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Let's suppose that the true model is: $$ y_t^* = x_t^* \beta + e_t^* $$ and suppose that data on $x_t^*$ is observed with error: $$ x_t = x_t^* + u_t $$ If we consider the regression $y_t^* = x_t \beta + e_t$, we have the issue that the regressor is endogenous, i.e. $E(e_t x_t) \neq 0$. So that the OLS estimator $\hat{\beta}_{OLS}$ will be biased and inconsistent.

I found that: $$ E[\hat{\beta}_{OLS}] = \beta \frac{Var(x_t^*)}{Var(x_t^*) + \sigma_u^2} $$ so the expected value of the estimator is downward biased.

What are common ways to correct for correct for the measurement error?

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  • $\begingroup$ Small correction: The estimator is not downward biased, but biased towards zero. $\endgroup$
    – E. Sommer
    Commented Apr 13, 2019 at 19:21
  • $\begingroup$ If the error is similar to random noise, then this particular problem can be mitigated by taking repeated measurements at each value of x. $\endgroup$
    – James Phillips
    Commented Apr 13, 2019 at 19:23
  • $\begingroup$ @E.Sommer Can you give us the difference between the two? $\endgroup$
    – Victor
    Commented Apr 13, 2019 at 20:14
  • $\begingroup$ If $\beta > 0$, $E[\hat{\beta}]< \beta$. If $\beta < 0$, $E[\hat{\beta}]> \beta$. $\endgroup$
    – E. Sommer
    Commented Apr 14, 2019 at 19:45

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Apart from obtaining a more reliable measurement of $x$, there is not much one can do from an econometric perspective. But if you find something despite measurement error, and the quantity of $\beta$ is not crucial, you are generally fine.

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