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, *δ _{i}*, 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 + b_{1}X_{1} + b_{2}X_{2}, where both X_{1} and X_{2} are continuous and there are no issues of selection, endogeneity, etc. to worry about. What precise causal effect does b_{1} estimate in this model? Morgan and Winship do this for the cases where X_{1} 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 b_{1} in most observational work using regressions (“holding X_{2} constant, a one-unit increase in X_{1} is associated with a b_{1} increase in Y”) with the potential outcomes way of discussing the same (“the something-something-something treatment effect of X_{1} is b_{1}“).

Since I don’t know how to think about this myself, I’m curious if you ever found a paper or textbook that does this… Continuous treatments under the potential outcomes framework are tricky. I’m aware of some papers on instrumental variables (Angrist and Imbens) and matching (Hirano and Imbens) that come somewhat close to what you want, but don’t work out the math or intuition for simple OLS.

[…] of circumstances. Dedicated readers will remember that I wondered about this very issue in my post Regression Estimates the Conditional-Variance-Weighted ZZZZzzzzzzz… from early 2012. From my perspective, the benefit of the Aronow and Samii paper is to focus tightly […]