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

Essay

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."