Disagreement Among Firm-Level Productivity Estimators and Its Consequences for Applied Inference

Firm-level total factor productivity is not observed: it is estimated, and an applied regression inherits that choice. We give four elementary propositions on the two measurement errors involved. On the left-hand side, the gap between any two residual-based measures in a downstream regression has a closed form in their implied factor elasticities. On the right-hand side, attenuation depends on the reliability of the variation that the estimator uses, and a two-rating design is exactly identified, so its uncorrelated-errors assumption has no testable implication. The propositions are applied to 29,434 firm-years across 4517 Chinese A-share firms (2015–2024) and to eleven measures: six parametric production-function estimators and five nonparametric frontier measures. The closed form predicts the realised gaps among the parametric measures with a correlation of 1.000. Under one specification, the six parametric measures give a positive ESG coefficient, significant at 1 percent, and the five nonparametric measures a coefficient indistinguishable from zero. The nonparametric measures penalise undesirable outputs, so the contrast is not estimator disagreement alone. A conventional slacks-based score on the same inputs and revenue, without those outputs, ranks with the baseline parametric measure at 0.70 rather than 0.23, and its ESG coefficient is positive.