For this exercise, you will take five data points and calculate the coefficients of the bivariate...

  

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For this exercise, you will take five data points and calculate the coefficients of the bivariate regression model by hand. This is not something you would typically do, but by doing so, I hope the relevant concepts will become more concrete. You will complete this exercise using 5 data points (n = 5) which are combinations of values of a Y variable and an X variable as indicated: i Y 1 2 3 2 1 4 7 1 5 1 So for instance: Y1 = 1 and X1 = 0. If it helps, I had in mind the idea that Y indicates the number of college applications sent and X is an indicator (dummy) variable for female. There are a few pieces to this problem outlined below: a) Use the formulas for the bivariate regression coefficients to calculate the coefficients a and ß in the regression model: Yi = a – BXi + ei The basic formulas are: B = Cov(Yi, Xi) /Var(Xi) and: 3. a = E[Yi] – BE[Xi] where you can find how to calculate the covariance, variance, and expectation in any resource about statistics. b) Find the difference in mean the mean value of Yi for individuals with Xi = 1 instead of Xi = 0. This should be the same as your coefficient B. C) Copy the table above and extend it by including columns for the fitted value of Yi and the resulting residual for Yi , and residual squared for Yi as well as three additional columns described below. d) Calculate the residual and the residual squared for each observation based on a value of B either one greater than or 1 less than the value of B you calculated. (If you found B = 8 then calculate the fitted values, residuals, and residuals squared based on B = 7 or ß = 9). Verify that the sum of squared residuals from the correct ß is less than the sum of squared residuals from this incorrect B. For this exercise, you will take five data points and calculate the coefficients of the bivariate regression model by hand. This is not something you would typically do, but by doing so, I hope the relevant concepts will become more concrete. You will complete this exercise using 5 data points (n = 5) which are combinations of values of a Y variable and an X variable as indicated: i Y 1 2 3 2 1 4 7 1 5 1 So for instance: Y1 = 1 and X1 = 0. If it helps, I had in mind the idea that Y indicates the number of college applications sent and X is an indicator (dummy) variable for female. There are a few pieces to this problem outlined below: a) Use the formulas for the bivariate regression coefficients to calculate the coefficients a and ß in the regression model: Yi = a – BXi + ei The basic formulas are: B = Cov(Yi, Xi) /Var(Xi) and: 3. a = E[Yi] – BE[Xi] where you can find how to calculate the covariance, variance, and expectation in any resource about statistics. b) Find the difference in mean the mean value of Yi for individuals with Xi = 1 instead of Xi = 0. This should be the same as your coefficient B. C) Copy the table above and extend it by including columns for the fitted value of Yi and the resulting residual for Yi , and residual squared for Yi as well as three additional columns described below. d) Calculate the residual and the residual squared for each observation based on a value of B either one greater than or 1 less than the value of B you calculated. (If you found B = 8 then calculate the fitted values, residuals, and residuals squared based on B = 7 or ß = 9). Verify that the sum of squared residuals from the correct ß is less than the sum of squared residuals from this incorrect B.