Assumption Lean Regression

Abstract

It is well known that with observational data, models used in conventional regression analyses are commonly misspecified. Yet in practice, one tends to proceed with interpretations and inferences that rely on correct specification. Even those who invoke Box’s maxim that all models are wrong proceed as if results were generally useful. Misspecification, however, has implications that affect practice. Regression models are approximations to a true response surface and should be treated as such. Accordingly, regression parameters should be interpreted as statistical functionals. Importantly, the regressor distribution affects targets of estimation and regressor randomness affects the sampling variability of estimates. As a consequence, inference should be based on sandwich estimators or the pairs (x–y) bootstrap. Traditional prediction intervals lose their pointwise coverage guarantees, but empirically calibrated intervals can be justified for future populations. We illustrate the key concepts with an empirical application.