I am attempting to estimate a multinomial logit/probit model for how a rebate is used. There are three outcomes: spending (1), debt (2), and savings (3). I need to account for selection bias in this model (i.e. account for receipt of rebate). My thought was to do a two-stage estimation where the first stage I estimate a probit model for the binary variable receipt (or not ) of rebate. Then calculate the Inverse Mills Ratio using the predicted values from the probit. Then estimate the multinomial model with the inclusion of the Inverse Mills Ratio. Does this seem like the correct procedure? I have done a brief review of the literature and it seems like using the Inverse Mills Ratio to correct for selection bias in a multinomial model, but I am not sure what other methods to employ. Thank you much for your time and help.
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