I am currently reading Introductory Econometrics by Wooldridge. Specifically, Chapter 9, in which he shows the attenuation bias that occurs due to the classical errors-in-variables assumption.
For the past few days I have been really struggling with the following:
I can get to the third equation, where
$plim(\hat{\beta_1})=\beta_1\left(1-\cfrac{(\sigma_{e_1})^2}{(\sigma_{x_1}^*)^2+(\sigma_{e_1})^2}\right)$
But I cannot see how the final term is derived from there. Wooldridge seemingly assumes that this is obvious to the reader, so no explanation is provided. Hopefully someone can see how he gets from the third equation to the fourth.
It may be worth noting that
$X_1=X_1^*+e_1$
And also that
$Cov(X_1^*,e_1)=0 \\ Cov(X_1,e_1)=(\sigma_{e_1})^2 \\ Cov(X_1,u-\beta_1 e_1)=-\beta_1 (\sigma_{e_1})^2$
Notice that you can rewrite $1$ as $$\frac{\left(\sigma_{x_1}^{*} \right)^2 + \left( \sigma_{\epsilon_1} \right)^2 }{\left(\sigma_{x_1}^{*} \right)^2 + \left( \sigma_{\epsilon_1} \right)^2 }. $$
Now note that $$\frac{\left(\sigma_{x_1}^{*} \right)^2 + \left( \sigma_{\epsilon_1} \right)^2 }{\left(\sigma_{x_1}^{*} \right)^2 + \left( \sigma_{\epsilon_1} \right)^2 } - \frac{\left( \sigma_{\epsilon_1} \right)^2 }{\left(\sigma_{x_1}^{*} \right)^2 + \left( \sigma_{\epsilon_1} \right)^2 } = \frac{\left(\sigma_{x_1}^{*} \right)^2 }{\left(\sigma_{x_1}^{*} \right)^2 + \left( \sigma_{\epsilon_1} \right)^2 }.$$ Just like the textbook suggests.