Assume the structural equation is

$Y_i={}_0+x_i{}_1+u_i$

where $E[u_i|x_i]=0.$ It was discovered that we observe tilde$(x{)}_i$ with a measurement error $w_i$ instead

of the real value $x_i$

tilde$(x{)}_i=x_i+w_i.$

It is known that $E[w_i]=0,V(w_i)={}_w^2,$cov$(x_i,w_i)=$cov$(u_i,w_i)=0.$ The OLS estimator

is based on regressing $Y_i$ on a constant and tilde$(x{)}_i.$

$($i$)$ Find the value to which the OLS estimator of ${}_1$ is consistent for.

$($ii$)$ Is the value equal to the true value ${}_1?$ If not, how is the bias related to the true value?

$($iii$)$ Assume we have a consistent estimator for $\frac{\text{h}\text{a}\text{t}({)}_w^2}{\text{h}\text{a}\text{t}({)}_x^2}.$ How would you make a correction to

consistently estimate ${}_1?$

$($iv$)$ Discuss other potential solutions when such estimator in $($iii$)$ is not available.