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I assumed you would need to do an IV probit in this scenario, but it was not used in this AER paper - Children and Their Parents' Labor Supply (Angrist and Evans, 1998).

The first stage is estimated using OLS:

  • more than 2 kids(1 or 0) = IV(1 or 0) + error

The second stage is:

  • labor force participation (1 or 0) = more than two kids (1 or 0) + error

These are all estimated using regular 2SLS, which involves OLS, and I thought that would not be appropriate?

The second stage uses different dependent variables which are usually continuous, but the first stage is always OLS with a dependent variable dummy. Why is this something that is okay for the first stage of a 2SLS?

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Using OLS on count and dummy variables is appropriate. The estimates are consistent. Using OLS instead of probit/logit is appropriate if the number of observations is sufficiently high. For example Bleakley (QJE, 2007) uses this approach on survey data.

I don't know why you would imagine that logit or probit needs to be used in the first stage either. The first stage is a simple projection of the instrumented variable on the rest. There is no room or necessity for a link function here.

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