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

Question

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?

Answer 1

It is not necessary to bootstrap in this case. The issue seems to be error in the X variable although that is not how you described the problem. If it is an error in variables problem then the remedy is to use the appropriate optimization criterion which is not least squares. Least squares is appropriate when all the error in the observed pairs is due to the Ys. The answer is to minimize the squared distance in a direction based on the ratio of the error variance in X to the error variance in Y. If the two error variances are equal this would be to minimize the squared error in the direction orthogonal to the line. Error in variables regression is covered in a number of texts.

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.

So I think you must not be stating your problem correctly. In either of these modeling situations you could bootstrap the samples and get bootstrap estimates but if the error distribution is normal and you have either error in variables regression or ordinary least squares regression there is no need to do so.

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.