Bias of ridge estimator

Solution 1:

As defined, the "squared bias" has no meaning since the bias is a vector. One should consider instead then squared norm of the bias, namely $B^2=[E(\hat{\beta}_R) - \beta]'[E(\hat{\beta}_R) - \beta]$. We have $$E(\hat{\beta}_R) - \beta= (X'X + k I)^{-1}(X'X - (X'X+kI))\beta=-k (X'X + k I)^{-1}\beta.$$ Therefore, $$B^2 = k^2 \beta'(X'X + k I)^{-2}\beta.$$

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