# Discover causal effects using OLS: does the treated and not treated group need to be similar on all observed variables?

I have one dummy variable, $$D$$, which equals 1 if the subject received treatment and $$0$$ otherwise.

My outcome of interest is $$Y$$. For example, $$D$$ tells me whether the subject took the drug or a placebo and $$Y$$ is a continuous variable measuring pain. I want to discover whether taking the drug reduces pain.

I have other variables that measure some features of the subjects, let's call them $$X_1$$ and $$X_2$$. For example, $$X_1$$ is the age of the subject and $$X_2$$ is the amount of physical activity the subject does each day.

By a t-test I discover that the mean of $$X_1$$ is different between the treated and not treated group, and the mean of $$X_2$$ is different between the two groups as well.

So I cannot use a naive estimator, and I understand that. It may be that group 1 experience less pain because the subjects in that group are younger, not because of my drug.

But if I write: $$Y = \beta_0 + \beta_1 D + \beta_2 X_1 + \beta_3 X_2$$ and run an OLS on it, will $$\beta_1$$ be the effect I am looking for?

Is this model correctly specified?

Yes, $$X_1$$ and $$X_2$$ are different between the two groups: the two groups are not the same (so there is no randomization). But, I'm controlling for the difference. I put the variables in the model, so I am accounting for the difference between the two groups.

Would this model work?

• The design could be this: the doctor send patients with pain to us. We administer the drug. Afterwards the patient go back to the doctor and have pain level measured. All patients go back to have the pain measured, but not all patient come to us for the drug. The ones that come to us are older ($X_1$) and less physically active ($X_2$). The young patient don't have the drug, neither the physically active. If $D$ is a dummy which equals 1 if the patient was given the drug, and $Y$ is pain level, does an OLS $Y=D+X_1+X_2$ will uncover the causal effect of the drug on the pain level? – robertspierre Apr 30 at 12:36