Different forms of Inverse Mills Ratio & their interpretation

I have noticed different authors using different forms of IMR, i.e., $\frac{f(x)}{F(-x)}$ or $\frac{f}{(1-F(x))}$ depending on whether they are modeling selection or non-selection in the first-stage model. I did some simulations in my case where a lot of observations were censored at $0$. In different runs, I tried $\frac{f(x)}{F(-x)}$, or $\frac{f}{(1-F(x))}$, $x\beta$, and the probit score, but to my surprise, the coefficient on my variable of interest hardly changed even though magnitude and signs of IMR changed drastically.

My questions go like this:

1. Is it true to say irrespective of the formula applied, $\beta_{k}$ on variable of interest adjust accordingly? I mean, as long as we are factoring selection bias, $\beta_{k}$ on variable of interest is unbiased and consistent.

2. How do I interpret IMR? I have observed that $\frac{f(x)}{F(-x)}$ is directly proportional to $x\beta$ (and hence the probit score), while $\frac{f}{(1-F(x))}$ is inversely proportional to $x\beta$. So, a positive sign on IMR should suggest that an increase in IMR (or decrease in $x\beta$ if we are using $\frac{f(x)}{F(-x)}$ leads to higher Y value. In other words, the smaller the likelihood of the first stage model, the greater the Y value. Is that correct?

There is a difference in interpretation but the math between the formulas should be the same.

Let's first recall some definitions.

For one, $f(.)$ generally refers to PDF which give the probability of an event; integrating that function yields $F(.)$, which tells us the probability that a value is less than or equal to the argument.

Thus, for a random variable $x$, the expression $\frac{f(x)}{F(-x)}$ indicates the likelihood of an event conditional on the probability being less than or equal $-x$. For a symmetric distribution like the Gaussian distribution, $1-F(x)$ and $F(-x)$ are identical. If you are confused about this, think about the standard normal distribution and a z-score of 1. The probability of having a z-score greater than 1 is the same as that of having a z-score less than or equal to -1.

To answer your questions directly now, 1.) if you used either of those formula - assuming your code and specifications are correct - then they should provide you with the same result. Otherwise, you are making an error somewhere. 2.) I answered another question about the interpretation of the Heckman term. See here where I described the interpretation.

For sources of my answers, I referenced my Greene Econometrics text and the simple z-score example I thought of myself. I hope this clarifies your confusion.

Edit: note they should be the same in your example because you made the assumption that the data is distribution normally via the probit

A simple answer to question 1: Your other coefficients and their SEs are not affected by choosing f/F or f/(1/F). It will only change the sign of the IMR's coefficient.

As for question 2: Better econometricians might correct me here, but I have never seen signs and coefficients of IMRs to be interpreted. Afaik, they are just an abstract variable containing a bias. If they are significant, they correct for the present selection bias (given a correct specification of the selection model) and that is how far the interpretation usually goes. Less applied scholars might see things differently though.