Introducing Ph.D. students to asymptotic inference for two‐stage M‐estimators: Easing analytic and coding demands via the use of numerical derivatives

Applications of two‐stage M‐estimators (2SMEs) abound in empirical economics. Asymptotic theory for 2SMEs (correct formulation of the asymptotic standard errors [ASE]) has been available for decades. Nevertheless, due to the daunting nature of the requisite matrix formulations, when conducting statistical inference based on two‐stage estimates, applied researchers often implement bootstrapping methods or ignore the two‐stage nature of the estimator and report the uncorrected second‐stage outputs from packaged statistical software. In the present paper, we offer teachers of econometrics a pedagogical approach for introducing Ph.D. students to asymptotic inference for 2SMEs, with a view toward easier software implementation and empirical application. We seek to demonstrate to students (and their teachers) that the analytic and coding demands for calculating correct ASEs for the 2SME need not be burdensome (or prohibitive). The main instructional (and practical) innovation that we offer in this regard is our suggested use of numerical derivative (ND) software for calculating the most challenging components of the ASE formulations. An exercise demonstrates to the student that, by implementing ND software, one can overcome the analytic and coding impediments to conducting inference based on 2SMEs, without abandoning rigor.