Correction for measurement error

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?

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

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.