Consider the Sample Regression Function $$Y _ { i } = \hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } X _ { i } + \hat { u } _ { i } .$$

Essay

Consider the sample regression function $$Y _ { i } = \hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } X _ { i } + \hat { u } _ { i } .$$ First, take averages on both sides of the equation. Second, subtract the resulting equation from the above equation to write the sample regression function in deviations from means. (For simplicity, you may want to use small letters to indicate deviations from the mean, i.e., $z _ { i } = Z _ { i } - \bar { Z }$ .) Finally, illustrate in a two-dimensional diagram with $S S R$ on the vertical axis and the regression slope on the horizontal axis how you could find the least squares estimator for the slope by varying its values through trial and error.