# just-identified model and over-identified model in the context of instrumental variable

According to this pdf, when number of instrument variable equals to the number of endogenous components, the model is said to be just-identified; if number of instrument variable is bigger than the number of endogenous components, the model is said to be over-identified.

I have these questions regarding this statement:

What exactly does "endogenous component" mean in this context? The variables which are correlated with the error term in the model (e.g. regression model)?

What would be an example for a just-identified model and an over-identified model?

Yes, the endogenous component is the number of regressors that are correlated with the error term.

From the pdf you link to:

Suppose we want to estimate returns to schooling in terms of earnings. Ability affects both earnings and schooling, so, if, as is often the case, ability is not measured in your dataset, you face omitted variable bias when using OLS correlated with the regressor schooling.

We hence need an instrument $z$ that is correlated with schooling and uncorrelated with ability.

One popular candidate for $z$ is proximity to college or university (Card, 1995). This clearly satisfies condition 2 as, for example, people whose home is a long way from a community college or state university are less likely to attend college.

It may satisfy 1, though since it can be argued that people who live a long way from a college are more likely to be in low-wage labor markets one needs to estimate a multiple regression for $y$ that includes as additional regressors controls such as indicators for non-metropolitan area.

A second candidate for the instrument is month of birth (Angrist and Krueger, 1991). This clearly satisfies instrument exogeneity as there is no reason to believe that month of birth has a direct effect on earnings if the regression includes age in years. Surprisingly correlation with the regressor may also be satisfied, as birth month determines age of first entry into school which in turn may affect years of schooling due to laws that specify a minimum school leaving age.

If you have access to one of the instruments to instrument the endogenous regressor schooling, your model is just identified. If you have access to both, you have two IVs for just one endogenous regressor, so you have overidentification.

• Thank you for your answer! There is one thing which is unclear to me: What does "one needs to estimate a multiple regression for yy that includes as additional regressors controls such as indicators for non-metropolitan area" mean? Jan 31, 2018 at 7:59
• In terms of the regression setup, what do you practically do with the two instruments? I'm familiar with 2SLS for one instrument. Would you first regress the endogenous regressor on both instruments (i.e., X ~ Z1 + Z2), take residuals, and then regress the outcome on the residuals? Apr 8 at 17:08