (ii) a system of moment conditions. In other words, the observed consumption pattern, captured by the random array $(\boldsymbol{\rho}_{t},\rand c_{t})_{t\in\mathcal{T}}$, can be $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalized under the restrictions on measurement error if and only if there exists a distribution of latent variables conditional on observables that satisfies the RP inequalities with probability 1, for the given supports $\Lambda$ and $\Delta$. Our notion of $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability makes clear that when one is dealing with measurement error, no RP test can decide whether a finite sample is consistent with model $\ensuremath{\mathrm{m}}$. We can decide only that the data set is asymptotically consistent with the model as the sample size goes to infinity. Moreover, even asymptotically, there is no way to differentiate between the notion of approximate $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability and the notion of exact $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability. Nonetheless, we can do traditional hypothesis testing and decide at a fixed significance level whether we reject the null hypothesis of (approximate) model $\ensuremath{\mathrm{m}}$ consistency under Assumption (ref) for a given sample. Conceptually, our notion of rationalizability corresponds to the extended notion of an identified set in schennach2014entropic. Note that the test is not yet formally established. We have a set of latent random variables $\rand e$ distributed according to an unknown $\mu\in\mathcal{P}_{E|X}$. This problem can be solved nonparametrically using the Entropic Latent Variable Integration via Simulation (ELVIS) of schennach2014entropic. The main advantage of the ELVIS approach is that it allows us to formulate a test that can be implemented in panel data sets suffering from measurement error of the type described only in terms of observables.

ELVIS and Its Implications for Testing and Inference

We start this section by showing how the nonparametric results of Theorem (ref) can be used to construct a set of (equivalent) parametric maximum-entropy moment conditions using schennach2014entropic. Next, we provide a semi-analytic solution to the these conditions. Finally, we propose a procedure to test for $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability. Following schennach2014entropic, we define the maximum-entropy moment as follows.

Note that \[ \left\{d\eta^*(\cdot|x;\gamma)=\dfrac{\exp(\gamma^{\prime} g(x,\cdot))d\eta(\cdot|x)}{\int_{e\in E|X}\exp(\gamma^{\prime} g(x,e))d\eta(e|x)},\:\gamma\in{\mathds{R}}^{k+q}\right\} \] is a family of exponential conditional probability measures. Thus, the maximum-entropy moment $h$ is the marginal moment of the function $g$, at which the latent variable has been integrated out using one of the members from the above exponential family. The importance of the maximum-entropy moment is captured in the following result.

The idea behind Theorem (ref) is that if there exists a distribution that satisfies the moment condition, then there must be another distribution that (i) belongs to a particular finite-dimensional exponential family and (ii) satisfies the same moment condition. Since we are only interested in the existence of the former distribution, instead of searching over the set of all possible distributions we are going to only search over this “smaller” exponential family. We emphasize that Theorem (ref) provides both necessary and sufficient conditions for the observed data to be (approximately) $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizable. This represents an important gain in power with respect to any of the averaging-based tests of RP models that are usually used in the presence of measurement error. High-level technical assumptions can ensure that the sequence of random latent variables that approximates model $\ensuremath{\mathrm{m}}$ converges to a proper random variable. Thus, this limiting random variable would ensure (i) that the infimum in Theorem (ref) is attained, and (ii) that the notion of approximate rationalizability collapses to exact rationalizability. However, this obscures the fact that any assumption made in that direction has no testable implications. The remarkable advantage of applying the results of schennach2014entropic to the RP approach is that it marginalizes out the latent random variables. More importantly, we have a robust statistical framework with which to test our models in the presence of measurement error. In particular, we have not made any strong distributional assumptions about $\boldsymbol{\lambda}$, $\boldsymbol{\delta}$ or $\rand u$ (the heterogeneous tastes). The only restrictions are the concavity assumption on the utility function, and a centering condition on measurement error.\footnote{At this point, we can use as an alternative the methodology presented by ekeland2010optimal to deal with latent variables in our moment conditions.}

Semi-analytic Solution for the Maximum-entropy Moment

One can directly employ the maximum-entropy moment in Theorem (ref) to test model $\ensuremath{\mathrm{m}}$. However, doing so is potentially problematic. One possible concern is the fact that the number of maximum-entropy moments corresponding to the $(g_I)$ conditions, $k=\left\lvert\mathcal{T}\right\rvert^{2}-\left\lvert\mathcal{T}\right\rvert$, grows quadratically with $\left\lvert\mathcal{T}\right\rvert$. Moreover, $\gamma_0$, the nuisance parameter value at which infimum is achieved, may be set-identified when unbounded (e.g., some of the components of $\gamma_0$ may be equal to infinity\footnote{By “equal to infinity” we mean that the infimum is achieved along the sequence of $\gamma$ that diverges to infinity along some coordinates.}), which would therefore lead to nonstandard testing procedures. Here we show that there exists a semi-analytic solution to the optimization problem where every component of $\gamma_0$ that corresponds to the RP inequality constraints is equal to $+\infty$, and every component of $\gamma_0$ that corresponds to the measurement error centering constraint is finite and unique. Thus, for testing purposes (under the null hypothesis of model $\ensuremath{\mathrm{m}}$), we can minimize an objective function over a parameter space of lower dimensionality.

Assumption (ref) rules out cases in which there is no measurement error and allows us to a have a unique minimizer of the objective function. It can be relaxed since our methodology still works for cases without measurement error, but in those cases, it is preferable to use the equivalent deterministic RP benchmark. Assumption (ref) is made to simplify the analysis and can be replaced by tail restrictions on the distribution of $\mathbf{x}$. Note that the user-specified distribution $\eta$ should obey the same restrictions as the unknown distribution of latent $\mathbf{e}$. Thus, we impose the following restrictions on $\eta$:

The first condition in Definition (ref) requires that the support of $\eta$ allows the inequalities to be satisfied. The second and third conditions are regularity conditions. This is a definition and not an assumption, as we can always construct an allowable $\eta$.\footnote{schennach2014entropic provides a generic construction that can be used here.} We are ready to present our main result.

The intuition behind Theorem (ref) is that the RP inequalities presented here restrict only the conditional support of the latent variables (including the measurement error). Hence, given the support restrictions captured by the RP inequalities, only the centering condition comes in the form of moments.\footnote{If Assumption (ref) is violated for some component $j$ of $g_M$ (e.g., $\sup_{E'}g_{M,j}(x,e)\geq0\:\ensuremath{\mathrm{a.s.}}$ for all $E'\subseteq E|X$), then since $\mathds{E}\left[g_{M,j}(\mathbf{x},\mathbf{e})\right]=0$ it has to be the case that $g_{M,j}(\mathbf{e},\mathbf{x})=0\:\ensuremath{\mathrm{a.s.}}$. Thus, $g_{M,j}$ becomes a condition similar to $g_{I}$ that can be replaced by a support restriction.} Theorem (ref) substantially simplifies the conclusion of Theorem (ref). First, we need to minimize the objective function over a much smaller parameter space (${\mathds{R}}^q$ instead of ${\mathds{R}}^{k+q}$). Thus, the problem becomes computationally tractable. Second, if the data is consistent with $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability, then the minimizer, $\gamma_{0,M}$, has to be finite and unique. Finally, to compute $\tilde{h}_{M}(x,\gamma_M)$ in applications, one may need to use Markov Chain Monte Carlo (MCMC) methods by sampling from $\eta$. Theorem (ref) implies that it suffices to sample from $d\tilde{\eta}(\cdot|x)=\mathds{1}\left(\,g_{I}(x,\cdot)=0\,\right)d\eta(\cdot|x)$. The straightforward way to sample from $\tilde\eta$ is to sample from $\eta$ and then reject the draw if it does not satisfy the RP inequalities captured by $\mathds{1}\left(\,g_{I}(x,\cdot)=0\,\right)$. The last part usually amounts to solving a linear program for $\ensuremath{\mathrm{R}}$-rationalizability. For the case where this rejection sampling is not efficient, in Appendix (ref), we describe a new hit-and-run algorithm to sample from $\tilde{\eta}(\cdot|x)$ directly.\footnote{The classical hit-and-run algorithm is an efficient MCMC method that generates uniform draws from a convex polytope. It initiates inside the polytope and proceeds by randomizing directions and then advancing a random distance while remaining inside the set.} This approach is particularly useful for testing $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{ED}}$-rationalizability in one of our applications.

