Suppose we have our standard DGP, $y=\alpha+\beta x+\varepsilon,$ where $x$ is binary. Let's say the observed $x$ is actually measured with error, so that the explanatory variable is misclassified for some fraction of the population. This is very non-classical measurement error.
I am trying to summarize the effects of this type of measurement error.
From Bound, Brown, and Mathiowetz (2001), I've learned that as long as the misclassification is non-differential (and none of the other classical measurement error assumptions are violated), the coefficient on $x$ will be attenuated. Here non-differential means the classifications errors are independent of the dependent variable $y$.
Two questions remain:
My simulation-based intuition is that size goes up and the power of the test goes down as the degree of misclassification rises. I am somewhat more confident of the second.
If $x$ is vector (but we only have measurement error in one variable), is the bias still unambiguously toward zero?
Bound, John, Charles Brown, and Nancy Mathiowetz, 2001. “Measurement Error in Survey Data.” In James J. Heckman and Edward E. Leamer (Eds.), Handbook of Econometrics, Volume 5. Amsterdam: Elsevier Science, 2001. Link to working paper version.