Identification of a Heckman Model through Nonlinearity

Question

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.

From Wikipedia:

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.[6]

Answer 1

For $X = Z$, the Wikipedia page you quoted gives the identifying equation $$ E[w \mid D = 1] = X\beta + \rho \sigma_u \gamma(X \gamma). $$ If the inverse Mill's ratio is approximately linear in $X\gamma$ then we can write $$ E[w \mid D = 1] \approx X\beta + X \theta = X(\beta + \theta). $$ Therefore, the OLS coefficient on X does not estimate the structural parameter $\beta$ (it estimates $\beta + \theta$). Of course, this does not mean that in this case the model is unidentifiable (if you know this is happening you might be able to correct for this). It does, however, imply that the Heckman identification strategy does not work. Since the linear approximation does not hold exactly, this is not a problem of theoretical identification. It is more a practical problem similar to identification in the presence of strong but not perfect colinearity.

According to Wikipedia, observations in the tail are important since a linear approximation of the inverse Mill's ratio works better at the origin than it does for large $X \gamma$ values. I haven't checked this but it should be relatively easy to verify.