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

\[plim b=β+Σ_{xx'}^{-1}⋅0=β\]