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8.1.1 Expectation of least squares estimator

Find the expectation of $\hat{\beta}$ in terms of $A$ , $X$ and $\beta$ .

\displaystyle\mathbb{E}[\hat{\beta}]=\mathbb{E}[AY]=A\mathbb{E}[Y]

by linearity of expectation, so $\mathbb{E}[\hat{\beta}]=AX\beta$ by definition of linear model.

Now

\displaystyle AX=(X^{\prime}X)^{-1}X^{\prime}X=I_{p}.

where $I_{p}$ is the $p\times p$ identity matrix. Consequently, {mdframed}

\displaystyle\mathbb{E}[\hat{\beta}]=AX\beta=I_{p}\beta=\beta,

so that the estimator is unbiased.