Causal Inference Using Multiple Regressions

In Chapter 3, we found that the OLS coefficients were ratios of particular covariances to variances. Variances are always positive. Hence, a coefficient, β^j generally has the same sign as the simple covariance between the associated regressor xj and y, and always has the same sign as the covariance between that part of xj unassociated with other xks and y. In Chapter 4, we found that OLS estimators could not reflect causal impacts unless E(μ|X) = 0 (assumption 3 held), which further implied that the error term was uncorrelated with the right-hand side regressors (or anything that should have been included in the model). We stress these two things in this chapter: first, a covariance (or correlation) does not indicate causality, and second, that causal estimates require that the treatment is either randomly assigned or can be treated “as if” it was randomly assigned.