Testing

Theorem (ref) provides moment conditions that are necessary and sufficient for the data $\{\rand x_{i}\}_{i=1}^n = \{(\boldsymbol{\rho}_{t,i},\rand c_{t,i})_{t\in\mathcal{T}}\}_{i=1}^n$ (where $n$ is the sample size), to be approximately consistent with $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability. Now, define the following sample analogues of the maximum-entropy moment and its variance:

Let $\Omega^{-}$ denote the generalized inverse of the matrix $\Omega$. The testing procedure we propose is due to schennach2014entropic and is based on this test statistic: \[ \mathrm{TS}_{n}=n\inf_{\gamma\in{{\mathds{R}}}^{q}}\hat{\tilde h}_M(\gamma)^{\prime}\hat{\tilde\Omega}^{-}(\gamma)\hat{\tilde h}_M(\gamma). \]

We conclude this section by noting that we can speed up computations of the test statistic by obtaining an initial guess for the minimizer of the objective function efficiently. We make use of the fact that, although the objective function $\hat{\tilde h}_M(\cdot)^{\prime}\hat{\tilde\Omega}^{-}(\cdot)\hat{\tilde h}_M(\cdot)$ may have several local minima, the quadratic form $\hat{\tilde h}_M(\cdot)^{\prime} B \hat{\tilde h}_M(\cdot)$, where $B$ is any conformable positive definite matrix, has a unique global minimum and has no other local minima.\footnote{Additional details about the computational aspects of our methodology can be found in Appendix (ref).}

Confidence Sets for Parameters of Interest

The above testing procedure can be modified for construction of confidence sets for parameter $\theta_0$ from Section (ref). In particular, recall that one just needs to extend the set of the original moment conditions (the centering condition $g_M$ and the RP inequalities $g_I$) by $g_R$ or $g_C$ and add (if needed) extra RP inequalities that correspond to $(\boldsymbol{\rho}^*_{T+1}, \mathbf{c}^*_{T+1})$. As in Section (ref) we can then define $\mathrm{TS}_{n}(\theta)$ as the value of the test statistic computed for a fixed value of $\theta$. Under assumptions similar to those of Theorem (ref), the confidence set for $\theta_0$ can be obtained by inverting $\mathrm{TS}_{n}(\theta_{0})$. That is, the $(1-\alpha)$-confidence set for $\theta_0$ is \[ \{\theta_{0}\in\Theta:\mathrm{TS}_{n}(\theta_{0})\leq\chi_{q_{\mathrm{ext}},1-\alpha}^{2}\}, \] where $\chi_{q_{\mathrm{ext}},1-\alpha}^{2}$ denotes the $(1-\alpha)$ quantile of the $\chi^{2}$ distribution with $q_{\mathrm{ext}}$ degrees of freedom ($\chi_{q_{\mathrm{ext}}}^{2}$). The number $q_{\mathrm{ext}}$ is determined by the number of the non RP moment conditions in the extended system that are not the support restrictions. Note that we do not pretest for $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability in order to construct the confidence set for $\theta_0$. If the data set is not $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizable, then the confidence set will be empty asymptotically.

Measurement Error in Different Data Sets

Recall that to apply our methodology we need measurement-error restrictions that come in the form of moments $g_{M}$ (Assumption (ref)). These moments capture the knowledge the user has about measurement error in a particular data set. In this section we provide examples of such centering conditions in different data sets relevant for the RP literature.

Measurement Error in Survey Data Sets

Measurement error in surveys may arise because of errors due to the respondent, the interviewer, or the survey design itself carroll2014introduction. Recall mistakes, social desirability, and recording errors, among other factors, may cause measurement error in surveys (see meyer2015household). Household surveys usually measure expenditure across different goods, but do not measure consumption or prices directly. Prices are sometimes price indexes constructed by a national statistics agency.\footnote{For instance, in data sets like the ones used in blundell2003nonparametric,adams2014consume, and kitamura2018nonparametric, price indexes are collected through direct observation in a given market and then merged with the expenditure survey data sets.} Consumption is generated by dividing reported consumption expenditures in surveys by the observed prices aggregated to a category of goods (e.g., blundell2003nonparametric,adams2014consume, and kitamura2018nonparametric). There is evidence that individuals, in either recall or self-reported surveys, can systematically overreport or underreport their expenditure on different categories of goods mathiowetz2002measurement, carroll2014introduction, pistaferri2015household. In sum, both prices and consumption may have measurement error. Consumption quantities are generated by dividing expenditures for a given good by its price index. This means that in the survey environment, measurement error in consumption is often nonclassical because it may combine measurement error of expenditures and prices in a nonlinear way attanasio2014nonclassical. Nonetheless, we will argue that average total expenditures is well-measured in some surveys. abildgren2018consistency find that mean total expenditures are usually well-measured in the Danish interview-based household budgets survey (DHS, 2015). They use administrative registry information of the same households interviewed in the survey of interest to conclude that there are no statistically significant differences between the mean total expenditure in Danish households as reported in the registry data set and in the household survey. In addition, kolsrud2017studying, using administrative registry consumption data from Sweden, also finds that mean household expenditures match those computed on the basis of household surveys. One potential caveat to this evidence is that registry data itself may be mismeasured. However, direct evidence on household expenditures from retail data shows that total household expenditure error in registry data has zero mean baker2018measurement.\footnote{This pattern of measurement error in expenditure is also indirectly supported by evidence on measurement error in income. For instance, angel2018differences, using data from Austria, show that the difference in household income reported from surveys and that from registry data displays mean-reverting behavior, as high income levels are underreported and low income levels are overreported. This evidence is consistent with the literature on measurement error in earnings bound_chapter_2001.} Based on this evidence for measurement error, the analyst can impose the following condition. For all $t\in\mathcal{T}$ it must be the case that

