Computing bias due to measurement error:

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 $\tilde d_i=d_i + v_i$ & $p_d=Pr[d_i=1]$

Therefore

$\tilde p_d=0.9p_d+0.1(1-p_d)$ & $(1-\tilde p_d)=(1-p_d)0.9+0.1p_d$

$Var(d_i)=\ p_d(1-p_d)$

$Var(\tilde d_i)=\tilde p_d(1-\tilde p_d)=0.64(p_d(1-p_d))+0.09$

Subbing into:

$$\hat\beta_1=\beta_1*\frac{Var(d_i)}{Var(\tilde d_i)}$$

$$0.205=\beta_1\frac{p_d(1-p_d)}{0.64p_d(1-p_d)+0.09}$$

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

• Please add the self-study tag if this is a homework/assignment question – Marquis de Carabas Apr 14 '16 at 22:21
• Apologies, wasnt aware of this. Done. – Nik-D Apr 15 '16 at 8:23