# 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.

• Could you please post a reference to some of the articles you are referring to? Jun 7 '15 at 4:33
• Hi! I am interested in this article: agencyft.org/wp-content/uploads/2013/11/important-papers-2.pdf. After reading for a while, I feel that the authors solve the bias by both differencing and running multiple regressions. But their writing is not very clear (or maybe it is just me).
– Khan
Jun 7 '15 at 10:05
• I think I would estimate both approaches to check for robustness Jun 8 '15 at 2:01
• thank you! what I am asking is that by running the two regressions (one for users and one for the whole sample) would the selection bias be gone. It makes sense intuitively but I still don't feel it is legitimate? Anyone helps? Besides, could anyone give some thought on the paper I posted above?
– Khan
Jun 8 '15 at 20:28
• You should present more thoroughly the model that you have in mind in the question above. My only guess is that you are not removing the selection-bias, but seeing if it is an issue in your estimares (by testing your model with the different samples). Jun 8 '15 at 21:07

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.

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.

• Thank you very much @Jamzy for your answer. However, I couldn't mark it as answering my question because: 1. (though you have provided an excellent sum-up of 2SLS), it is not directly answering (how selection bias in the example should be handed) and 2. It is not necessarily, and in fact not, 2SLS being used here. In fact, the paper I posted mistakenly used two stage regression with the first stage of probit and the second stage of OLS (famously christened as forbidden regressions, which is wrong). After about 2 months reading the paper, I comes to a conclusion that the paper is not strong.
– Khan
Oct 13 '15 at 23:13
• And their conclusions should not be taken as its face value. However, I do really appreciate your dedicated help and your amazing knowledge. Thank you @Jamzy.
– Khan
Oct 13 '15 at 23:14
• I do agree with you regarding the paper. The methodology seemed a little bit strange to me but I didn't go through it in too much detail. I guess my answer is only partially complete, multiple regressions can definitely solve selection bias issues, but I cannot comment on validity of the model used. I do suggest you answer this question here yourself though since you have gone through it in detail. Oct 13 '15 at 23:19