Yes, you need an exclusion restritcion/instrumental variable. Nevertheless, it is quite hard to find any variable that might affect entering the labour force but not unemployment. In this respect, try to review the literature. Even in the more common framework consisting in studying wages/employment, it is cumbersome to find a variable that reasonably might affect employment and not wages. You cannot test those correlations as you do not know the true data generating model. You mostly have to discuss it. A possibility could be to look at some IVs in the literature and to find that the 2-equation model and the model without selection provide similar results.

The main model should be a probit model because the variable is binary and the selection equation is also a probit. It would not make much sense to put together a probit and a logit model. With two probits, you should estimate the modelo using maximum likelohood methdos. If I am not wrong, Stata should allow carrying out such a model.

I hope it helps.

As far as I am concerned, what you did is Heckman two-step method rather than Instrumental variables.

If you just want to tacklethe problem of sample bias from unemployed people, only using Heckman two-step method is enough. For detailed information, you can read "Microeconometrics: Methods and Applications" by Cameron and Trividi. If you want to estimate using stata, please type "help heckman" to see how to use this code.

For IV, it depends on your story. The IV should only affect Y through X, and cannot directly affect Y. I do not think we can find proper IV from LFS dataset. Maybe you have to look for other datasets, and try to find an IV from other datasets, and combine those 2 datasets.

Thank you very much José-Ignacio Antón and Kaizhao Guo

Dear José

Is it possible to run the model without an instrumental variable? How do I know the results are incorrect ? Wage though seems to be a bad instrument since it affects both dependent variables.

Another question is: why do we do the process of excluding ? In other words, how is the exclusion done and why?

I am doing Heckman 2-step but I think i still need an instrument. Let me explain more so you can have a clear idea.

Dear Kaizhao

I am doing Heckman 2-step but I think i still need an instrument. Let me explain more so you can have a clear idea.

In the first equation, probability of joining the labour market ( 1 if the individual is in the labour market, and zero otherwise)

Those with value of zero are not observed in the second equation, so the model calculate only those who are in the labour market (value 1). Implementing an instrument will make everyone in the sample observed.

The second equation is also binary outcome variable ( 1 if the individual unemployed and zero otherwise)

You mentioned that "If you just want to tackle the problem of sample bias from unemployed people, only using Heckman two-step method is enough." Based on my elaboration above, does your statement remains valid ? I just want to double check that I got your idea right.

Thank you very much again to both of you

In the Heckman procedure, you need a variable in the first equation not present in the second (i.e., an instrument).

Theoretically, you perform the Heckman procedure without "the instrument", as the modelo would be identified by the functional form. In practise, any referee will ask for a model with an exclusion restriction.

Thank you very much for this

You explained the issue well.

I do have a question though about coding the dependent variables since I have 2 binary.

I am trying to do the analysis without an IV (no exclusion restriction applied for now) and faced an issue with coding the variables.

The first participation dummy variable will have say, 100 people that are in the labour market (either employed or not employed) and 1000 out of the labour market. Participation in Labour market will take (1) – out of the LM take (0) dummy variable (the dependant variable. This will be determined by say X1, X2, X3 and say no IV variable is used.

The unemployed dummy variable will have say, 30 people in unemployment  (1) and 70 people in employment (0) dummy variable. This will be determined by the same above X1, X2, X3 variables.  

Now, the question is how to code the previous equations? The first equation has 1100 participants. However, the 2nd equation has only 100 participants.

Do I code the 1st equation In Labour market (1) and Out (0); The 2nd Unemployed (1) and Employed (0) and leave the rest blank?

Thank you very much in advance

Check this link:

https://www.stata.com/manuals13/rheckprobit.pdf

Stata will deal adequately with the samples. Just be sure that in the first equation, all the values of the outcome are 0/1 and, in the second equation, the values of the outcomes are 0/1 for those observations with 1 in the first equation and "." for the ones with 0 in the first eqqation.

Thank you very much.

That was very helpful

One thing though, the p-value for (rho) is not reported?

How one can determine the significance of the correlation between the 2 residuals? In other words, how can we determine if the rho is significant or otherwise?

AND does it matter if the rho is negative, say ( -0.85 )

Thank you again for the great and valuable comments.

There is a LM test that indicates you if the rho is sgniciantly different from zero or not (see the attachment, please).

The sign of the rho indicates the direction of the correlation between the error terms in both equations. If that sign makes sense, it will depend on the problem. Assuming that something like ability can be in the error, it makes sense that participation in the labour market is positively correlation with ability a unemployment is negatively correlation, so it could make sense.

If you do not use an exclusion restriction, referees are very likely to question the value of the analysis, unless it is just a robustness check.

I guess that one could argue that something like housing tenure should correlated with unemployment but not participation, but I cannot imagine one in the opposite sense: I would just survey the applications in order to see which variables they use.

Best,

José-Ignacio Antón