The variance of regression coefficients when the population is finite

Recent work has returned attention to the role of finite -population corrections in empirical settings. It is well established that if the only source of variation arises from the sampling design, then the asymptotic variance of regression estimators must include the proportion of the finite population that is sampled. If there is, in addition, a random shock to each element of the finite population, then it is commonly observed that the resulting super -population renders the finite -population correction moot. We explore this setting and find that this common observation does not fully capture the richness of the result. The fraction of the finite population that is sampled defines bounds on the variance of regression estimators. Ignoring the finite -population correction yields the upper bound, which can be quite conservative.