How to do regression with error in variables and known correlations among the errors? After the very satisfying answer to:
How to do regression with known correlations among the errors?
I take the question to my next point of interest:
What can you do when you have a regression with error in variables problem:
$$\begin{cases}
   Y_t = \beta_1x_t^* + \beta_0 + \varepsilon_t \\
          Y_t = y_t^* +\varepsilon_t\\
          X_t = x_t^* +\eta_t\\
 \left(\varepsilon_1,\eta_1,\ldots,\varepsilon_n,\eta_n\right)\sim\mathcal{N}_{2n}\left(\mathbf{0},D \right) \\
 1\leq t \leq n
\end{cases}$$
you observe $(Y_t,X_t)$ and you know $D$, the covariance matrix of the errors? 
I have been reading Wayne and Fuller Measurment Error Model, it's a bit cumbersome and only deals with special structures for $D$.
Is there any well established estimator for $(\beta_0,\beta_1)$ ? Something like Generalized least squares when there are no error in the variables.
 A: If you know the covariance matrix of the measurement errors, one strategy is to estimate the model by OLS, calculate the bias, and then correct for the bias.  Write (going to matrix notation):
\begin{align}
Y &= X^*\beta + \epsilon\\
  &= X\beta + \left(X^*-X\right) \beta + \epsilon\\
  &= X\beta - \eta \beta + \epsilon
\end{align} 
So, the OLS estimator is:
\begin{align}
\hat{\beta}_{OLS} = \beta &+ \left( X'X\right)^{-1}X'\left( -\eta \beta + \epsilon \right)\\
                  = \beta &- \left( X'X\right)^{-1}X'\eta \beta + \left( X'X\right)^{-1}X' \epsilon\\
 = \beta &- \left( X'X\right)^{-1} X^* \, '\eta \beta + \left( X'X\right)^{-1} X^* \, ' \epsilon\\
&- \left( X'X\right)^{-1} \eta'\eta \beta + \left( X'X\right)^{-1} \eta' \epsilon
                 \end{align}
The first term is $\beta$.  The second and third terms go in probabability to zero by the usual arguments, just requiring that $X^*$ be uncorrelated with both $\eta$ and $\epsilon$.  So, the OLS estimator goes to:
\begin{align}
\hat{\beta}_{OLS} &\xrightarrow{P} \beta - Q^{-1}D_{\eta\eta}\beta + Q^{-1}D_{\eta\epsilon}\\
                  &= \left( I-Q^{-1}D_{\eta\eta}\right)\beta + Q^{-1}D_{\eta\epsilon}
\end{align}
If it is not clear from context, $D_{\eta\eta}$ is the part of the matrix $D$ which pertains only to the $\eta$, and $D_{\eta\epsilon}$ is the part of the matrix $D$ which pertains only to the covariances between the $\eta$ and the $\epsilon$.  Also, the matrix $Q$ is the probability limit of $\hat{Q}=\frac{1}{N}X'X$.
There are two sources of bias, here.  The term $\left(  I-Q^{-1}D_{\eta\eta} \right)$ represents something like the classical attenuation bias.  The true coefficient gets multiplied by (1-expression).  Expression is like a noise-to-signal ratio: it has the variance of $\eta$ in its "numerator" and the variance of $\eta$ plus the variance of $X$ in its "denominator" --- the quotes because these are matrix operations, not real operations so there are not really numerators and denominators.  When only one element of $X$ has measurement error, this formula does literally give the classical attenuation bias formula.  Here it gives something akin to it.  The second term is like a classical endogeneity bias arising from the correlation between $X^{*}$, the variable we are using on the RHS, and $\epsilon$---where the correlation arises through $\eta$.
Finally, a consistent estimator is:
\begin{align}
\tilde{\beta} = \left(I-\hat{Q}^{-1}D_{\eta\eta} \right)^{-1}\left(\hat{\beta}_{OLS} - \hat{Q}^{-1}D_{\eta\epsilon} \right)
\end{align}
It is a pretty good bet that this estimator is not efficient since I have done nothing about adjusting for heteroskedasticity or serial correlation which the assumptions given do not rule out.  You could calculate the variance of the estimator by pushing even more algebra, but I am not interested in doing it.
If anyone wants to check/correct my algebra, much obliged.