Equation ((ref)) provides a measurement error moment that allows for nonclassical measurement error in consumption in the survey environment. In particular, it requires only that total expenditure measurement error be nonsystematic. Equation ((ref)) implies that measurement error in consumption does not alter the mean value of total expenditures. In other words, it captures the idea that consumers, on average, may remember the total expenditure level better than the actual details. Other surveys, such as the Spanish Continuous Expenditure Survey (1985-1997), used in beatty2011howdemanding, adams2014consume, and in one of our applications, have documented both over- and underreporting of income. Unfortunately, there is no validation data on expenditure itself (the object of our interest). However, gradin2008inequality report that for the time period of interest the evolution of the mean household income and the mean household expenditure for Spain are very similar. Spanish, Swedish, and Danish households self-report expenditures for a number of good categories in their respective household surveys. We see no reason to believe that Spanish subjects will behave differently from Danish ones, given the similar format of the surveys. For that reason, we believe it is reasonable to assume that the mean expenditure is usually well measured in the Spanish household survey. However, in practice, the condition in Equation ((ref)) may not be restrictive enough (i.e., the joint hypothesis of this assumption and $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationality may not be rejected in general). We remind the reader that the measurement error moment is not the result of a modeling decision but should capture additional information that is usually obtained from validation exercises. Consequently, if we know too little about measurement error, then our data will not suffice to ensure a reasonable statistical power. A natural solution to this problem is to provide additional restrictions on measurement error by collecting new information about it. These additional restrictions will necessarily apply to specific settings or data sets. A particular case of the condition captured by Equation ((ref)), with the additional support constraint that prices are perfectly measured (i.e., $\boldsymbol{\rho}^*_t=\boldsymbol{\rho}_t\:\ensuremath{\mathrm{a.s.}}$ for all $t\in \mathcal{T}$), is:

\setcounter{assumption}{1}

Assumption (ref).(ref) requires that measurement error in consumption be orthogonal to the effective prices. We maintain the assumption that prices are well measured in the rest of this subsection. We have three reasons for doing this. (i) This assumption is a common requirement in most of the work dedicated to the study of consumption decisions in households using survey data. In fact, we follow several works interested in modeling consumption in assuming that total expenditures and prices are measured better (i.e., with a higher signal-to-noise ratio) than the generated consumption variable.\footnote{For examples of papers assuming that prices are perfectly measured and that measurement error is present only in consumption, see cochrane1991simple, ventura1994note, carroll2001death, ludvigson2001approximation, alanbrowning2009estimating, adams2014consume, carroll2014introduction, toda2015double, and alan2018euler.} (ii) It is empirically plausible that prices are better measured than consumption. In fact, there is evidence castillo1999medicion,guerrero2008sesgos that the price index in Spain (which we use in one of our applications) is usually well measured with an estimated upward bias of the total price index of $0.07$ percent for the time window $1981-1991$ castillo1999medicion, which is arguably small.\footnote{The bias is defined as the difference between the Laspeyres price index as used in Spain and the ideal Fisher price index diewert1998index. This is usually known as the substitution or aggregation bias.} For context, the estimated bias for the US in the well-known Boskin report was $0.40$ percent for a similar time period boskin1998consumer. In contrast, there is agreement that consumption measurement in surveys is very noisy alan2018euler.\footnote{For instance, alan2010estimating estimate that roughly $80$ percent of the variation in consumption growth rate is due to measurement error (this is for the PSID in the US).} (iii) We show that $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}-$rationalizability is robust to measurement error when it is local (Appendix (ref)). Hence, we will focus on the main source of measurement error that is consumption in the case of surveys. In sum, in the self-reported expenditures survey panel data sets, we argue that the main source of measurement error is in the consumption variable. The mean-budget neutrality assumption is still compatible with nonclassical measurement error in consumption, such as $\mathbf{w}^{c}_{l,t}\leq0\:\ensuremath{\mathrm{a.s.}}$ ($\mathbf{w}^c_{l,t}\geq0\:\ensuremath{\mathrm{a.s.}}$) for some good category $l$ and all decision tasks $t$. That is, this condition allows agents on average to underreport (overreport) consumption on some categories of goods as long as they on average equally overreport (underreport) consumption on other categories of goods. Mean-budget neutrality will fail if measurement error in expenditures is systematic (i.e., every consumer overreports expenditures or underreports expenditures simultaneously). We want to highlight the fact that despite its simplicity, Assumption (ref).(ref) is a generalization of commonly held parametric assumptions about measurement error in the study of consumption. Here we provide some examples.

We conclude this section by noting that even if prices are measured correctly, there is a potential source of error coming from the fact that price indexes are the result of aggregation of commodities into categories. This aggregation implies that price indexes do not exactly reflect actual prices faced by consumers. Thus, the budget sets faced by the consumers may be different from the ones implied by price indexes.\footnote{We thank a referee for pointing this out.} This problem is common to all demand analysis using survey data sets. Nonetheless, separability of the utility function or homotheticity within a category are examples of conditions under which commodity aggregation produces consumption and price indexes that are consistent with utility maximization jerison1994commodity, lewbel1996aggregation. If one is willing to impose such conditions, then a rejection of utility maximization, even in the presence of commodity aggregation error, means that rationality can be rejected at the disaggregated level.\footnote{Recent work by sato2020large provides nonparametric RP evidence in favor of weak separability and validity of several widely used price indexes for food and beverages categories using data from Japan. In addition, a survey by shumway2001does finds evidence in favor of the conditions in lewbel1996aggregation.} In addition, a nonrejection of the UMT means that price indexes and consumption behave as if they are rational. That is, the substitution patterns observed in the data after commodity aggregation and their relation with relative changes in price indexes can be summarized by the first-order condition approach.\footnote{Another possible interpretation of our analysis in survey environments is that we are testing for utility-maximizing behavior in a budgeting problem. In the budgeting problem, the consumer allocates her income into different good categories to maximize her utility at that level of aggregation. Then, the consumer maximizes utility within different categories. This nested maximization problem is similar to mental accounting thaler1985mental in its modern interpretation by montgomery2019structural. } The aggregation error is not the only additional source of error in prices in survey data. For instance, since prices and expenditures are measured in different surveys, the observed prices in the first survey most likely are not those faced by some of the consumers in the second survey.\footnote{For instance, gaddis2016 documents that prices are sometimes collected by statistical offices in urban areas only, and that food prices can differ between urban and rural areas. However, price indexes constructed from these prices are matched with household expenditures surveys from both urban and rural areas.} Further investigation of these additional sources of error are left for future research.

Measurement Error in Consumption or Prices in Experimental Data Sets

Measurement error in consumption or prices in experimental data sets arise due to difficulties in eliciting the intended choices of the experimental subjects. Indeed, the experimental elicitation of choice may be subject to random variation due to (i) the subject's misperception of some elements of the task, (ii) the level of understanding of the experimental design, and (iii) nonsystematic mistakes in implementing intended choice. In general, there is an imperfect relation between the elicited choice and the intended choice behavior that the experiment tries to measure gillen2015experimenting.

We consider two sources of measurement error in the experimental environment in the context of the budget allocation task due to choi2014more and ahn2014estimating (which we use in our second application). The first one is the possibility of measurement error in consumption due to trembling-hand errors and the second one is measurement error in prices due to misperception. We capture the relation between the true intended choices and the measured choices with the following assumption that allows for trembling-hand errors.

