# Choosing a robust estimator to account for measurement error in dependent variable

I have cross-sectional regression model $\hat{Y}_i = a + bX_i + e_i$ estimated over 200 cross-sectional observations. The $\hat{Y}_i$'s were generated in 200 time series regressions, and so has measurement error. Suppose that the measurement error can be modeled by $\hat{Y}_i = Y_i - u_i$, with $u_i \sim N(0,\sigma^2)$. Then we have $Y_i - u_i = a + b X_i + e_i$ ..$\Rightarrow$.. $Y_i = a + b X_i + (e_i + u_i) = Y_i = a + b X_i + \epsilon_i$, where $(e_i + u_i) = \epsilon_i$. This means the diagonal of the residual variance covariance matrix is larger. The variance covariance matrix is given by $\Sigma = E[\epsilon \epsilon^T]$; under this expression, every element of the diagonal in $E[\epsilon \epsilon^T]$ is larger than every diagonal element of $E[e e^T]$. Given the formula for the coefficient variance covariance matrix, the coefficient SEs will be larger from $E[\epsilon \epsilon^T](X^T X)^{-1}$ than from $[e e^T](X^T X)^{-1}$, leading to more abundant type 2 errors.

Another way to express this problem is as follows. Assume that $u_i$ and $e_i$ are uncorrelated, we have $var(e_i + u_i) = var(\epsilon_i) = \sigma_e^2 + \sigma_u^2 > var(e_i) = \sigma_e^2$ ... Our coefficient standard errors will be incorrectly large..

Slide 4 of THIS describes the issue (but gives no prescriptions).

So, what estimator do I use to reduce the instance of weak power/large type 2 errors in this econometric model?

But the way you describe it all the measurement error is in Y and even though you claim to have two sets of errors e$_i$ and u$_i$ each with 0 mean, the residual variance from least squares will estimate the variance of u$_i$ + e$_i$ and you would do nothing different from what you would ordinarily do.
Having looked at the slides I see that my explanation is correct. But they claim that they want to estimate the variance of e$_i$ rather than e$_i$ + u$_i$. Unfortunately there is no way to decompose the error to get the estimate. It is a problem without a solution.