# Unbiased estimator of regression coefficient in high dimension

Is there any unbiased estimator for the regression coefficient $$\beta \in \mathbb{R}^p$$, p >> 1, where $$y_k = x_k^T\beta + \epsilon \in \mathbb{R}?$$ Note that $$x_k \in \mathbb{R}^p$$ and $$\epsilon \sim \mathcal{N}(0, \sigma^2)$$, for some $$\sigma > 0$$.

• It would be more standard to use $p$ instead of $n$ for the dimension and then use $n$ to denote sample size; consider updating your post. With this new notation, how large is $p$ relative to $n$? If $p<n$, you can use OLS. May 12 at 6:03
• It is written as it is. In your notation, p = n, n = 1. May 12 at 15:47
• Would the gauss-markov-theorem not apply?
– Dave
May 12 at 16:00
• How large is $n$? Is it larger or smaller than $p$? If $p<n$, you can use OLS. May 12 at 20:06
• Please read the question description as it is. I have specified that p >> n = 1. I am less confident on the existence of unbiased estimator in this setting. May 12 at 21:18