Assumption (ref).(ref) requires that measurement error in consumption be nonsystematic or equivalently centered around zero. Formally, for all $t\in\mathcal{T}$ it must be that \[ \mathds{E}\left[\mathbf{w}^c_t\right]=0. \] We use data from kurtz2019neural to provide direct empirical evidence supporting Assumption (ref).(ref). In a setup with two goods kurtz2019neural implement a motor task by asking their subjects to click on a visual target located on a budget line. Their experimental interface is exactly the same as in choi2014more. kurtz2019neural record the target coordinates and the actual coordinates that subjects choose on a screen using a mouse. The difference between the target coordinates and the actual choice is the trembling-hand error. Using this data set, we verified that the average trembling-hand error is statistically not different from zero.\footnote{Formally, we did a t-test of Assumption (ref).(ref) across $23$ subjects for $27$ trials. The null hypothesis is not rejected at the $5$ percent significance level for all except for one good at one trial, but the null hypothesis is not rejected in any instance at the $1$ percent significance level. We did not perform a joint test because there are missing values.} We believe this is reasonable evidence in favor of Assumption (ref).(ref) for budget allocation experiments. Failure to account for the possibility of subjects' misperception of the experimental task may affect the elicitation of true consumer behavior. In particular, the experimental design of choi2014more and ahn2014estimating relies on a graphical representation of the budget hyperplane to elicit consumption choices. It is therefore a visual task and an economic task at the same time. We now consider the possibility of misperception of prices that can be thought of as measurement error in prices due to experimental design.

Assumption (ref).(ref) relaxes the implicit requirement in the deterministic RP framework that subjects perceive the budget constraints without any distortion. Note that the graphical experimental device used by choi2014more and ahn2014estimating may make it difficult for consumers to correctly understand the true prices. We believe that it is desirable to have an RP test of the validity of the UMT that is robust to the misperception of prices when the distortion is (i) attributable to the experimental design and (ii) nonsystematic. We use data from kurtz2019neural to provide empirical evidence supporting Assumption (ref).(ref). In contrast to the case of trembling-hand errors, there are no direct measurements on price misperception. However, kurtz2019neural collected data on visual misperception of coordinates that we can use to test Assumption (ref).(ref) indirectly. In particular, we find evidence for a special case of this assumption that is compatible with the misperceived price being $\boldsymbol{\rho}_{l,t}=\boldsymbol{\rho}_{l,t}^*/\boldsymbol{\epsilon}_{l,t}$, for all goods $l\in \{1,\cdots,L\}$, where $\boldsymbol{\epsilon}_{l,t}$ captures misperception. Assumption (ref).(ref) holds if the misperception error has mean 1 and is independent of true prices. In a setup with two goods, kurtz2019neural implement a visual task by giving their subjects a particular numerical target $z_t^*=(z_{1,t}^*,z_{2,t}^*)$. Next, the experimenters ask their subjects to locate this coordinate on a budget line. As a result, the point $z_t=(z_{1,t}^*\epsilon_{1,t},z_{2,t}^*\epsilon_{2,t})$ is observed, where we treat $\epsilon_{t}$ as the misperception error. Since prices can be computed from two observed points on the same budget line, misperceived prices will be $\rho_{l,t}=\rho_{l,t}^*/\epsilon_{l,t}$ for $l\in\{1,2\}$.\footnote{Formally, we assume that the perception error realization is the same when the subject observes a second point given the same budget line.} A sufficient condition for Assumption (ref).(ref) is that the multiplicative error is independent of true prices and that the mean perception error across individuals is $1$. We cannot check the independence condition from this validation data. Nonetheless, we cannot reject the null hypothesis that the mean (multiplicative) perception error is equal to $1$ across the $25$ subjects in this experiment, for each of the $27$ trials (at the $5$ percent significance level).\footnote{We performed a t-test analogous to the case of the trembling-hand error.} In the budget allocation tasks implemented by choi2014more and ahn2014estimating, the subject is forced to choose a point on the budget line. Given that in these experimental environments the total income or wealth is known, the support of measurement error $E|X$ must be such that $\boldsymbol{\rho}_t^{*\prime}\mathbf{c}_t^*=\boldsymbol{\rho}_t^{\prime}\mathbf{c}_t\:\ensuremath{\mathrm{a.s.}}$.\footnote{In our application, we want to test separately for the validity of the static UMT together with (i) trembling-hand error in consumption or (ii) misperception of prices. Hence, we will assume that prices are perfectly measured in the former case (i.e., $\mathbf{w}_t^p=0\:\ensuremath{\mathrm{a.s.}}$), while assuming that consumption is perfectly measured in the latter case (i.e, $\mathbf{w}_t^c=0\:\ensuremath{\mathrm{a.s.}}$).} Note that in both the trembling-hand errors and in the misperception case we have $d_M=|\mathcal{T}|\cdot L$. The number of centering conditions grows with the number of commodities and the number of decision tasks.

Measurement Error in Scanner Data Sets

Even though our main focus is on survey and experimental data sets, our methodology can be used in other types of data sets with their corresponding measurement error constraints. Here we provide a quick overview of the case of scanner data sets because of its relevance for RP practitioners. Scanner consumer panel data sets are usually of high quality; thus, measurement error concerns may be less important. However, in some cases, like in the well-known Nielsen Homescan Scanner Data Set (NHS), there is evidence of measurement error in prices due to classical misreporting but also due to imputation done in the data collection stage einav2010recording.\footnote{Similarly, in the older Stanford Basket Scanner Data Set echenique2014testable, there could be measurement error in prices due to unobserved coupons or discounts.} einav2010recording use validation data from a retailer and compare it to a subsample of the self-reported NHS (for $2004$); they conclude that consumption is measured rather precisely with roughly $90$ percent of all records being exactly reported. On the other hand, prices are likely to be recorded with an upward bias. In fact, prices are measured precisely in only $50$ percent of records in the sample of interest. The sample mean of the logarithm of prices\footnote{The mean is taken across all records in the time window of interest.} in the NHS seems to be slightly above the same quantity for the validation data.\footnote{It seems that Nielsen generates the price in cases in which they have access to retailer-level price data. The reason for the overreporting is that this imputation process ignores the discounts that consumers may get einav2010recording.} Statistically, the difference between the logarithm of prices in the total sample of interest in the NHS and in the validation data is not different from zero (at the $5$ percent significance level), as reported in einav2010recording.\footnote{In the subsample of records from the NHS in which the consumers did not get a sales discount, the measurement error in the logarithm of prices seems classical (i.e., centered around zero and symmetric). In contrast, in the subsample of records in which the consumers got a sales discount, the distribution of measurement error in the logarithm of prices has a fat right tail einav2010recording. As a result, the total measurement error in the logarithm of prices is not symmetric.} We believe that the findings of einav2010recording support the conclusion that consumption measurement error in the NHS can be treated as local perturbations (see Appendix (ref)). Hence, we assume that consumption is measured precisely (i.e., $\mathbf{w}_t^c=0\:\ensuremath{\mathrm{a.s.}}$) and impose the following centering condition for the NHS:

The above centering condition for measurement error allows for nonsystematic overreporting or underreporting of the logarithm of prices on average. Assumption (ref).(ref) implies that the number of measurement error conditions is $d_M=\left\lvert\mathcal{T}\right\rvert\cdot L$. The source of measurement error in the NHS is very different from the one in survey data. In the NHS, households report quantities and prices, while in the survey data set, households report expenditures. For this reason survey data sets are usually used for aggregate analysis (at the level of the category of goods), in cases where price indexes computed by the national statistical agencies are available. As a result, the main mismeasured object is consumption. In contrast, scanner data sets have rich disaggregated information on prices and quantities. But misreported or imputed prices lead to measurement error. For that reason, we impose our centering condition on the main source of measurement error in each of these cases.

