$1.$ Suppose the true structural equation is

Y; $=$ Bo$+$ XiBI $+$ X$7$B$2+$ Vi

but by mistake, a researcher ran least square regression without the X$?,$ term as in

Y; $=$ Bo $+$ XiB$1+$ Vi$.$

Assume cov $($Xi$,$ U;$)=0,$ E $[$Xi$]=0$ and E $|$ X$3]=1.$ Is his$/$her estimate consistent for B$1?$ If

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

$2.$ Assume the structural equation is

Yi $=$ Bot XiBi $+$Ui

where E $[$u; $|$X;$]=0.$ It was discovered that we observe X$,$ with a measurement error w; instead

of the real value Xi

X; $=$ X; $+$ Wi$.$

It is known that E $[$w;$]=0,$ V $($wi$)=$ ow$,$ cov $($Xi$,$ wi$)=$ cov $($Ui$,$ Wi$)=0.$ The OLS estimator

is based on regressing Yi on a constant and Xi$.$

$($i$)$ Find the value to which the OLS estimator of B$,$ is consistent for.

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

$($iii$)$ Assume we have a consistent estimator for consistently estimate B$1?$

How would you make a correction to

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

$3.$ Exercise #$9\mathrm{.}7($Stock and Watson$)$

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

$"$An ordinary least squares regression of Y onto X will be internally inconsistent if X is correlated with the error term."

$"$Each of the five primary threats to internal validity implies that X is correlated with the error term."