## Solving for a Matrix

Preamble:
The coefficients obtained from an OLS regression are estimated with errors that covary. The exact covariance you can expect depends on the covariance between the regressors, and the covariance between the "true" residuals; since the regression coefficient errors covary, so do the residuals covary incorrectly. I am interested in estimating unbiased residual covariance from observed residual covariance.

Actual Math:
I am trying to solve for $E$ using:

$E &= M-X(X'X)^-^1X'EX(X'X)^-^1X'$

• Where:
• E is an nxn matrix; it is the "true" covariance of residuals
• X is an nxm matrix, and is known; it describes the regressors
• M is an nxn matirx, and is known; it describes the measured covariance of residuals
Thanks for any help!