OLS Unbiasedness

Recovering population parameters for β\pmb \beta

Given the linear model

y=Xβ+ϵ\begin{align} \pmb y = X \pmb \beta + \pmb \epsilon \end{align}

One of the assumption that must be satisfied to recover an unbiased estimate of β\pmb \beta is that

E[ϵi∣xi]=0\begin{align} E[\epsilon_i | \pmb x_i] = 0 \end{align}

That is to say, given our prediction some xi\pmb x_i we would have no better guess for ϵi\epsilon_i other than 00.

This condition implies that Cov(xi,ϵi)=0Cov(\pmb x_i, \epsilon_i) = \pmb 0.

\begin{align} \end{align}