# Regression Estimates the Conditional-Variance-Weighted ZZZZzzzzzzz….

As part of my Comparative Methods course, I assigned the book Counterfactuals and Causal Inference by (Cornell sociologist) Stephen Morgan and Christopher Winship. That means—do students know this?—that I had read it again myself before we discuss it next Wednesday. On my long plane ride back from Japan, I read the entire book, which is full of delicious quotes like “the OLS estimate represents the conditional-variance-weighted estimate of the underlying causal effects of individuals, δi, where the weights are a function of the conditional variance of D” and “It turns out that the Wald estimator is indeed consistent for δ in this scenario, but only when is δ considered an invariant structural effect.” I slept pretty well.

But I must emphasize, this book is really useful, and should be read more widely by political scientists than it is. I want the students to understand how to relate the potential outcomes framework to their own research, but my greatest personal interest in the book lies in its treatment of OLS regression, which is really complete. I must confess that I was unaware how difficult it is to interpret OLS regression coefficients within the potential outcomes framework, even with a binary treatment and the “proper” covariates and no selection or endogeneity. We almost never estimate the effect of X on Y. (And this is not just that there isn’t one causal effect out there waiting to be estimated, my point goes deeper.) What we do instead is calculate statistics that if we are lucky have some causal interpretation, one that we usually cannot state with much precision.

This leads to a question. Is there any paper out there that describes simple multivariate OLS with continuous treatments in terms of the potential outcomes framework? I mean really simple: Y = a + b1X1 + b2X2, where both X1 and X2 are continuous and there are no issues of selection, endogeneity, etc. to worry about. What precise causal effect does b1 estimate in this model? Morgan and Winship do this for the cases where X1 is binary, and suggest ways to generalize this to many-valued treatments, but I’m interested in how to match the way that we talk about b1 in most observational work using regressions (“holding X2 constant, a one-unit increase in X1 is associated with a b1 increase in Y”) with the potential outcomes way of discussing the same (“the something-something-something treatment effect of X1 is b1“).