Question $I$

Suppose that the true model $is$

where $E(u_i|x_{1i},x_{2i})=0.$ However, $to$ estimate ${}_1,$ you run a regression without $x_{2i}$ :

$y_i={}_0+{}_1{x}_{1i}+$tilde$(u{)}_i,$ where tilde$(u{)}_i={}_2{x}_{2i}+u_i$

$(a)(4pts)$ Explain why $itis$ not advisable $to$ run the short regression $(2)to$ estimate ${}_1$

$in$ general.

$(O)(4pts)$ Discuss how the bias $of$ the OLS estimator $of{}_1$ obtained from the short

regression $(2)is$ determined.

For the rest $of$ the questions, define $as$ the correlation coefficient between $x_{1i}$ and

$x_{2i}.$ Define $r=\frac{\text{V}\text{a}\text{r}(x_{1i})}{\text{V}\text{a}\text{r}(x_{2i})}$ which $is$ the ratio $of$ the variance $of{x}_{1i}$ and $x_{2i}.$

$(4pts)$ Suppose that ${}_2$ and $r$ are fixed. Some people argue that $if$ the $absolute$

value $ofis$ small, then the $absolute$ value $of$ the bias $is$ small. $Do$ you agree?

Explain briefly.

$(7)(4pts)$ Suppose that $if{}_2$ and $are$ fixed. Some people argue that $if$ the $absolute$

value $ofris$ small, then the $absolute$ value $of$ the bias $is$ small. $Do$ you agree?

Explain briefly.

$(4pts)$ Find two special cases where the estimator $of{}_1$ from the short regression

$(2)$ actually has zero bias. Discuss intuitively why the estimator has $no$ bias $in$ these

two cases.