Finding correct elasticities in log-linear and exponential models allowing heteroskedasticity

Log-linear models are popular in practice because the slope of a log-transformed regressor is believed to give an unit-free elasticity. This widely held belief is, however, not true if the model error term has a heteroskedasticity function that depends on the regressor. This paper examines various mean – and quantile-based elasticities (mean of elasticity, elasticity of conditional mean, quantile of elasticity, and elasticity of conditional quantile) to show under what conditions these are equal to the slope of a log-transformed regressor. A particular attention is given to the ‘elasticity of conditional mean (i.e., regression function)’, which is what most researchers have in mind when they use log-linear models, and we provide practical ways to find it in the presence of heteroskedasticity. We also examine elasticities in exponential models which are closely related to log-linear models. An empirical illustration for health expenditure elasticity with respect to income is provided to demonstrate our main findings.