In the regression through the origin model $Y _ { i } = \beta _ { 1 } X _ { i } + u _ { i }$ the OLS estimator is
$\widehat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } } { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } }$ Prove that the estimator is a linear function of $Y _ { 1 } , \ldots , Y _ { n }$ and prove
that it is conditionally unbiased.

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The Answer of In the regression through the origin model $Y _ { i } = \beta _ { 1 } X _ { i } + u _ { i }$ the OLS estimator is
$\widehat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } } { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } }$ Prove that the estimator is a linear function of $Y _ { 1 } , \ldots , Y _ { n }$ and prove
that it is conditionally unbiased.

In the Regression Through the Origin Model Yi=β1Xi+uiY _ { i } = \beta _ { 1 } X _ { i } + u _ { i }Yi​=β1​Xi​+ui​

In the Regression Through the Origin Model $Y _ { i } = \beta _ { 1 } X _ { i } + u _ { i }$