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 $\alpha$?
 A: Why wouldn't you? The goal is to get good predicted probabilities, and surely you will get better predicted probabilities with an intercept in the model. But you don't have to wonder; you can try your way, assess balance after matching or weighting on the resulting propensity scores, try the other, and compare performance. I really can't imagine too many scenarios in which you wouldn't want an intercept. It removes 1 extra degree of freedom but will likely dramatically improve the performance of the model.
A: 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 $\hat{p}$.
