Consistency of the OLS estimator
Problem
Given:
- a linear regression model \(y=Xβ+ε\) with a random sample
- the Gauss-Markov assumptions, \(E\left( ε \right|X)=0\) and \(Var\left( ε \right|X)=σ^2I\)
- \(E\left( x_ix'_i \right)=Σ_{xx'}\) is invertible
- \(b={\left( X'X \right)}^{-1}X'y\) is the OLS estimator of \(β\)
- Show that \(b\) is a consistent estimator of \(β\) , \(plim b=β\) .
- Hint: start from the statistical formula, \(b=β+{\left( X'X \right)}^{-1}X'ε\) . Rewrite this as
\[b=β+{\left( \frac{1}{n}X'X \right)}^{-1}\left( \frac{1}{n}X'ε \right)\]
- Then use plim rules.
Solution
\[plim b=plim \left( β+{\left( \frac{1}{n}X'X \right)}^{-1}\left( \frac{1}{n}X'ε \right) \right)\]
By the first plim rule (see Plim-rules) we have
\[plim \left( β+{\left( \frac{1}{n}X'X \right)}^{-1}\left( \frac{1}{n}X'ε \right) \right)=β+plim \left( {\left( \frac{1}{n}X'X \right)}^{-1}\left( \frac{1}{n}X'ε \right) \right)\]
By the fourth plim rule we have (plim of products is products of plims)
\[plim \left( {\left( \frac{1}{n}X'X \right)}^{-1}\left( \frac{1}{n}X'ε \right) \right)=plim \left( {\left( \frac{1}{n}X'X \right)}^{-1} \right)plim \left( \left( \frac{1}{n}X'ε \right) \right)\]
By the second plim rule we have (plim goes inside the inverse)
\[plim \left( {\left( \frac{1}{n}X'X \right)}^{-1} \right)={\left( plim \left( \frac{1}{n}X'X \right) \right)}^{-1}=Σ_{xx'}^{-1}\]
Since
\[plim \left( \left( \frac{1}{n}X'ε \right) \right)=0\]
we have
\[plim b=β+Σ_{xx'}^{-1}⋅0=β\]