Other Forms of Measurement Error: Moment Inequalities and Instruments

Our methodology also allows for imposing moment inequality restrictions on measurement error. Following schennach2014entropic, we can handle conditions of the type \[ \mathds{E}\left[g_M(\mathbf{x},\mathbf{e})\right]\geq 0 \] by introducing an additional slack positive random vector $\mathbf{s}=(\mathbf{s}_j)_{j\in \{1,\cdots,d_M\}}$ such that \[ \mathds{E}\left[g_M(\mathbf{x},\mathbf{e})-\mathbf{s}\right]= 0. \] Moment inequalities may be particularly useful for taking into account bounds on measurement-error averages (e.g., $\mathds{E}\left[\mathbf{w}_t^p\right]\leq 0$ for all $t\in \mathcal{T}$). Imposing support constraints on measurement error (e.g., by rounding: $\left\lvert\mathbf{w}_{l,t}\right\rvert\leq 1/2\:\ensuremath{\mathrm{a.s.}}$) can be handled automatically by setting the support $E|X$ appropriately. Measurement error moments can also capture exclusion/orthogonality restrictions. In other words, the analyst may have information or be willing to assume that a particular observed variable is orthogonal to measurement error in consumption, prices or expenditures. The literature of demand estimation in both the static and dynamic setups, which use moments, usually handles measurement error through exclusion restrictions.\footnote{See lewbel2009tricks and alan2018euler.} Note that Assumption (ref).(ref) can be understood as an orthogonality restriction between prices and consumption measurement error (when prices are observed without error). Consider an instrumental variable $\mathbf{z}_t$ supported on ${\mathds{R}}^{L}$ that is orthogonal to the consumption measurement error; this can be expressed as: \[ \mathds{E}\left[\mathbf{z}_t^{\prime}\mathbf{w}^c_t\right]=0, \] for all $t\in \mathcal{T}$. For Assumption (ref).(ref), the variable $\mathbf{z}_t=\mathbf{p}_t\:\ensuremath{\mathrm{a.s.}}$. These additional restrictions will increase the power of the test simply because they contain more information about measurement error.

Asymptotic Power for s/ED-Rationalizability: Illustrative Example

In previous sections we discussed different centering conditions that can be imposed on measurement error in different data sets. Since our characterization of $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationality is necessary and sufficient, we have asymptotic power of one against the alternate hypothesis of inconsistency with $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{m}}$-rationalizability. Nonetheless, we still have to show that the alternative hypothesis space is nonempty. That is, we need to make sure that the restrictions on measurement error we provide are falsifiable. Here we build an illustrative example for $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{ED}}$-rationality and the centering condition captured by Assumption (ref).(ref) with no price measurement error. Similar examples can be built for other moments used in our applications (see Appendix (ref)). We also provide simulation evidence in Appendix (ref). Consider the random array $(\boldsymbol{\rho}_t,\mathbf{c}_t)_{t\in \mathcal{T}}$ such that $\mathcal{T}=\{0,1\}$, $\boldsymbol{\rho}_{0}=(1,1)^{\prime}$, $\boldsymbol{\rho}_1=(2,2)^{\prime}$, $\mathbf{c}_0=(1,1)^{\prime}$, and $\mathbf{c}_1=(2,2)^{\prime} \:\ensuremath{\mathrm{a.s.}}$. This data set requires that prices and consumption be the same for all consumers in the population. Moreover, it is easy to see that in deterministic terms $\ensuremath{\mathrm{ED}}$-rationality fails. We will now show that $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{ED}}$-rationality also fails with appropriate centering conditions. Assume towards a contradiction that this random array is $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{ED}}$-rationalizable. Then, there have to be random variables $\mathbf{d}\in(0,1]$, $\{\mathbf{w}^{c}_t\}_{t=0,1}$, and $\{\mathbf{v}_t\}_{t=0,1}$ such that

Combining these two inequalities, we get \[ \dfrac{\mathbf{d}}{2}(\mathbf{v}_1-\mathbf{v}_0)\geq \mathbf{v}_1-\mathbf{v}_0\:\ensuremath{\mathrm{a.s.}}. \] Thus, since $1\geq\mathbf{d}>0\:\ensuremath{\mathrm{a.s.}}$, it follows that $\mathbf{d}(\mathbf{v}_1-\mathbf{v}_0)\leq0\:\ensuremath{\mathrm{a.s.}}$. However, this implies that $\mathbf{v}_0\geq\mathbf{v}_1\:\ensuremath{\mathrm{a.s.}}$. If Assumption (ref).(ref) holds, then it must be the case that: \[ \mathds{E}\left[\rho_0\mathbf{w}_0\right]=\mathds{E}\left[\rho_1\mathbf{w}_1\right]=0. \] As a result, applying the centering condition, we obtain a contradiction: \[ 0\geq\mathds{E}\left[\mathbf{d}(\mathbf{v}_1-\mathbf{v}_0)\right]\geq 4 +\mathds{E}\left[\rho_1^{\prime}(\mathbf{w}_1-\mathbf{w}_0)\right]=4+\mathds{E}\left[\rho_1^{\prime}\mathbf{w}_1\right]-2\mathds{E}\left[\rho_0^{\prime}\mathbf{w}_0\right]=4. \] This contradiction means that the random array $(\boldsymbol{\rho}_t,\mathbf{c}_t)_{t\in \mathcal{T}}$ is not $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{ED}}$-rationalizable under the centering conditions we described above. We highlight that potential lack of statistical power of some centering conditions is not a defect of our methodology. The reason is that if the quality of the data set at hand is not good enough to credibly discern whether observed behavior is consistent with a given model, then no methodology can address this issue.

Empirical Application (I): Testing the Dynamic UMT with Exponential Discounting in Survey Data

In our first application, we apply our methodology to a consumer panel data set gathered from single-individual and couples' households in Spain to test for $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{ED}}$-rationalizability. This important model is under increasing scrutiny because experimental evidence tends to find that the behavior of experimental subjects is time-inconsistent.\footnote{See, for instance, andreoni2012estimating, montielolea2014axiomatization, and echenique2014testable.} Nonetheless, it is important to explore to what extent this finding has external validity. To address this issue, some researchers have turned to survey data in the form of household consumption panels. Most of this work has found evidence against exponential discounting blownonparametric2017. However, the existing literature has not yet addressed the issue of measurement error in the consumption reported by households in a way that allows us to perform traditional hypothesis testing. (Some additional problems with the existing evidence are (i) the strong parametric assumption on preferences, and (ii) homogeneity restrictions on the discount factor and preferences.) One solution to some of the problems in the literature can be found in the work on deterministic RP by browning1989anonparametric. In particular, Browning's work avoids making parametric assumptions about the functional form of instantaneous utility. However, this work does not take into consideration the fact that consumption can be mismeasured. In our Monte Carlo experiment, the deterministic test in browning1989anonparametric rejects the correct null hypothesis of exponential discounting behavior in $79.4$ percent of the cases, while our methodology correctly fails to reject the null hypothesis that all households are consistent with exponential discounting at the correct $5$ percent significance level (Appendix (ref)). In addition, when we applied browning1989anonparametric deterministic methodology to our single-individual households data set, we also obtained a very low success rate. However, this low success rate of the deterministic test for exponential discounting may be due to measurement error. In our empirical application, we found support for exponential discounting behavior in single-individual households, while at the same time, support for the negative finding in blownonparametric2017 in the case of couples' households. This fact indicates that deterministic tests may not be very informative about the behavior in a population due to measurement error. Small violations of the deterministic RP inequalities will lead to big rejection rates. Introducing measurement error into the analysis takes these small violations into account.\footnote{In Appendix (ref), we also establish that our test fails to rejects implications of the collective household consumption problem presented in adams2014consume.}

