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Consider a multiple regression model

$$ y = X\beta + \varepsilon $$

with $K$ regressors in $X$.

If the model is correctly specified, the OLS estimator $\hat\beta_{OLS}=(X'X)^{-1}X'y$ will be the minimum-variance (or best) linear unbiased estimator (MVLUE or BLUE) of $\beta$.

When it comes to minimizing the mean squared error (MSE), ridge regression and lasso may provide estimators $\hat\beta_{ridge}(\lambda_{ridge})$ and $\hat\beta_{lasso}(\lambda_{lasso})$, respectively, with smaller MSEs than that of OLS for some intervals of penalty intensity $\lambda_{ridge}$ and $\lambda_{lasso}$.

As far as I understand, this concerns the entire parameter vector $\beta$ rather than each individual parameter $\beta_k$ for $k=1,\dotsc,K$.

Question: Can something concrete and useful be said about the individual parameter estimates $\hat\beta_{k,OLS}$ versus $\hat\beta_{k,ridge}$ and $\hat\beta_{k,lasso}$ in terms of MSE?

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    $\begingroup$ Nitpick: it's not minimizing MSE "instead of [minimizing] variance", it's minimizing MSE instead of minimizing variance and guaranteeing unbiasedness. $\endgroup$ – amoeba Aug 1 '16 at 16:29
  • $\begingroup$ @amoeba, good point. I have now omitted the whole "instead" clause. $\endgroup$ – Richard Hardy Aug 1 '16 at 17:35

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