19.11 Nonlinear IV

  • assumption so far: linear model
  • cannot do a double-logit: forbidden regression (Woolridge, 2010)
  • one possibility: linear probability model (OLS)
    • but: poor performance of OLS vs. probit/logit when the mean of the binary variable is near 0 or 1
    • estimates that are less precise than models that properly take into account nonlinearity

Binary treatment

Woolridge (2010)

  1. estimate 1st stage with nonlinear regression (probit): treatment on instruments and controls
  2. get the predicted values
  3. use predicted values in place of instrument in 2SLS

treatment effect regression

  • avoids 2SLS
  • directly models the binary data structure
  • estimate the probit first stage and linear second stage at the same time
  • sampleSelection::treatReg

Binary Outcome: Control function approach

  • don’t isolate the explained part of \(X\) and use that in the 2nd stage
  • instead: use \(X\), but also control for the unexplained part of it
  • linear IV: same results as 2SLS
  • ivprobit

Both are binary: Bivariate probit

  • 2SLS can be especially imprecise when both stages are binary
  • ML estimate: two probit models at the same time
  • dependent variables of the models are correlated
  • GJRM::gjrm(Model = "B", margins = c('probit','probit'))