Our empirical application also contributes to the literature on estimating the discount factor distribution in survey data sets and in a classical consumer theory environment. This has been the topic of a large body of work which, however, has reached little or no consensus.\footnote{We refer the reader to the survey by frederick2002timediscounting for its extensive references.} This lack of consensus can be attributed in some degree to a failure to identify the parameters of interest. Here, we show that the discount factor distribution cannot be identified solely from prices, interest rates, and consumption observations in a data set that suffers from measurement error. (For details see Section (ref).) However, our methodology allows us to test for exponential discounting behavior even in this setting (i.e., without identifying the discount factor distribution).\footnote{In order to learn more information about the discount factor distribution, one needs additional data. One notable example is mastrobuoni2016criminal, which uses a quasi-experiment to pin down criminals' time preferences.}

If one ignores the issues of measurement error, the Euler equation allows one to estimate the discount factor and the marginal utility either parametrically or semi-parametrically.\footnote{Examples of estimators of the Euler equation and similar models include hall1978stochastic, hansen1982generalized, dunn1986modeling, gallant1989seminonparametric, chapman1997approximating, campbell1999force, ai2003efficient, chen2009land, darolles2011nonparametric, chen2014local, and escanciano2016.} Since our objective is not to estimate but to test the exponential discounting model, we follow a different path.

In particular, we work with the data set used in adams2014consume: the Spanish Continuous Family Expenditure Survey (Encuesta Continua de Presupuestos Familiares). The data set consists of the expenditures for 185 individuals and 2004 couples, as well as prices for 17 commodities (categories of goods) recorded over four time periods. Each household was interviewed for four consecutive quarters between 1985-1997. We construct a panel data set of consumption and prices by pooling household's quarterly data points. The categories of goods are: all food and nonalcoholic drinks, all clothing, cleaning, nondurable articles, household services, domestic services, public transport, long-distance travel, other transport, petrol, leisure (four categories), other services (two categories), and food consumed outside the home. The data set also contains information on the nominal interest rate on consumer loans faced by the household in any particular quarter.\footnote{We spare the reader more details and refer them instead to adams2014consume for further information on the data set.} Formally, we test for $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{ED}}$-rationalizability with (i) effective prices equal to the discounted spot prices, $\boldsymbol{\rho}_t=\boldsymbol{\rho}_t^{\ensuremath{\mathrm{ED}}}$ (defined in Table (ref)), (ii) random marginal utility of income equal to the discounted value of one unit of wealth, $\boldsymbol{\lambda}_t=1\:\ensuremath{\mathrm{a.s.}}$, and (iii) $\boldsymbol{\delta}_t=\mathbf{d}^{t}$ where $\mathbf{d}$ is interpreted as the (time-invariant) random discount factor supported on or inside $(0,1]$.\footnote{When the discount factor is $0$, it is easy to see that $\ensuremath{\mathrm{ED}}$-rationality becomes equivalent to $\ensuremath{\mathrm{R}}$-rationality. In practice, we pick the interval $[0.1,1]$ as the largest possible support for discount factor. This interval contains most reasonable values in the literature frederick2002timediscounting,montielolea2014axiomatization.} Imposing the additional Assumptions (ref).(ref), (ref), and (ref), we can apply our testing methodology developed in Section (ref). Recall that Assumption (ref).(ref) indicates that measurement error does not alter the mean value of total expenditure, $\mathds{E}\left[\boldsymbol{\rho}_{t}^{\prime}\mathbf{c}_{t}\right]=\mathds{E}\left[\boldsymbol{\rho}_{t}^{\prime}\mathbf{c}_{t}^{*}\right]$.

The Results

Single-Individual Households

We apply the deterministic methodology of browning1989anonparametric to single-individual households. Our initial conclusion is that $84.3$ percent of the single-individual households behave inconsistently with exponential discounting (even when allowing for substantially more heterogeneity than previous works).\footnote{We search for each individual household discount factor $d$ in the grid $\{0.1,0.15,\cdots,1\}$. See crawford2010habits,adams2014consume, and blownonparametric2017 for discount factor ranges close to $[0.9,1]$.} Next, we revisit this conclusion using our methodology, which addresses measurement error, while allowing a heterogeneous discount factor. We find that we cannot reject exponential discounting. Formally, we find at the $5$ percent significance level that the random array $\mathbf{x}=(\boldsymbol{\rho}_t,\mathbf{c}_t)_{t\in\mathcal{T}}$ is $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{ED}}$-rationalizable with a random discount factor $\mathbf{d}$ supported on or inside $[0.975,1]$ ($\mathrm{TS}_{n}=6.480$, $\text{p-value}=0.166$). We believe that the lower bound value of $0.975$, for the quarterly discount rate, is reasonable (it corresponds to a annualized discount rate of $0.9$). Although there is no agreement in the literature about what appropriate values for the discount factor are frederick2002timediscounting, a common benchmark in applied work is to set the discount factor according to the real interest rate in the economy.\footnote{dejong2011structural suggests setting the discount factor value to $d=1/(1+\overline{r})$, where $\overline{r}$ is the average (across individuals) annual real interest rate.} In our case, the lower bound of the quarterly discount factor is $0.975$. This discount factor bound corresponds to an annual real interest rate of $10.7$ percent. This is roughly consistent with this benchmark for the average real interest rates observed in our sample.\footnote{Using the results from Section (ref) we also tested several candidates for the average quarterly discount factor $\theta_0=\mathds{E}\left[\mathbf{d}\right]$: $0.995$ ($\mathrm{TS}_{n}=14.071$, $\text{p-value}=0.015$), $0.996$ ($\mathrm{TS}_{n}=5.105$, $\text{p-value}=0.403$), and $0.997$ ($\mathrm{TS}_{n}=2.967$, $\text{p-value}=0.705$). The smallest $\theta_0$ that is not rejected at the $5$ percent significance level, $0.996$, corresponds to the annual average discount factor of $0.984$ and the average annual real interest rate of $1.6$ percent.} However, at the $5$ percent significance level, we cannot reject exponential discounting, when discounting is set at $1\:\ensuremath{\mathrm{a.s.}}$ ($\mathrm{TS}_{n}=6.140$, $\text{p-value}=0.189$). Of course, this does not mean that this is the value of the discount factor. In fact, our sample and our knowledge about measurement error are not enough to elicit the support of the distribution of discount factors. Nonetheless, we can differentiate between behavior consistent with exponential discounting and systematic departures from it (as seen in the power analysis of our method). In sum, our findings provide evidence supporting exponential discounting for singles under reasonable discount factors. Using our methodology in data sets with more time periods or with additional information about measurement error will improve the information we can obtain about discount factors. However, we will see that identifying the support of the discount factor is not essential to provide informative bounds on average demand (see Appendix (ref)).\footnote{As robustness check we conducted the test for the discount factor supported on or inside $[0.1,1]$, $[0.5,1]$, and $[0.9,1]$. As expected, since all three intervals contain $[0.975,1]$, the null hypothesis is not rejected. The test statistic $\mathrm{TS}_{n}$ is equal to $6.476$ with $\text{p-value}=0.166$ for all three intervals.}

