What would be the intuitive interpretation of the ratio of probability and cummulative density function during the first stage of Heckman selection model?
Is it the likelihood of selection for the sample?
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The Inverse Mills Ratio - the ratio of the PDF to the CDF - is a hazard function for censoring data. I've listed the Wikipedia page for Survival Analysis below, which is directly related to hazard functions.
To clarify with an example, if we're doing a Heckman Sample selection for wages and there is a decision to work or not the IMR is capturing said truncation. The intuition is that $\lambda = \frac{\phi(z'\theta)}{1-\Phi(z'\theta)}$ represents the likelihood that an agent makes it to the second stage given that with their characteristics, i.e. $z'\theta$, they have yet to be truncated.
Note: I have made the first stage decision one typically seen in these Heckman models. Formally, the decision to work in this example is: $d= \mathbb{1}[z'\theta + \mu_{1} \geq 0 ]$.
So, if by likelihood of sample selection you mean the likelihood we observe their second stage data given their first stage characteristics, then yes, you'd be correct. The inclusion of an estimated value $\hat{\lambda}$ purges the endogeneity (sample selection) from the second stage regression.