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Heterogeneous Elasticities, Aggregation, and Retransformation Bias

Ellen Munroe, Alexander Newton, Meet Shah

Abstract

Economists often interpret estimates from linear regressions with log dependent variables as elasticities. However, the coefficients from log-log regressions estimate the elasticity of the geometric mean of $y_i|x_i$, not the arithmetic mean. The unbounded difference between the two is known as retransformation bias and can take either sign. We develop a specification-robust debiased estimator of the average arithmetic elasticity and re-estimate 50 results from top 5 papers published in 2020. We find that 19 are significantly different, with the median absolute difference being 65% of the OLS elasticity estimate. Furthermore, we show standard instrumental variables assumptions with log dependent variables do not identify the elasticity. We specify a control function approach and re-estimate papers that use 2SLS with log dependent variables. We find that 13 of 19 results from top 5 papers are significantly different between the two approaches. Retransformation bias arises as a result of heterogeneous responses. The geometric mean elasticity corresponds to the average response. Arithmetic and geometric means are elements of the power mean family. We show power mean elasticities are sufficient statistics for a common class of decision problems.

Heterogeneous Elasticities, Aggregation, and Retransformation Bias

Abstract

Economists often interpret estimates from linear regressions with log dependent variables as elasticities. However, the coefficients from log-log regressions estimate the elasticity of the geometric mean of , not the arithmetic mean. The unbounded difference between the two is known as retransformation bias and can take either sign. We develop a specification-robust debiased estimator of the average arithmetic elasticity and re-estimate 50 results from top 5 papers published in 2020. We find that 19 are significantly different, with the median absolute difference being 65% of the OLS elasticity estimate. Furthermore, we show standard instrumental variables assumptions with log dependent variables do not identify the elasticity. We specify a control function approach and re-estimate papers that use 2SLS with log dependent variables. We find that 13 of 19 results from top 5 papers are significantly different between the two approaches. Retransformation bias arises as a result of heterogeneous responses. The geometric mean elasticity corresponds to the average response. Arithmetic and geometric means are elements of the power mean family. We show power mean elasticities are sufficient statistics for a common class of decision problems.
Paper Structure (43 sections, 21 theorems, 164 equations, 6 figures)

This paper contains 43 sections, 21 theorems, 164 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Motivating Example: Marginal Value of Public Funds
  3. Related literature
  4. Model
  5. Primitives
  6. Results
  7. Discussion
  8. Identification
  9. Debiased Estimation
  10. Ordinary estimator
  11. Instrumental variables estimator
  12. Empirical evidence
  13. IV re-estimations
  14. Conclusion
  15. Proofs
  16. ...and 28 more sections

Key Result

Lemma 2.4

For any $\phi \in \mathbb{R}$,

Figures (6)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 1 more figures

Theorems & Definitions (40)

  • Definition 2.2: Power-mean aggregators
  • Definition 2.3: Power-mean elasticity
  • Lemma 2.4: Power-mean elasticity representer
  • Proposition 2.5: Non-equivalence under heterogeneity
  • Proposition 2.6: Wedge decomposition of power-mean elasticities
  • Corollary 2.7: No Wedge
  • Example 2.1: Closed-form power-mean elasticities under Gaussianity
  • Example 3.1: Demand, profits, and welfare
  • Example 3.2
  • Proposition 3.6: Representation
  • ...and 30 more