# How to choose instruments for GMM estimation?

I have a linear regression with one dependent variable and ten INdependent variables. I want to estimate this relationship using GMM, but I need at least ten instruments.

Can I use the same dependent variables as instruments for the regression? If not, how would I go about choosing instruments in such a case?

EDIT: my research studies reported generalized trust levels in 77 nations and has covariates such as lnGDP2013 lnPopulationSize2014 dummy variables for history of legal institutions from: Germany, Scandinavia, Britain, France, Germany, a dummy variable for whether or not a nation was involved in the transatlantic slave trade and a dummy variable for whether or not a nation was colonized by Europe in the last 300 years.

I have transformed the dependent variable from a percentage TRUST = Percentage of people who answered that 'most people can be trusted' when surveyed to ln(TRUST/1-TRUST) which gives residuals whose estimated kernel density is roughly normal.

In addition I have a few other covariates such as probability of two individuals chosen from a nation being of the same ethnic group.

EDIT2: I've parsed the model down so I am using GMM to explain the impact of whether or not a nation send or received slaves during the transatlantic slave trade on reported generalized trust levels using Africa as an instrument for whether or not a nation sent or received slaves. I can argue that being an African nation in and of itself should not having a dampening impact on trust but would be correlated with the likelihood that a nation was involved in the transatlantic slave trade.

• This is a typo: one dependent variable and ten dependent variables – jmbejara Nov 20 '16 at 3:50
• Also, to be able to better answer your question, you need to be more specific about the context. Could you add some more details to your question? Choosing instruments requires exogeneity, which we can only discern through context. – jmbejara Nov 20 '16 at 3:51
• shouldn't it be 1 dependent, and ten INdependent?? – An old man in the sea. Nov 21 '16 at 10:47
• yup, my apologies – MHall Nov 21 '16 at 19:17
• @MHall <- Use @ to tag a certain user. In that way we get a message warning of any reply. – An old man in the sea. Nov 21 '16 at 21:07

You have a linear model $y = \beta_0 + \beta_1 x_1 + \cdots + \beta_{10} x_{10} + u$, where $x_1, x_2, \ldots, x_{10}$ are endogenous. Then, yes, you need at least ten instruments.
Can you use the same dependent variables as instruments for the regression? No, you can't. Instruments should be exogenous (uncorrelated with $u$) and relevant (strongly correlated with the endogenous regressors). But if some of the regressors are (believed to be) exogenous, then you use those exogenous ones as instruments. But you still need extra instruments for endogenous regressors. The rule is that you need at least $k$ extra instruments if there are $k$ endogenous regressors.