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I have an estimator for the coefficients of the model $$ y=X\beta+\varepsilon $$

with $y_{n\times1}$, $X_{n\times p}$, $\beta_{p\times1}$, $\varepsilon_{n\times1}$. The estimator is in the form

$$ \hat{\beta}=\left(\frac{1}{n}X^{T}X+I\right)^{-1}\left(\frac{1}{n}X^{T}X+kI\right)\hat{\beta}^{ols} $$ for $0\le k\le 1$. It is obvious that $\hat{\beta}$ is a consistent estimator if $k=1$.

My question?

Is the boundedness of $k$ in $[0,1]$ an obstacle to define $k$ as $k_{n}$?

If I can apply $k\to k_{n}$, I think I can write if $k_{n}\to1$, $n\to\infty$ then $\hat{\beta}$ is consistent for $\beta$. (In this case $k$ is not fixed to zero, instead, goes to zero.)

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    $\begingroup$ If, in principle, $k$ may depend on $n$, then boundedness would not be an issue. Take $k_n=\epsilon/n$ for some $0<\epsilon<1$. $\endgroup$ Dec 4 '20 at 7:46
  • $\begingroup$ Could you explain to us what "define $k$ as $k_n$" means? $\endgroup$
    – whuber
    Dec 7 '20 at 18:02