I often read that the standard Heckman selection model is identified from the non-linearity of the Mill’s ratio.
I don’t fully understand why linearity (or lack of) determines identification.
The model obtains formal identification from the normality assumption when the same covariates appear in the selection equation and the equation of interest, but identification will be tenuous unless there are many observations in the tails where there is substantial nonlinearity in the inverse Mills ratio. Generally, an exclusion restriction is required to generate credible estimates: there must be at least one variable which appears with a non-zero coefficient in the selection equation but does not appear in the equation of interest, essentially an instrument. If no such variable is available, it may be difficult to correct for sampling selectivity.