Given the (multivariate) linear regression model $\mathbf y=\mathbf X \mathbf\beta_0 + \epsilon$ and $\mathbb E[\epsilon|\mathbf X]=0$ for $\beta_0 \in \mathbb R^k$, determine if the following estimator is unbiased a/o consistent:
$\hat\beta_j = (\mathbf X^T \mathbf X + \lambda \mathbf I_k)^{-1} \mathbf X^T \mathbf y$
where $\lambda>0$
I notice that this is basically just the OLS estimator, but with the addition of a scaled identity matrix inside the inverse. I just do not know how to get rid of this, I feel like there is an algebra trick that I'm missing, or perhaps it involves spectral decomposition.
For now, this is where I have arrived:
$\hat\beta_j = (\mathbf X^T \mathbf X + \lambda \mathbf I_k)^{-1} \mathbf X^T \mathbf X\beta_0 + (\mathbf X^T \mathbf X + \lambda \mathbf I_k)^{-1} \mathbf X^T \epsilon$
and $\mathbb E[\hat\beta_j] = (\mathbf X^T \mathbf X + \lambda \mathbf I_k)^{-1} \mathbf X^T \mathbf X\beta_0$
This would seem to suggest that the estimator is biased, but I'm not sure this is correct.
I know that it is possible to be both biased and consistent, but I also have no idea how to prove this because I can't isolate $\beta_0$ algebraically in order to take the limit necessary.
\begin{align} \hat \beta = \frac{n}{n} \hat \beta = \left(\frac{1}{n}X^TX + \frac{1}{n}\lambda I \right) ^{-1} \frac{1}{n}X^Ty \end{align} Using the multivariate WLLN $$ \frac{1}{n} X ^T X \xrightarrow{p} E[\mathcal X \mathcal X ^T], \quad \frac{1}{n} X ^Ty \xrightarrow{p} E[\mathcal Y^T \mathcal X], $$ and $\lambda/n I \to 0I = O$, hence using the continuous mapping theorem, you conclude $$ \hat \beta \xrightarrow{p} (E[\mathcal X \mathcal X ^T])^{-1}E[\mathcal Y^T \mathcal X] = \beta $$