Usage of estimated coefficients of a first regression as explanatory variables in the second stage of a regression

[Williams Ahouakan]

Dear Statalists
I m working on the relationship between schooling decisions, child labor and the regional rates of returns to school in my country. I so first estimate these regional rates of return to school using Lewbel (2012) approach. And secondly, I would like to use the regional estimated rate of returns to school as explanatory variables to see if they impact on the parents decisions to send their children at school or to work.
So in the first equation, I have estimated Mincer equation using Lewbel (2012) approach to solve the potential problem of endogeneity biais link to school. And in the second step, I would like to use a Bivariate Probit model to examine if these regional rates of return to school impact on parents decisions to send their children at school or make them work. But after the step estimation, I face with 2 problems:
1- Some coefficients of my first step estimation are not significant. Should I consider them as 0 in the second stage of my estimation (I speak about the bivariate probit model)
2-The first stage coefficients (rate of returns to school) are of higher level compared to the second stage estimation. Is there any problem with this?
Thank you to answer me!

[Richard Williams]

With regards to 1, I would never do that. The coefficients could be huge but the standard errors even bigger. If you want effects fixed at zero, then rerun the model with those variables omitted, and then use those coefficients.

[Williams Ahouakan]

Thank you Sir Richard Williams for your answer!
I have to give an other explanation. I have 58 regions in which I have estimated the rate of returns to school! And they are almost all non significant (the rate of return to school are significant only in 4 regions). So I will be obliged to not consider a large part of my sample if I omit them!
Thnaks!

[Clyde Schechter]

Reiterating Richard's thoughts in a slightly different way, just because your effects are not statistically significant, it by no means follows that they are zero. It just means that they are sufficiently imprecisely estimated that we can't tell if they are zero or not. And even though the 54 effects are individually not statistically significant, it is worth noting the possibility that a joint test of them all being zero might still be resoundingly rejected. So for all of these reasons, setting them to zero in your second stage analysis is not appropriate unless you have some unrelated, compelling reasons to do so.

[Williams Ahouakan]

Thank you Sir Clyde Schechter for your answer!
Should I so use these estimates as such (i.e. without remplace them by 0) in the second stage of my regression?
Thanks!

[Clyde Schechter]

In a word, yes.

[Williams Ahouakan]

Thank you Sir Clyde Schechter!

[Williams Ahouakan]

Sirs Clyde Schechter and Richard Williams, what do you think about the usage of the ratio of the coefficient and its standard errors (i.e. the Student t) to take into account of both its magnitude and its precision. A kind of indicator.

[Stephen Jenkins]

The fact that that the first-stage coefficients are estimates should be taken into account when assessing the estimates at the second stage. One relatively "well-known" paper in economics about this issue is: Murphy, K. and R. Topel (1985). Estimation and inference in two step econometric models. Journal of Business and Economic Statistics 3, 370–9. There have been Stata implementation of these methods: type findit topel and look at the material that is revealed.

[Williams Ahouakan]

Thank you for your answer Sir Stephen Jenkins!

[Williams Ahouakan]

Dear Statalists, Sirs Stephen Jenkins, Clyde Schechter and Richard Williams
This is an article (working paper) showing why it is not easy to use the Murphy and Topel (1985)'s approach to assess the problem I try to solve. I would like to share it with you http://crm.sem.tsinghua.edu.cn/Uploa...8225827697.pdf