Selection bias - Will multiple regressions solve the problem? Now I have a data from an African country concerning (i) levels of corruption across public sectors and (ii) perceptions of service quality from households (bad, medium and high). The data consist two types of household: those who have used the public services and those who have NOT. 

I have read several papers from some respected journals (World Development, Journal of Development Economics) that address the problem of selectivity bias: those who have not engaged in public services might do so because they knew they would have to bribe (corruption) or they had bad experience from the past, and they would end up feeling bad. 

Authors from the papers, however, do not use Heckit models, that I have learned from my degree. Instead, they argue that by running two regressions: (i) using data on those who actually used the services and (ii) all households in the sample, regardless of the service usage. I feel this approach is not correct. 

I wish to understand further how selection bias, particularly in this example, should be handled. Some problems with the data are that the data might be subjective, quite small (around 500 households), and prone to measurement errors. Do you have any suggestions on dealing with the problems? 

Many thanks.
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.