Consider the Sample Regression Function $Y _ { i } ^ { * } = \hat { y } _ { 0 } + \hat { y } _ { 1 } X _ { i } ^ { * } + \hat { u _ { i } }$

Essay

Consider the sample regression function $Y _ { i } ^ { * } = \hat { y } _ { 0 } + \hat { y } _ { 1 } X _ { i } ^ { * } + \hat { u _ { i } }$ ,
where * indicates that the variable has been standardized. What are the units of measurement for the dependent and explanatory variable? Why would you want to transform both variables in this way? Show that the OLS estimator for the intercept equals zero. Next prove that the OLS estimator for the slope in this case is identical to the formula for the least squares estimator where the variables have not been standardized, times the ratio of the sample standard deviation of X and Y, i.e., $\hat { \gamma } _ { 1 } = \hat { \beta } _ { 1 } * \frac { S _ { X } } { S _ { y } }$