Suppose the true structural equation is

$Y_i={}_0+x_i{}_1+x_i^2{}_2+U_i$

but by mistake, a researcher ran least square regression without the $x_{i1}^2$ term as in

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

Assume cov$(x_i,U_i)=0,E[x_i]=0$ and $E[x_i^3]=1.$ Is his$/$her estimate consistent for ${}_1?$ If

not, show which OLS assumption fails and discuss potential solutions.

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.

Exercise #$9\mathrm{.}7($Stock and Watson$)$

Are the following statements true or false? Explain your answer.

a$."$An ordinary least squares regression of $Y$ onto $x$ will be internally inconsistent if $x$ is

correlated with the error term."

b$.$ "Each of the five primary threats to internal validity implies that $x$ is correlated with the

error term."