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Goodness-of-fit and utility estimation: what's possible and what's not

Yujian Chen, Joshua Lanier, John K. -H. Quah

TL;DR

The paper addresses whether goodness-of-fit indices for testing utility-maximization can be both continuous and accurate. It formalizes five loss-function–based indices (Afriat, Varian, Swaps, nonlinear LS, Houtman-Maks) and proves an impossibility result: for well-behaved utilities, no index can be both continuous and accurate while yielding a minimizable, accurate loss function. It introduces essential accuracy, a robust welfare criterion, and a framework for bounded rationality (consideration-sets), plus an empirical demonstration using scanner data to compare welfare implications under soda reductions. The findings clarify fundamental limitations of standard fit measures, propose a robust welfare analysis approach when best-fitting utility functions do not exist, and characterize utility classes that admit nicer properties. Collectively, the work informs both theoretical understanding and practical welfare analysis in empirical revealed-preference settings.

Abstract

A goodness-of-fit index measures the consistency of consumption data with a given model of utility-maximization. We show that for the class of well-behaved (i.e., continuous and increasing) utility functions there is no goodness-of-fit index that is continuous and accurate, where the latter means that a perfect score is obtained if and only if a dataset can be rationalized by a well-behaved utility function. While many standard goodness-of-fit indices are inaccurate we show that these indices are (in a sense we make precise) essentially accurate. Goodness-of-fit indices are typically generated by loss functions and we find that standard loss functions usually do not yield a best-fitting utility function when they are minimized. Nonetheless, welfare comparisons can be made by working out a robust preference relation from the data.

Goodness-of-fit and utility estimation: what's possible and what's not

TL;DR

The paper addresses whether goodness-of-fit indices for testing utility-maximization can be both continuous and accurate. It formalizes five loss-function–based indices (Afriat, Varian, Swaps, nonlinear LS, Houtman-Maks) and proves an impossibility result: for well-behaved utilities, no index can be both continuous and accurate while yielding a minimizable, accurate loss function. It introduces essential accuracy, a robust welfare criterion, and a framework for bounded rationality (consideration-sets), plus an empirical demonstration using scanner data to compare welfare implications under soda reductions. The findings clarify fundamental limitations of standard fit measures, propose a robust welfare analysis approach when best-fitting utility functions do not exist, and characterize utility classes that admit nicer properties. Collectively, the work informs both theoretical understanding and practical welfare analysis in empirical revealed-preference settings.

Abstract

A goodness-of-fit index measures the consistency of consumption data with a given model of utility-maximization. We show that for the class of well-behaved (i.e., continuous and increasing) utility functions there is no goodness-of-fit index that is continuous and accurate, where the latter means that a perfect score is obtained if and only if a dataset can be rationalized by a well-behaved utility function. While many standard goodness-of-fit indices are inaccurate we show that these indices are (in a sense we make precise) essentially accurate. Goodness-of-fit indices are typically generated by loss functions and we find that standard loss functions usually do not yield a best-fitting utility function when they are minimized. Nonetheless, welfare comparisons can be made by working out a robust preference relation from the data.
Paper Structure (30 sections, 19 theorems, 31 equations, 4 figures, 2 tables)

This paper contains 30 sections, 19 theorems, 31 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Utility maximization and goodness-of-fit indices
  3. Goodness-of-fit indices
  4. Essential accuracy
  5. Accuracy versus continuity
  6. Loss Functions and Goodness-of-fit Indices
  7. The Afriat Index.
  8. The Varian Index.
  9. The Swaps Index.
  10. Non-Linear Least Squares.
  11. The Houtman-Maks Index.
  12. Loss functions to estimate utility
  13. Popular Loss Functions
  14. Welfare when there are no best-fitting utility functions
  15. Loss Functions and Consideration Sets
  16. ...and 15 more sections

Key Result

Theorem 1

A purchase dataset $D = (\boldsymbol{q}^t,\boldsymbol{p}^t)_{t \leq T}$ can be rationalized by a well-behaved utility function if and only if $D$ satisfies GARP.

Figures (4)

  • Figure 1: The dataset on the left, referred to as $D_1$ in Example \ref{['example:bad-news']}, satisfies GARP. The sequence $\boldsymbol{q}_n = (4 - \tfrac{1}{3n}, 4 + \tfrac{2}{3n})$ tends to the location where the two budget lines cross (i.e. where the arrow is pointing in the figure to the left). The dataset which contains this limiting bundle, depicted on the right, does not satisfy GARP.
  • Figure 2: The GARP-violating dataset $\widebar D$. Depicted are various binary relations over the bundles $\boldsymbol{q}$, $\boldsymbol{q}'$, and $\tilde{\boldsymbol{q}}$.
  • Figure 3: Weak and strong compensation levels for a 25% reduction in soda consumption using the Varian index
  • Figure 4: Weak and strong compensation levels for a 25% reduction in soda consumption using the Varian Index

Theorems & Definitions (42)

  • Theorem 1: Afriat
  • Definition 1
  • Example 1
  • Proposition 1
  • Definition 2
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Example 1: continued
  • Proposition 5
  • ...and 32 more