Include constant to estimate propensity scores? Should we include the constant when we estimate propensity scores?
We model the probability that $D_i=1$ using a set of controls $x$
$$ p (D_i=1 | X=x) = \alpha + x \beta + \varepsilon $$
to created predicted values $\hat{p}$. Should we include the $\alpha$?

I agree with @Noah. I just want to add the following. Suppose we have the outcome $y$ and predictor variables $X_i$ in a regression model, where $i = \{1, 2, ...\}$. Then we have the option to standardize the predictor variables $X_i$ before fitting the regression model and regress $y$ on $X_i - \bar{X_i}$ instead of $X_i$. Therefore, if we use propensity score $\hat{p}$ (estimated from $X_i$) instead of $X_i$, e.g., in PS adjustment, then we could as well "standardize" $\hat{p}$ if that gives us some desirable propriety, e.g., to achieve some interpretation of model parameters, etc. So my view is that it depends on what we want to accomplish with $\hat{p}$.
