The instrumental variables (IV) estimator, asymptotics
Summary
- 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 \(k×1\)
- 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 \(k×k\) and invertible
- \(Var\left( ε_i|z_i \right)=σ^2\) .
- Statistical formula
\[b_{IV}=β+{\left( Z'X \right)}^{-1}Z'ε\]
- Result: since \(Z'X=\sum_{i=1}^{n}{ z_ix'_i }\)
\[E\left( \frac{1}{n}Z'X \right)=Σ_{zx'}\]
- where \(Σ_{zx'}=E\left( z_ix'_i \right)\)
- Results: The variance of every element in \(E\left( \frac{1}{n}Z'X \right)\) will go to zero as \(n→∞\) .
- Result:
\[plim \frac{1}{n}Z'X=Σ_{zx'}\]
- which is invertible.
- Result: If \(Z\) are valid instruments then
\[E\left( \frac{1}{n}Z'ε \right)=0\]
- and
\[Var\left( \frac{1}{n}Z'ε \right)= \frac{σ^2}{n}Σ_{zz'}→0\]
- where \(Σ_{zz'}=E\left( z_iz'_i \right)\)
- Result:
\[plim \frac{1}{n}Z'ε=0\]
Note that \(plim \frac{1}{n}X'ε\) is not zero which is why least squares will fail.
- Consistency of the IV estimator:
\[plim b_{IV}=β\]
- Result:
\[E\left( \frac{1}{\sqrt{n}}Z'ε \right)=0\]
- and
\[Var\left( \frac{1}{\sqrt{n}}Z'ε \right)=σ^2Σ_{zz'}\]
- Result: (central limit theorem)
\[ \frac{1}{\sqrt{n}}Z'ε→N\left( 0,σ^2Σ_{zz'} \right)\]
- Result:
\[\sqrt{n}\left( b_{IV}-β \right)→N\left( 0,σ^2Σ_{zx'}^{-1}Σ_{zz'}Σ_{xz'}^{-1} \right)\]
- Alternatively,
\[\sqrt{n}\left( b_{IV}-β \right)→N\left( 0,σ^2{\left( Σ_{xz'}Σ_{zz'}^{-1}Σ_{zx'} \right)}^{-1} \right)\]
- where \(Σ_{xz'}=E\left( x_iz'_i \right)=Σ'_{zx'}\)
- The asymptotic variance \(σ^2{\left( Σ_{xz'}Σ_{zz'}^{-1}Σ_{zx'} \right)}^{-1}\) can be consistently estimated by
\[ns^2{\left( X'P_ZX \right)}^{-1}\]
- where \(P_Z=Z{\left( Z'Z \right)}^{-1}Z'\) and \(s^2= \frac{1}{n}\sum{ e_i^2 }\) where \(e_i=y_i-x_i'b_{IV}\) are the IV residuals. As an estimator of \(Var\left( b \right)\) we therefore use
\[s^2{\left( X'P_ZX \right)}^{-1}\]
- All inference based on the normal distribution of \(b\) is therefore approximately correct.