Consider the classical measurement error model:
$$Y= \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \varepsilon$$
where $W=X+U$ is observed. X is the 'true' quantity and U is the measurement error. Var$(X) = \Sigma_x$, Var$(U) = \Sigma_U$.
The 'naive' estimator $\hat{\beta}_{\text{naive}} = (W^TW)^{-1}W^TY$ is the OLS estimator of $\beta = (\beta_1, \beta_2)^T$. A well-known result is that E$(\hat{\beta}_{\text{naive}}) = \Lambda \beta$ where $\Lambda = \Sigma_x (\Sigma_x + \Sigma_u)^{-1}$
I am interested in the asymptotic variance of the 'naive' estimator where $\Sigma_x$ and $ \Sigma_u$ are known. I am fairly confident based on a result in Measurement error in nonlinear models - Carrol (2006) that the asymptotic variance in the $p=1$ case is given by $\frac{\sigma^2_\epsilon + \beta^2 \sigma^2_x \frac{\sigma^2_u}{\sigma^2_u + \sigma^2_x}}{n (\sigma^2_u + \sigma^2_x)}$ if this is helpful.
