Consider the simple regression model

$y={}_0+{}_{1}D+u,$

where $D$ is a binary variable.

a$)$ Let $y_{i1}$ be $y_i$ with $D_i=1,y_{i0}$ be $y_i$ with $D_i=0,$ and ${\displaystyle \underset{i=1}{\overset{N}{}}}{y}_i={\displaystyle \underset{i=1}{\overset{N_1}{}}}{y}_{i1}+{\displaystyle \underset{i=1}{\overset{N_0}{}}}{y}_{i0},$ where $N_1+N_0=N.$ Show that $\frac{?}{\text{b}\text{a}\text{r}}(y)=\frac{N_1}{N}\frac{{\displaystyle \underset{i=1}{\overset{N_1}{}}}{y}_{i1}}{N_1}+\frac{N_0{\displaystyle \underset{i=1}{\overset{N_0}{}}}{y}_{i0}}{N}\frac{N_0}{N_0}\frac{?}{\text{b}\text{a}\text{r}}(y{)}_1+\frac{1\frac{-}{b}ar(D)}{b}ar(y{)}_0.$

b$)$ Show that the OLS estimator: hat$({)}_1=\frac{{\displaystyle \underset{i=1}{\overset{N}{}}}(D_i-(\frac{?}{\text{b}\text{a}\text{r}}(D)))(y_i-(\frac{?}{\text{b}\text{a}\text{r}}(y)))}{{\displaystyle \underset{i=1}{\overset{N}{}}}(D_i-(\frac{?}{\text{b}\text{a}\text{r}}(D)){)}^2}$ can be written as $\frac{?}{\text{b}\text{a}\text{r}}(y{)}_1-\frac{?}{\text{b}\text{a}\text{r}}(y{)}_0.$

$[$Hint: Note that $D_i{y}_i=y_{i1}$ and $D_i^2=D_i.$ Therefore hat$({)}_1=$

$\frac{{\displaystyle \underset{i=1}{\overset{N}{}}}{D}_i{y}_i-{\displaystyle \underset{i=1}{\overset{N}{}}}(\frac{?}{\text{b}\text{a}\text{r}}(D))y_i-{\displaystyle \underset{i=1}{\overset{N}{}}}(\frac{?}{\text{b}\text{a}\text{r}}(y))D_i+{\displaystyle \underset{i=1}{\overset{N}{}}}(\frac{?}{\text{b}\text{a}\text{r}}(D))(\frac{?}{\text{b}\text{a}\text{r}}(y))}{{\displaystyle \underset{i=1}{\overset{N}{}}}{D}_i^2-\frac{N}{b}ar(D{)}^2}=\frac{{\displaystyle \underset{i=1}{\overset{N}{}}}{y}_{i1}-(\frac{?}{\text{b}\text{a}\text{r}}(D)){\displaystyle \underset{i=1}{\overset{N}{}}}{y}_i-(\frac{?}{\text{b}\text{a}\text{r}}(y)){\displaystyle \underset{i=1}{\overset{N}{}}}{D}_i+\frac{N}{b}ar(Dy)}{{\displaystyle \underset{i=1}{\overset{N}{}}}{D}_i-\frac{N}{b}ar(D{)}^2}.$Can you write down the detailed steps and answers with typing or handwriting. Thank you so much, I$$ll give you a thump up