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: \(y_i=x'_iβ+ε_i\)
  1. Find elementary zero functions if \(x\) is exogenous
  2. Find elementary zero functions if \(x\) is endogenous while \(z\) are instruments
  3. If \(x\) is exogenous and \(z\) are instruments, will the functions in b. still be elementary zero functions?

Solution

  1. \(f\left( w_i,β \right)=x_i\left( y_i-x'_iβ \right)\)
  2. \(f\left( w_i,β \right)=z_i\left( y_i-x'_iβ \right)\)
  3. Yes, certainly. However the ones in a are better.