In a previous post we looked at the properties of the ordinary least squares linear regression estimator when the covariates, as well as the outcome, are considered as random variables. In this post we’ll look at the theory sandwich (sometimes called robust) variance estimator for linear regression. See this post for details on how to use the sandwich variance estimator in R.
Linear regression
Linear regression with random regressors, part 2
Previously I wrote about how when linear regression is introduced and derived, it is almost always done assuming the covariates/regressors/independent variables are fixed quantities. As I wrote, in many studies such an assumption does not match reality, in that both the regressors and outcome in the regression are realised values of random variables. I showed that the usual ordinary least squares (OLS) estimators are unbiased with random covariates, and that the usual standard error estimator, derived assuming fixed covariates, is unbiased with random covariates. This gives us some understand of the behaviour of these estimators in the random covariate setting.
Regression inference assuming predictors are fixed
Linear regression is one the work horses of statistical analysis, permitting us to model how the expectation of an outcome Y depends on one or more predictors (or covariates, regressors, independent variables) X. Previously I wrote about the assumptions required for validity of ordinary linear regression estimates and their inferential procedures (tests, confidence intervals) assuming (as we often do) that the residuals are normally distributed with constant variance.