Couples' Households

For the couples' households, the deterministic test of browning1989anonparametric rejects the exponential discounting model for $89.8$ percent of the observations. Although this number seems large, one should keep in mind that for single-individual households the same deterministic test rejects the model in $84.3$ percent of the cases. At the same time, for $\mathbf{d}\in[0.1,1]$, our method does not reject the exponential discounting model for single-individual households. But we do reject the model for couples' households. In the case of couples' households, the test statistic is $\mathrm{TS}_{n}=71.015$ ($\text{p-value}<10^{-12}$).\footnote{As a robustness check, we also tested the model assuming that $\mathbf{d}$ is supported on or inside $[0.5,1]$, $[0.9,1]$, and $\mathbf{d}=1\:\ensuremath{\mathrm{a.s.}}$. As expected, since these intervals are contained in $[0.1,1]$, the null hypothesis is rejected. The test statistic $\mathrm{TS}_{n}$ ($\text{p-value}$) is equal to $71.015$ ($<10^{-12}$), $70.964$ ($<10^{-12}$), and $101.579$ ($<10^{-12}$), respectively.}

Discussion and Related Work on Testing the Exponential Discounting Model

One possible concern about our methodology is that its power is low in survey data sets (due to a small $T$ dimension of the data) given its nonparametric nature. However, in Appendix (ref) we report, for $1000$ trials with a sample size of $2000$, a rejection rate greater than or equal to $72$ percent (with a data generating process consistent with the collective model as in adams2014consume).

Our results for couples' households provide the first nonparametric evidence which is robust to measurement error and which demonstrates that not all couples' households manifests behavior consistent with exponential discounting. In Appendix (ref), we establish that a suitable extension of our methodology fails to reject the collective household consumption problem presented in adams2014consume.\footnote{mazzocco07 also provides evidence in favor of the collective model using a parametric methodology.} This should convince practitioners about the importance of modeling intrahousehold decision-making when dealing with intertemporal choice. The rejection of exponential discounting behavior for couples' households can be better understood given theoretical results that show that aggregating time-consistent preferences may lead to time-inconsistent behavior jackson2015collective.

The deterministic methodology of browning1989anonparametric concludes that the fraction of households that is inconsistent with exponential discounting under the deterministic test is similar for both single individuals and couples, but our statistical test rejects in the latter case while reaching the opposite conclusion in the former case. The difference in conclusions is due to the fact that our test implicitly takes into account the severity of the violations of exponential discounting, and imposes the mean budget-neutrality assumption on the measurement error corrections.\footnote{We have also tested $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{ED}}$-rationality under Assumption (ref).(ref), which requires that consumption quantities measurement error is centered around zero as in varian1985non. We strongly reject the null hypothesis in this case, providing an illustration of the importance of using empirically-backed centering conditions.} Our main empirical finding is robust to price measurement error. Adding an additional source of measurement error would make the rationalizability notion less demanding.\footnote{See Appendix (ref) for results on robustness of our methodology to local perturbations in prices.}

Empirical Application (II): Static UMT in Experimental Data Sets with Trembling-Hand Errors and Misperception of Prices

In this section, we use our methodology to test the static UMT in the widely known experimental data set by ahn2014estimating. The experimental task consists of $T=50$ independent decision trials with $n=154$ subjects. Each decision is a portfolio problem. The subjects face three states of the world $\sigma\in\{1,2,3\}$. The subjects are given $100$ tokens per task and they have to choose a bundle of Arrow securities, $c_t\in {\mathds{R}}^{3}_{+}$, for a randomly drawn price vector $p_{t}\in {\mathds{R}}^{3}_{+}\setminus{\{0\}}$. The subjects are forced to choose a bundle that satisfies Walras' law such that for every decision task it must be that $p_{t}^{\prime} c_{t}=100$. The subjects receive a payment in tokens according to the probability of each state of the world at the end of each round. At the end of the experimental task one of the rounds was selected using a uniform distribution and the tokens payment corresponding to that round is paid in dollars.\footnote{The subjects are informed that the probability of state $\sigma=2$ is $1/3$, and the joint probability of the states $\sigma\in \{1,3\}$ is $2/3$.} The exchange rate is $0.05$ dollars per token and the participation fee is $5$ dollars. This ingenious experimental device (due to choi2014more) has allowed the RP practitioners to collect a large number of observations per individual with high price variation. beatty2011howdemanding highlighted the importance of rich price variation to have enough power in the experimental design to detect violations of the UMT. The deterministic RP test for the static UMT in this data set concludes that only $12.99$ percent of the experimental subjects pass the test. At first sight, this is a striking result, because the majority of subjects seems to be inconsistent with the core consumer model in economics. We reexamine the robustness of this result to measurement error in consumption due to errors in the elicitation of the intended behavior of consumers. In our application, we found support for the static UMT in this experiment in stark contrast with the findings from the deterministic RP test. Measurement error in the experimental environment may arise due to the nature of the design. Subjects are presented with a graphical representation of a $3$ dimensional budget hyperplane, and they must choose the consumption bundle by pointing to a point in this hyperplane using a computer mouse or the arrow keys in a keyboard. We must note that there is a mechanical measurement error due to the resolution of the budget hyperplane which is $0.2$ tokens. More important factors, such as a lack of expertise in the decision task, could lead the consumers to make implementation mistakes when trying to choose their preferred alternative. kurtz2019neural provide direct evidence of trembling-hand error in a budget allocation task similar to ahn2014estimating, as well as indirect evidence of visual misperception, as we have previously discussed. Nonetheless, the actual reason why the experimental design fails to elicit the intended decision task is not our main concern. We take the stand that a desirable test of the static UMT has to be robust to possible nonsystematic mistakes arising from any experimental design. We consider both trembling-hand errors in consumption and nonsystematic misperception of prices. Formally, we test for $\ensuremath{\mathrm{s}}/\ensuremath{\mathrm{R}}$-rationalizability with (i) effective prices equal to the prices at each trial $\boldsymbol{\rho}_t=\boldsymbol{\rho}_t^{\ensuremath{\mathrm{R}}}$ (defined in Table (ref)), (ii) marginal utility of wealth $(\boldsymbol{\lambda}_t)_{t\in \mathcal{T}}$ supported on $\Lambda={\mathds{R}}^{\left\lvert\mathcal{T}\right\rvert}_{++}$, and (iii) random discount factor equal to $1$ ($\boldsymbol{\delta}_t=1\:\ensuremath{\mathrm{a.s.}}$ for all $t\in\mathcal{T}$). We note that there is evidence for trembling-hand errors and misperception errors in the budget allocation task we consider here (see Section (ref)). However, we fail to reject the null hypothesis of the static UMT when allowing only for price misperception. Evidently, it follows that we fail to reject the null hypothesis of the static UMT when allowing for both price misperception and trembling hand errors. In addition, we reject the null hypothesis of the static UMT when allowing only for trembling hand errors. Thus, we can conclude that the main source of error in this type of experimental tasks is most likely price misperception.

