Does the regression intercept capture the effect of the treatment year? I read in a notes that runs the following regression
$Y_{it}=\alpha+\beta_1D^{-1}_{it}+\beta_2D^{+1}_{it}+\gamma X_{it}+\epsilon_{it}$
$Y_{it}$ is some health index for individual $i$ in year $t$, $i=1,...,1540$ represents individuals(patients) and $t=1,2,3$ represents years and a treatment happens at $t=2$. 
$D^{-1}_{it}$ and $D^{+1}_{it}$ are dummy variables: in particular, $D^{-1}_{it}$ equals one if the individual is in one year before the treatment (individuals with $t=1$), $D^{+1}_{it}$ equals one if the individual is in one year after treatment.
$X_{it}$ is some individual or time varying control variable, and $\epsilon_{it}$ is the regression error term.
My question is: does the OLS estimate of $\alpha$ capture the effect of $t=2$ (the treatment year) on the average value of $Y$? Why or why not? It would be great if you could show it either intuitively or mathematically. Thanks!
 A: When $t=2$, both your $D$'s are zeroes, so equation reduces to 
$$Y_{it} = \alpha + \gamma X_{it} + \epsilon_{it} $$
Now, see that setting $X_{ij}=0$ leaves only $\alpha$ and error term (that have 0 mean) in you model. So $\alpha$ is a mean of $Y$ for $t=2$ and patients with $X_{ij}=0$ resulting from your model.
You asked 

does the OLS estimate of $\alpha$ capture the effect of $t=2$ (the treatment
  year) on the average value of $Y$?

There is no such thing as "effect of $t=2$". It is always effect of $t=2$ compared to something (in your case to $t=1$ or $t=3$).
Let's see effect of $t=2$ compared to $t=1$. Imagine two patients with the same $X$, one on $t=2$ other on $t=1$. Or easier: one patient in $t=1$ and $t=2$ with the same $X$ on those days so for him/her $X_{i1}=X_{i2}$.
Then $Y_{i2}=\alpha+\gamma X_{i2}$ and $Y_{i1}=\alpha+\beta_1+\gamma X_{i1}$. 
So $Y_{i2}-Y_{i1}=-\beta_1$. So effect of $t=2$ compared to $t=1$ is $-\beta_1$.
And, per analogiam, effect of $t=2$ compared to $t=3$ is $-\beta_2$.
