Selection bias - Will multiple regressions solve the problem?
What you are referring to is two-stage least squares. This is an instrumental variable commonly applied to correct for endogeniety and selection bias. It is a pretty hot topic in economics at the moment and, when applied correctly, can be very useful and will remove the selection bias.

There are a few conditions and assumptions -
Suppose you want to estimate this equation

$y_i = \beta x_i +\epsilon_i$
where


$i$ indexes observations,
$y_i$ is the dependent variable,
$x_i$ is an independent variable,
$\epsilon_i$ is an unobserved error term representing all causes of $y_i$ other than $x_i$
$\beta$ is an unobserved scalar parameter.


we suspect $x_i$ is endogenous


($cov(x_i, \epsilon)\neq 0$)


but we have a variable $z_i$ which is correlated to $x_i$ but uncorrelated with $\epsilon$


($cov(z_i, x_i) \neq 0$ and $cov(z_i, \epsilon)=0$). 


This is a candidate for 2SLS:

Stage one is to estimate $x_i$:

$\hat x_i= \hat \gamma z_i + \epsilon$ 

Stage 2 is to estimate $y_i$

$y_i= \beta \hat x_i+ \epsilon$

The difference between this and the first regression is that $\hat x_i$ is uncorrelated with the error term and this selection bias has been corrected for. There are lot's of different uses of this. In principle, the two-stage regression used in the paper could correct for some of the biases discussed in the paper. 

Further Reading:

This ppt (Slide 20 onwards) descibes an excellent (and very famous) example of IV.

These notes are also pretty solid as well.

This question provides some textbooks which are also very helpful. My personal favourite is "Mostly Harmless Econometrics" by Angrist and Pischkes (Written by the guy who did that famous example example above) and "Econometric Analysis of Cross Section and Panel Data" by Wooldridge. It is worth noting that both of these are graduate level books.