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I am running a two-variable linear regression. Only one variable is measured with error; however, the second variable is interacted with the first, giving rise to correlated measurement errors.

$y = X_1^*\beta_1 + X_1^*X_2\beta_2 + \epsilon$

Suppose the measurement error of $X_1^*$ has mean $0$ and standard deviation $\sigma_{x_1}$. How can I get corrected estimates of the coefficients? (Either proofs or Monte Carlo simulation is fine)

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  • $\begingroup$ Rarely should you include x1*x2 but not x2. $\endgroup$
    – David Lane
    Jan 15, 2017 at 21:20
  • $\begingroup$ Let's assume (for simplicity) that this is one such rare case. $\endgroup$
    – wwl
    Jan 16, 2017 at 0:34
  • $\begingroup$ The Bayesian approach provides a straightforward solution. Let me know if you are interested. $\endgroup$
    – mef
    Aug 16, 2017 at 16:33

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