Misperception of Prices

We consider the possibility of measurement error in prices arising from misperception. We investigate the case where consumers behave as if they are trying to maximize a utility function subject to a misperceived vector of prices. In this regard, we take the point-of-view of gillen2015experimenting that points out that misperception in a low stakes experimental environments may affect the external validity of the conclusions drawn from an experimental data set. kurtz2019neural provide indirect evidence of errors induced by the visual task in the budget allocation problem in the very closely related interface of choi2014more. We require that consumers' average perception of prices is unbiased, namely, for all $t\in \mathcal{T}$ it must be that \[ \mathds{E}\left[\boldsymbol{\rho}_t\right]=\mathds{E}\left[\boldsymbol{\rho}_t^*\right]. \] This is captured in Assumption (ref).(ref). In order to isolate the effect of misperception on the observed violations of the static UMT, we assume that consumer behavior is measured perfectly ($\mathbf{w}_t^c=0\:\ensuremath{\mathrm{a.s.}}$). In addition, due to the experimental design in ahn2014estimating, it must be that true prices are such that $\mathbf{p}^{*\prime}_{t} \mathbf{c}^*_t=100\:\ensuremath{\mathrm{a.s.}}$ (i.e., Walras' law holds). The value of the test statistic is $\mathrm{TS}_{n}=17.879$ ($\text{p-value}> 1-10^{-10}$). This is below the conservative critical value $\chi_{150,0.95}^{2}=179.581$.\footnote{We used $900$ draws in the Monte Carlo computation of the maximum-entropy moment of this problem. We chose this number on the basis of the trembling-hand error exercise in this experimental data set in the next section.} We do not reject the null hypothesis of the static UMT in the presence of misperception of prices, when the average vector of prices is equivalent to the true vector of prices.\footnote{Assumption (ref).(ref) has empirical bite. See Appendix (ref).} More importantly, this finding puts in perspective the rejection of the static UMT in experimental data sets that use the graphical representation of budget hyperplanes choi2014more,ahn2014estimating. In particular, we find evidence that prices misperception matters. When we account for this possibility, we no longer reject the null hypothesis of the static UMT.

Trembling-Hand Errors in Consumption

We say that measurement error in consumption is the result of a trembling-hand when it is centered at zero. This idea is captured in Assumption (ref).(ref) that requires that for all $t\in \mathcal{T}$ it must be that $\mathds{E}\left[\mathbf{w}^c_t\right]=0$. Also, we keep the restriction that the true prices and consumption satisfy Walras' law $\mathbf{p}^{*\prime}_{t} \mathbf{c}^*_t=100\:\ensuremath{\mathrm{a.s.}}$, and that prices are measured perfectly $\mathbf{w}_t^p=0\:\ensuremath{\mathrm{a.s.}}$. As we discussed before, we have obtained direct evidence for this centering condition using the replication data of kurtz2019neural. We strongly reject the null hypothesis of the static UMT when allowing for measurement error in the elicitation of the true consumer behavior due to trembling-hand errors. The test statistic is $\mathrm{TS}_{n}=299.137$ ($\text{p-value}<10^{-11}$). This is above the conservative critical value $\chi_{150,0.95}^{2}=179.581$.\footnote{We used $2970$ draws in the Monte Carlo computation of the maximum-entropy moment of this problem. We also tried $580$ and $900$ with test statistics with values of $301.654$ and $306.882$. Which is evidence that moderate size of draws for the Monte Carlo integration step do relatively well in this setup.}

Relation to the Literature

afriat1967construction shows that it is sufficient to know shape constraints on the utility function (i.e., concavity) to bypass the need to know the utility function when testing for the UMT. We generalize this insight by allowing measurement error in consumption and prices. Among authors using the deterministic RP approach, the immediate antecedents to our work using the first-order approach are browning1989anonparametric, blownonparametric2017, and brownquasilinear. Important advances have been made on testing and doing counterfactual analysis under rationalizability or random utility.\footnote{Relevant examples are blundell2014bounding, dette2016testing, lewbel2017unobserved, and kitamura2018nonparametric.} However, the majority of these results assume that observed quantities are measured accurately. varian1985non is possibly the first work to introduce the subject of measurement error into the RP approach. Varian's methodology is the closest to that of our own work; he considers precisely measured (albeit random) prices to study measurement error in consumption. Varian's work is compatible with standard statistical hypothesis testing under the strong assumptions of normality (with known variance) and additivity of consumption measurement error. In contrast, our methodology is fully nonparametric. We are able to improve upon Varian's methodology and relax its core assumptions by using a moments approach to measurement error in the RP framework. Other papers have dealt with measurement error under different parametric assumptions about measurement error or about the heterogeneity of preferences.\footnote{gross1995testing assumes that random consumption is generated by consumers with similar preferences. tsur1989testing imposes a log-normal multiplicative measurement-error structure in expenditures. hjertstrand2013simple proposes a generalization of varian1985non, but requires knowing the distribution of measurement error. echenique2011money assume that measurement error in prices is a normal random variable independent across households and prices with a fixed mean and known variance.} deb2017revealed consider a nonparametric model of “price preference.” They propose an RP test of their model that is robust to small measurement error in prices. boccardi2016predictive considers a case of demand with error and establishes a way to account for the trade-off between the fit of the model and its predictive ability (which is a generalization of beatty2011howdemanding).\footnote{The fit of an RP model indicates whether the a data set is consistent with the model. The predictive ability of an RP model is a measure of how easy it is for a data set generated at random to be consistent with the model. See beatty2011howdemanding for further details.}

In practice, the RP theorists (e.g., adams2014consume and cherchye2017household) have dealt with measurement error by perturbing (minimally) the observed individual consumption in order to satisfy the conditions of an RP test. For instance, adams2014consume find the additive perturbation with a minimal norm that renders the individual consumption streams compatible with the RP restrictions. Then a subjective threshold is imposed on the maximum admissible norm of the measurement error vector. If the computed norm is above the threshold, then the model is rejected. However, their methodology has one important drawback: every data set can be made to satisfy their test or, equivalently, the test has no power. In addition, the test in adams2014consume has no size control.

Among researchers using the RP approach, blundell2003nonparametric,blundell2014bounding are the first to provide consumer demand bounds, under the assumption of static utility maximization in a semiparametric environment (with additive heterogeneity) in which income changes continuously. Our work differs from theirs mainly in that we allow for unrestricted heterogeneity in preferences, do not require that income be observable, and do not impose semiparametric assumptions on wealth effects to provide bounds for demand, given new prices. In addition, nonclassical measurement error is not compatible with their approach.

Conclusion

We propose a new stochastic and nonparametric RP approach (suitable for an environment with measurement error in consumption or prices) that is useful to test for several consumer models that can be characterized by their first-order conditions. In particular, our work can be used (but is not limited) to test for static utility maximization afriat1967construction, and for dynamic rationalizability with exponential discounting browning1989anonparametric.

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