Assume that the data looks as follows: $$\boldsymbol { Y } = \left( \begin{array} { l }
Y _ { 1 } \
Y _ { 2 } \
\vdots \
Y _ { n }
\end{array} \right) , \boldsymbol { U } = \left( \begin{array} { l }
u _ { 1 } \
u _ { 2 } \
\vdots \
u _ { n }
\end{array} \right) , \boldsymbol { X } = \left( \begin{array} { c }
X _ { 11 } \
X _ { 12 } \
\vdots \
X _ { 1 n }
\end{array} \right) \text {, and } \boldsymbol { \beta } = \left( \beta _ { 1 } \right)$$ Using the formula for the OLS estimator $\hat { \boldsymbol { \beta } } = \left( \boldsymbol { X } ^ { \prime } \boldsymbol { X } \right) ^ { - 1 } \boldsymbol { X } ^ { \prime } \boldsymbol { Y }$, derive the formula for $\widehat { \beta } _ { 1 }$, the only slope in this "regression through the origin."

Question

Assume that the data looks as follows: $$\boldsymbol { Y } = \left( \begin{array} { l } 
Y _ { 1 } \
Y _ { 2 } \
\vdots \
Y _ { n }
\end{array} \right) , \boldsymbol { U } = \left( \begin{array} { l } 
u _ { 1 } \
u _ { 2 } \
\vdots \
u _ { n }
\end{array} \right) , \boldsymbol { X } = \left( \begin{array} { c } 
X _ { 11 } \
X _ { 12 } \
\vdots \
X _ { 1 n }
\end{array} \right) \text {, and } \boldsymbol { \beta } = \left( \beta _ { 1 } \right)$$ Using the formula for the OLS estimator $\hat { \boldsymbol { \beta } } = \left( \boldsymbol { X } ^ { \prime } \boldsymbol { X } \right) ^ { - 1 } \boldsymbol { X } ^ { \prime } \boldsymbol { Y }$, derive the formula for $\widehat { \beta } _ { 1 }$, the only slope in this "regression through the origin."

Quizplus · Accepted Answer

The Answer of Assume that the data looks as follows: $$\boldsymbol { Y } = \left( \begin{array} { l } 
Y _ { 1 } \
Y _ { 2 } \
\vdots \
Y _ { n }
\end{array} \right) , \boldsymbol { U } = \left( \begin{array} { l } 
u _ { 1 } \
u _ { 2 } \
\vdots \
u _ { n }
\end{array} \right) , \boldsymbol { X } = \left( \begin{array} { c } 
X _ { 11 } \
X _ { 12 } \
\vdots \
X _ { 1 n }
\end{array} \right) \text {, and } \boldsymbol { \beta } = \left( \beta _ { 1 } \right)$$ Using the formula for the OLS estimator $\hat { \boldsymbol { \beta } } = \left( \boldsymbol { X } ^ { \prime } \boldsymbol { X } \right) ^ { - 1 } \boldsymbol { X } ^ { \prime } \boldsymbol { Y }$, derive the formula for $\widehat { \beta } _ { 1 }$, the only slope in this "regression through the origin."

Assume That the Data Looks as Follows Using the Formula for the OLS Estimator