📈 economics

A Frequentist Approach to Revealed Preference Analysis

This paper develops a frequentist framework to quantify the statistical power of revealed preference tests, demonstrating how sample size and data richness determine the ability to detect economically meaningful deviations from theoretical models and enabling practical inference on demand functionals and welfare effects.

Original authors: Charles Gauthier, Raghav Malhotra, Agustin Troccoli Moretti

Published 2026-02-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Charles Gauthier, Raghav Malhotra, Agustin Troccoli Moretti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to figure out if a suspect is following a strict rulebook (like "always choose the cheapest option" or "always maximize happiness"). You have a notebook with a list of their choices: what they bought when prices were high, what they bought when prices were low, and so on.

The Old Way (The "Pass/Fail" Test)
Traditionally, economists use a method called Revealed Preference. It's like a strict math test.

If the suspect's choices fit the rulebook perfectly, they pass.
If there's even one tiny contradiction, they fail.

The Problem:
Imagine the suspect is actually cheating (making irrational choices), but you only have a notebook with three entries.

The detective says, "They passed! They are rational!"
But really, they just got lucky. With only three data points, a cheater can easily slip through the cracks. It's like a student guessing on a 3-question quiz and getting a perfect score by chance. You can't tell if they actually know the material or just got lucky.

The New Paper's Solution: "How Big a Lie Can We Catch?"
This paper asks a smarter question: "How many observations do we need to be sure we catch a cheater?"

Instead of just saying "Pass" or "Fail," the authors use a framework from Statistical Learning (think of it as a "lie detector" for data) to measure the power of the test.

Here is the breakdown using simple analogies:

1. The "Smoothness" Assumption

The paper assumes that people's choices are smooth.

Analogy: Imagine a car driving on a road. If you turn the steering wheel a tiny bit, the car doesn't suddenly teleport to the other side of the highway; it moves a little bit. Similarly, if a person's budget changes slightly, their shopping list shouldn't change wildly.
Why it matters: Because choices are smooth, if you see enough data points (enough "snapshots" of the car's path), you can draw a very accurate line connecting them. If the person is actually cheating (driving erratically), that line will eventually look jagged and obvious once you have enough snapshots.

2. The "Detective's Confidence"

The authors calculate exactly how many snapshots ( $n$ ) you need to catch a specific amount of cheating ( $\epsilon$ ).

Scenario A: You want to catch a huge cheater (someone who buys luxury cars when they can't afford them). You only need a few snapshots to catch them.
Scenario B: You want to catch a tiny cheater (someone who makes a 1% irrational choice). You need thousands of snapshots to be sure.
The Takeaway: If a study only has 10 data points and says "The consumer is rational," it might just mean the study wasn't looking hard enough. The "lie" was too small to be seen with so little data.

3. The "NP-Hard" Problem (The Maze)

Some rules are incredibly complex to check.

Analogy: Imagine trying to solve a maze. Some mazes are easy (like checking if someone buys apples or oranges). Others are like a giant, twisting labyrinth with millions of paths (checking if someone's choices fit a complex "weak separability" rule).
The Old Problem: Checking these complex mazes exactly is computationally impossible for computers (it takes too long).
The New Solution: The paper says, "Let's stop trying to solve the maze perfectly. Instead, let's build a fence around the maze."
- They create a functional test. Instead of checking every single path, they check if the person is close enough to the rule.
- The Trade-off: You might let a tiny bit of cheating slip through (a small "false negative"), but you get a result that a computer can actually calculate in seconds, and you know exactly how confident you can be in that result.

4. Measuring "Welfare" (The Cost of Living)

The paper also shows how to estimate how much better off a person is, even with limited data.

Analogy: Imagine you want to know how much a person's "happiness" changed when gas prices went up.
The Result: You can't know the exact number, but the paper gives you a Confidence Interval.
- Instead of saying "You lost $50," you say, "We are 95% sure you lost between $40 and $60."
- The more data you have, the tighter that range becomes (e.g., $49 to $51).

Summary: What Does This Mean for the Real World?

"Passing" isn't enough: Just because a dataset passes a standard test doesn't mean the person is rational. It might just mean the dataset was too small to catch the irrationality.
Sample Size Matters: If you want to detect small, subtle irrationalities, you need a massive amount of data. If you only have a little data, you can only detect big, obvious mistakes.
Better Tools for Complex Rules: For very complex economic rules that are too hard to calculate exactly, this paper gives us a way to test them approximately, with a clear understanding of how accurate the test is.

In a nutshell: This paper is about upgrading the "lie detector" for economists. It moves from a simple "Pass/Fail" sign to a sophisticated "Probability Meter" that tells you exactly how much data you need to trust your conclusion.

1. Problem Statement

Traditional Revealed Preference (RP) analysis, rooted in the work of Afriat and Varian, tests whether a finite dataset of choices is consistent with a specific class of preferences (e.g., utility maximization) using exhaustive inequalities (like GARP).

The Limitation: Standard RP tests are deterministic. If data passes the test, the analyst cannot distinguish between a truly rational agent and an irrational agent whose deviations were simply too small or the dataset too sparse to detect.
The Gap: There is no standard way to assess the statistical power of RP tests. Existing approaches (e.g., Bronars, 1987) rely on arbitrary alternative hypotheses (like random choice), which do not capture the nuance of "small" deviations from rationality.
The Core Question: How many observations ( $n$ ) are required to detect a deviation from rationality of a specific magnitude ( $\varepsilon$ ) with a given confidence level ( $1-\delta$ )?

2. Methodology

The authors propose a frequentist framework that integrates Statistical Learning Theory (specifically PAC learning) with economic theory.

