I was reading the Rubin: Causal inference and Angrist, J.D. and Pischke: most harmless econometrics. Both of them are great textbooks. During my reading, I have the following question: what is the correct interpretation of the coefficient of a covariate variable (rather than the treatment variable) in a multiple linear regression model?
Let's look at a simple random experiment setting. Let $D_i$ be the treatment variable of subject $i$, with $D_i=1$ means treatment and $D_i=0$ means control. Let $X_i$ be a covariate variable, e.g., age. Let $Y_i(1)$ be the value of the response variable if subject $i$ is in treatment group, i.e., $D_i=1$ and similarly, we can define $Y_i(0)$. Then we write $Y^{obs}_i=D_iY_i(1)+(1-D_i)Y_i(0)$, which is the observed value of the response variable for subject $i$. So we are working under the Rubin potential outcome framework.
As the data comes from a simple random experiment, one could directly run a linear regression without including the covariate variable like this \begin{equation} Y^{obs}_i=\alpha+\tau D_i+\epsilon_i. \end{equation} One could show that $\tau=E[Y^{obs}_i|D_i=1]-E[Y^{obs}_i|D_i=0]$. Note that this is always true, even if the data does not come from random experiment. So $\tau$ means: if you could draw infinitely number of samples from the joint distribution of $(D_i,Y^{obs}_i)$, then you obtain the average in the observed response variable for the group of people with $D=1$ and the group of people with $D=0$, then you take the difference, that's $\tau$. Now if the data does come from random experiment, then we further have $\tau=E[Y_i(1)-Y_i(0)]$, which is the treatment effect. Whether the data comes from random experiment or not, the OLS estimator $\hat{\tau}$ is unbiased and consistent of $\tau$.
Now, we know that we could include the covariate $X_i$ in our regression and get \begin{equation} Y^{obs}_i=\alpha'+\tau D_i+\beta X_i+\epsilon_i. \end{equation} Then on page 122 of Rubin: Causal Inference says "...irrespective of whether the regression function is truly linear in the covariates in the population, the OLS $\hat{\tau}$ is consistent for $\tau$". I understand that. But the book never say anything about $\beta$, the coefficient of the covariate $X_i$. In practice, if someone tell you $\beta$ is negative and significant, what exactly does that mean? Think about it, what if $Y$ is the wage and $D$ is whether the subject attends a specific training program. Now if $\beta$ is negative and significant, can we say there is a discrimination against age? What is the correct meaning of $\beta$? What if the data does not come from random experiment, then will there be anything different for $\beta$? just like the difference in the interpretation of $\tau$?