# Simultaneous equation model without instrumental variables

Very short question: are there tools (by preference in R or Stata) to solve a simultaneous equation model, without needing instrumental variables?

In my case, I would like to model irrigation and croptype. A farmer needs more or less irrigation depending on the croptype. However, the croptype that he chooses also depends on the possibility to irrigate.

Therefore:

Irrigation = climate + Croptype + .......
Croptype   = geography + climate + Irrigation + ......


In my understanding, I need instruments for irrigation and croptype to solve this endogeneity problem (created by simultaneous equations). Is this correct, or can I also solve this without instruments?

The problem namely is that I do not find good instruments. There are several reasons for this:

1. Croptype and irrigation are endogenous on there own because they are farm management variables. All variables on farm level are endogenous and have the same problem as the decisions are interdependent on each other.
2. In the equations, there are a lot of exogenous variables such as climate, land value, geography, policy factors, cultural factors, economic factors... any search for additional variables seems to give high correlation problems with the other variables. Nevertheless, correlation with the endogenous variable stays Low.
3. If this information might help: I have a short period of panel data but I would Prefer not to use them as I would like to examine the climate impact on the system.

Maybe as well important to know is that both irrigation and croptype are categorical variables.

Crossposted! I noticed a similar question which has not yet been answered. Therefore I also crossposted my question here.

• I am not sure whether I am misinterpreting the question, but would not a structural equation model also do the work? – Peter Birch Jan 20 '17 at 10:08
• Hi Peter, I am still looking into it. Tomorrow I have an appointement with somebody who knows more about structural equation models. I will let you know what she says! Thanks for the suggestion! – user33125 Jan 23 '17 at 8:06
• Hi Peter, it is not possible to use structural equation models for this type of problems. As far as the experience on my university goes, nobody would know how that would work. So in case you do have experience with that, I would be happy to hear from you, or anybody else. But for now, this is a dead end to me. Thank you very much for your suggestion anyway! – user33125 Jan 24 '17 at 19:45

What you have for your two key variables is a problem of simultaneity, as Irrigation and Croptype codetermine each other. This means that in both of your equations you have a problem of endogeneity, meaning that your covariates are correlated with the error term of the respective model. If you estimate the models as you wrote them down via OLS and without instruments, they will be biased and (possibly) produce inconsistent estimates.

Therefore, as you suggested already, an Instrumental Variable approach would be recommended.

• Yes, I understand all that. But, I do not have appropriate instruments. So my question therefore was: can I do it without instruments? – user33125 Jan 19 '17 at 18:01
• I suppose looking for appropriate instruments will be the "easiest" thing to do. Why is it that you don't have instruments? Is it due to data restrictions? When it comes to finding appropriate instruments, researchers tend to get creative. Your key variables relate to agriculture, so what researchers usually do (I think) is to use some information on geography for IV. Note however, that the instrument should not have explanatory power in the original equation... Finding appropriate instruments requires solid knowdledge of the relevant subject, so I'm afraid I cannot really help you there – Tartan Leaves Jan 19 '17 at 18:17
• Thanks for your suggestion but I really don't find good instruments. I have a huge amount of data on farmers, geography, climate, policy, farm management... but most of these variables are already in the Original equation, or they are highly correlated with other exogenous variables, or they only have a very Low correlation with 'irrigation or crop type, or (finally) the instrument is endogenous itself. I really posted this question because I believe I cannot find à good IV. – user33125 Jan 19 '17 at 18:22
• Ok. Can you then maybe edit your question to clarify this? Both, that you are aware of the endogeneity problem and that you think that there are no proper instruments. Then the next answer may be more helpful ;-) – Tartan Leaves Jan 19 '17 at 18:26
• Okay thanks! I was probably very desparate when I wrote the question. I Will edit! – user33125 Jan 19 '17 at 18:27

It is a system of two equations with two unknown variables. You can solve the system to get crotype and irrigation in function of the other variables. It is their reduced form that you can estimate. EDIT: It is explained in wikipedia.

I am still working it out, but the solution can be found in STATA, in the gsem software. gsem has a lot of new functions in the most recent version of STATA.

Even though gsem focusses on structural equation models, it has extensions to use it for simultaneous equation models. (See full structural equation model in the following presentation).