Endogenous Sample Selection and Heckit Correction in relation to Research Question

Question

I am researching the funding amounts of start-ups. I am calculating two models:

Model a) Funding Chances (binary dependent variable funding_yesno) on full sample 

Model b) Log(Funding Amount) on subsample of *funded* start-ups

The IV/CV on both models are identical.

The classic example is estimating "wage offer" and subsampling on labor market participation. In that classic example the "wage offer" can not be observed when not participating in the labor market. However, in our example, we know that the funding amount actually is 0 if a start-up is not funded.

Therefore, I am wondering, is Model B a case of endogenous sample selection, what induces a sample selection bias I need to correct for? Or does the evaluation of such depend on my research question / hypothesis? So, could a hypothesis concerning the funding amount simply state "among funded start-ups" to overcome a potential bias?

Futhermore, Wooldridge (2016, p. 556) states on Heckman correction: "Intuitively, if we do not have a variable that affects selection but not y, it is extremely difficult, if not impossible, to distinguish sample selection from a misspecified functional form". Therefore, I am wondering: can't I simply apply the Heckit approach, with the same IV/CV for both the Model A (Selection Model) and Model B?

Thanks a lot for your most valued opinions on those questions!

Answer 1

From my understanding, the outcome variable you want to model is funding amount. But, your sample consists of firms that do receive funding and firms that do not receive funding. Modeling funding amount on the whole sample might suffer from selection bias as those firms that seek/receive funding might be systematically different from firms that do not seek/receive funding. I emphasize might because this is purely a hypothesis, and determination about the potential for sample selection must come from your theory.

Your theory must therefore predict what factors determine the likelihood of a firm receiving any amount of funding in the first place. These are the variables you will include in the first stage of your Heckman correction. You need to have at least one instrument variable in the first stage that predicts the likelihood of receiving any funding in the first place and not affecting variation in the amount of funding received. This will help you develop a valid exclusion restriction in the selection equation. See this paper for a robust explanation of why your predictors in both stages must differ to get more accurate estimates.