I'm currently doing an exercise and faced with the following question:
We have the true form: $y_i=\beta_0 +\beta_1 d_i +u_i $
Where $d_i$ is a dummy variable. We have measured $d_i$ with measurement error such that 10% of those for whom $d_i=1$ have been recorded to have $d_i=0$ and similarly 10% of those for whom $d_i=0$ have been recorded to have $d_i=1$.
We have estimated $\hat\beta_1 =0.205$ with $SE=0.015$.
Compute the bias due to measurement error in the OLS regression.
I'm aware with measurement error in dummy variables we have a form of non-classical measurement error, where the measurement error is negatively correlated with the true value.
I've played around with it for a while but haven't really been able to figure out how to go about this. I attempted to use the reliability ratio:
Define $\tilde d_i=d_i + v_i$ & $p_d=Pr[d_i=1]$
$\tilde p_d=0.9p_d+0.1(1-p_d)$ & $(1-\tilde p_d)=(1-p_d)0.9+0.1p_d$
$Var(\tilde d_i)=\tilde p_d(1-\tilde p_d)=0.64(p_d(1-p_d))+0.09$
I'm aware this must be wrong, I imagine because perhaps this isn't classical measurement error. Any guidance on how to calculate the bias would be greatly appreciated. Thanks