Elementary zero function
Problem
Setup:
- 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 \(k×1\) .
- LRM: \(y_i=x'_iβ+ε_i\)
- Find elementary zero functions if \(x\) is exogenous
- Find elementary zero functions if \(x\) is endogenous while \(z\) are instruments
- If \(x\) is exogenous and \(z\) are instruments, will the functions in b. still be elementary zero functions?
Solution
- \(f\left( w_i,β \right)=x_i\left( y_i-x'_iβ \right)\)
- \(f\left( w_i,β \right)=z_i\left( y_i-x'_iβ \right)\)
- Yes, certainly. However the ones in a are better.