A. The Setup

Model: A decision maker (DM) has a true choice function $g$ mapping prices/income to demand.
Assumptions:
- The class of possible choice functions $\mathcal{G}$ satisfies a Lipschitz condition (demand is smooth; small changes in prices lead to bounded changes in demand).
- Data is generated by sampling prices/income randomly from a distribution $\mu$ .
- The analyst observes a finite dataset $D_n = \{(p_k, x_k)\}_{k=1}^n$ .
Hypothesis Testing:
- $H_0$ : The true model $g$ belongs to a class of interest $\mathcal{F}$ (e.g., rational preferences).
- $H_1$ : The true model $g$ is "far" from $\mathcal{F}$ . Specifically, $g \in \mathcal{G} \setminus \mathcal{F}(\varepsilon)$ , where $\mathcal{F}(\varepsilon)$ is the $\varepsilon$ -ball around $\mathcal{F}$ defined by a generalization error metric (integral distance between functions).

B. Theoretical Foundation: PAC Learnability

The paper leverages the concept of Probably Approximately Correct (PAC) learning.

Lemma 1: The class of Lipschitz demand functions is PAC learnable. This means that given a sample size $n$ growing polynomially with $1/\varepsilon$ and $1/\delta$ , any algorithm can approximate the true demand function $g$ within error $\varepsilon$ with probability $1-\delta$ .
Key Insight: If the true demand $g$ is far from the null hypothesis class $\mathcal{F}$ (distance $>\varepsilon$ ), and the sample size is large enough, any function consistent with the data (any rationalization) will also be far from $\mathcal{F}$ . Consequently, an RP test will reject the null with high probability.

C. Functional Restrictions

The framework extends beyond standard RP inequalities (which are often NP-hard to compute, e.g., for weak separability) to functional restrictions.

A restriction is a map $R: \mathcal{G} \to \mathbb{R}_{\geq 0}$ such that $R(g) = 0 \iff g \in \mathcal{F}$ .
To ensure valid finite-sample tests, $R$ must be uniformly continuous (stability under small perturbations) and regular (detectable separation).
This allows for polynomial-time tests even when exact RP characterizations are computationally infeasible.

3. Key Contributions

1. Explicit Power Guarantees

The authors derive explicit bounds linking sample size ( $n$ ), detection threshold ( $\varepsilon$ ), and confidence ( $1-\delta$ ):

Theorem 3.1 (Forward Bound): Given $\varepsilon$ and $\delta$ , there exists a sample size $n(\varepsilon, \delta)$ such that the RP test has power $\geq 1-\delta$ against any alternative separated by $\varepsilon$ .
Theorem 3.2 & 3.3 (Inverse Bounds): Given a fixed sample size $n$ , one can compute the achievable confidence level for a given $\varepsilon$ , or the minimal detectable separation $\varepsilon(n, \delta)$ for a given confidence.

2. Functional Testing Algorithm

They provide Algorithm 1, a constructive procedure for testing functional restrictions.

It constructs a test statistic based on the restriction $R$ .
It sets a rejection threshold based on the modulus of continuity of $R$ .
Result: The test has size $\leq \delta$ and power $\geq 1-\delta$ for alternatives separated by $\varepsilon$ . This solves the computational intractability of tests like Weak Separability (which is NP-hard) by allowing a small, controlled Type I error (size) in exchange for polynomial-time computation.

3. Confidence Intervals for Welfare

Theorem 4.2 demonstrates how to construct confidence intervals for smooth functionals of demand, such as welfare changes (e.g., Equivalent Variation).

If the true demand is approximated within $\varepsilon$ , and the welfare functional is uniformly continuous, the estimated welfare lies within a specific interval of the true welfare with probability $1-\delta$ .
The width of the interval depends directly on the sample complexity of learning the underlying preference class.

4. Simulation Results

The authors simulate power curves for four models:

SUM: Static Utility Maximization (General case).
H: Homotheticity.
WS: Weak Separability.
EU: Expected Utility.

Finding: Standard sample sizes generate widely different power across models. For instance, Expected Utility (EU) is highly sensitive to small deviations (high power with small $n$ ), whereas Weak Separability (WS) requires massive sample sizes to detect similar deviations. This contextualizes why some conditions "rarely reject" in practice—it may be a lack of power, not evidence of rationality.

4. Results and Significance

Reframing RP Analysis: The paper shifts the focus from "Does this dataset pass the test?" to "Is the dataset rich enough to detect irrationality?" It provides a rigorous way to quantify the "resolution" of an RP test.
Computational Tractability: By moving from exact, zero-size tests to functional restrictions with controlled size, the authors make it possible to test complex preference structures (like weak separability) in finite samples without resorting to NP-hard algorithms.
Policy Relevance: The ability to construct confidence intervals for welfare measures (like cost-of-living changes) based on finite data is a significant contribution to applied microeconomics.
Interpretation of Empirical Failures: The simulation results suggest that the frequent failure to reject GARP in empirical data might not imply strong rationality, but rather that standard datasets are too small to detect the specific types of irrationality present. Conversely, the frequent rejection of Expected Utility in experiments may reflect its high sensitivity (high power) to even negligible deviations.

5. Conclusion

This paper establishes a theoretical foundation for power analysis in revealed preference. By linking the geometry of demand functions (Lipschitz continuity) with statistical learning theory (PAC bounds), the authors provide a framework to determine exactly how much data is needed to distinguish between rational and irrational behavior. This framework is general, applicable to various preference classes, and offers practical tools for constructing tests and welfare confidence intervals even when exact characterizations are computationally prohibitive.