Standard Errors for Regression-Based Causal Effect Estimates in Economics Using Numerical Derivatives

The aim of nearly all empirical studies in economics is to provide scientific evidence that can be used to assess policy relevant cause-and-effect. In the context of the general potential outcomes framework, we review how a causal effect parameter can be rigorously but tractably specified, identified and estimated along with its asymptotic standard error. For cases in which the analytic and computational requirements for calculation of the ASE are challenging, we suggest the use of numerical derivatives (ND). We detail the specific type of ND software required for this purpose, and note that it is offered as a feature in most statistical packages. As an illustration, we analyze the causal effect of wife's high school graduation on family size using the Stata/Mata deriv command. Code for this example is supplied in an appendix.