Removing the bias from some unknown measurement error Imagine I have two variables X and Y which have a statistical relationship. However I cannot observe X. I can only observe X* = X + U where U is some 0-centered random noise. I don't know U but I roughly know its aggregate variance var(U). If I regress Y on X* I get a biased effect b*, which is smaller than the true effect b. Given this information, how can I debias my estimate b*?
X and U are independent, and U and Y are also independent.
I assume we can apply measurement error theory here.

I think I found the answer in this 1992 paper
$$b = b^*(I + \Sigma\Sigma^{-1}_X)$$
where I is the identity matrix, $\Sigma$ is the covariance of the noise (matrix) and $\Sigma_X$ is the covariance of the original X.
