Overidentifying restrictions test

Summary

Random sample \(\left( y_i,x_i,z_i \right)\) for \(i=1, \ldots ,n\) where \(x_i\) is \(k×1\) and \(z_i\) is \(r×1\) with \(r≥k\)
A linear regression model, \(y_i=x'_iβ+ε_i\)
Exogeneity fails, \(E\left( ε_i \mid x_i \right)≠0\)
z-variables are instruments, \(E\left( ε_i|z_i \right)=0\) and \(E\left( z_ix'_i \right)=Σ_{zx'}\) is \(r×k\) with rank \(k\) .
\(Var\left( ε_i|z_i \right)=σ^2\) .

We have \(r\) equations \(E\left( {z_iε}_i \right)=0\) but only \(k\) unknowns. If \(r>k\) then we say that we have overidentification .
We can test if all instruments are exogenous using an overidentifying restrictions test (also called a Sargan test). The null hypothesis is that all instruments are exogenous.
To perform the test in Stata, do “estat overid” after estimating the model using ivregress (with strictly more instruments than endogenous variables)