I am using probability scoring on a data set where one variable has very small area of common support between treated and control (a small area in the middle exists). When I run a logit regression in R, I get the error message that "fitted probabilities numerically 0 or 1 occurred". A basic assumption in propensity scoring is that the probabilities lay between 0 and 1. What should be my next step to address a problem like this?
As you said, an assumption of causal inference is positivity, which is that all units have a nonzero probability of being in either treatment group. If this condition is violated, you essentially cannot make valid causal effect estimate without heroic assumptions. If you are willing to make an assumption, which is that the effect of that variable does not moderate treatment in any way, then after matching on the other variables, you can run a linear model including treatment and the covariate. The estimated coefficient for the treatment will be the causal effect of treatment assuming the effect of that covariate is the same across groups. This is essentially the logic of a regression discontinuity, where the treatment effect is only truly valid in the small area of common support but it is assumed that its effect would be constant across the range of the covariate.