What is a standard error? (And how should we compute it?)

Abstract

I review the definition of a standard error from a frequentist perspective, including both exact analysis and asymptotic analysis. Using the linear model for illustration, I discuss the model-based, design-based, and sampling-based approaches to uncertainty in obtaining standard errors. The model-based approach is widely applicable and produces reasonable measures of estimator precision in many settings. In some situations, particularly in the context of clustering, the model-based approach can suffer from ambiguity, and can lead to standard errors that are systematically biased. A combination of the design-based and sampling-based approaches requires the researcher to think about the variation in key explanatory variables when computing standard errors, and it can even apply to cases where the entire population is observed.