Meet Shah

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.

Reforms which reduce misallocation incentivise distorted firms to expand, but doing so requires finance. I show theoretically that removing one distortion can therefore worsen another: firms which would gain the most from reform may be unable to finance their desired expansion, as pre-reform they have the lowest internal funds as they are the most distorted. I test this mechanism using two contemporaneous policy changes in India. The introduction of the Goods and Services Tax (GST) reduced input-tax cascading, while the Asset Quality Review and subsequent revised Prompt Corrective Action framework generated a credit-supply contraction among distressed banks. I construct firm-level exposure to both shocks using pre-reform indirect tax payments and the composition of firms’ local bank branch networks. Firms more exposed to the credit shock borrow less and experience lower sales growth. The average estimated gain from GST is around 15% of sales over 2017-2021, but declines sharply with exposure to the credit shock. Firms one standard deviation above mean credit exposure lose around 6 percentage points of the reform gain, while those two standard deviations above lose up to 12 percentage points. At the highest exposures, the estimated gain is statistically indistinguishable from zero. These results show that the gains from removing distortions depend on the financing environment available. The loss is persistent: allocations were worse for constrained firms for at least 6 years after the reforms.