Heterogeneous Elasticities, Aggregation, and Retransformation Bias

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.