6. In class we discussed how different criteria for minimizing errors may lead to different regression lines. To revisit this concept, consider a sample of 10 data points as follows (where each point is of the form (X, Y)): seven separate observations of (4,4) one observation each of (0,0), (0, 4) and (4,0) The following diagram will assist us in this exercise: (4,4) arbitrary line > 4-6 -4- (0,0) (4,0) In the diagram, an arbitrary line has been constructed. The labels a, 4-a, b and 4-b represent the absolute values that the various points are away from the line. So, if the sample had been only one each of the different points (i.e., 4 sample observations in total instead of 10 sample observations) then the sum of the absolute value of errors for the arbitrary line shown would be a +(4- a) + b + (4- b) = 8. (a) Consider a sample consisting of the 10 data points listed above and assume the criterion we use to determine the regression line is to minimize the sum of absolute errors. (1) Looking at the above diagram, write out the total sum of the absolute errors using a's and b's. () Based on part (a)t), if we wish to minimize the sum of absolute errors, what value should 'a' have in the above diagram? What value should 'b' have? (As suggested in the diagram, you may assume 0 sa s 4 and 0 sb s 4.) (iii) Based on your response to part (a)(ii) what would be the ensuing regression line using this criterion? Is the regression line unique? Determine the total sum of absolute errors under the enguing regression line (b) Determine the regression line if our criterion is to minimize the sum of squared errors. That is determine the OLS regression line using 12 - Cov(X,Y) and b = Y-box Var(x) (c) Again suppose that our regression line criterion is to minimize the sum of squared errors. This criterion is being applied to the sample of 10 data points originally cited in this exercise (1) Looking at the above diagram, write out the total sum of squared errors for an arbitrary line using a's and b's. (ii) Determine the value of 'a' which minimizes the expression you found in part (c)(i). Likewise, determine the value of 'b' which minimizes the expression. Argue that this produces the exact same regression line you just found in part (b). Determine the total sum of squared errors using this regression line 6. In class we discussed how different criteria for minimizing errors may lead to different regression lines. To revisit this concept, consider a sample of 10 data points as follows (where each point is of the form (X, Y)): seven separate observations of (4,4) one observation each of (0,0), (0, 4) and (4,0) The following diagram will assist us in this exercise: (4,4) arbitrary line > 4-6 -4- (0,0) (4,0) In the diagram, an arbitrary line has been constructed. The labels a, 4-a, b and 4-b represent the absolute values that the various points are away from the line. So, if the sample had been only one each of the different points (i.e., 4 sample observations in total instead of 10 sample observations) then the sum of the absolute value of errors for the arbitrary line shown would be a +(4- a) + b + (4- b) = 8. (a) Consider a sample consisting of the 10 data points listed above and assume the criterion we use to determine the regression line is to minimize the sum of absolute errors. (1) Looking at the above diagram, write out the total sum of the absolute errors using a's and b's. () Based on part (a)t), if we wish to minimize the sum of absolute errors, what value should 'a' have in the above diagram? What value should 'b' have? (As suggested in the diagram, you may assume 0 sa s 4 and 0 sb s 4.) (iii) Based on your response to part (a)(ii) what would be the ensuing regression line using this criterion? Is the regression line unique? Determine the total sum of absolute errors under the enguing regression line (b) Determine the regression line if our criterion is to minimize the sum of squared errors. That is determine the OLS regression line using 12 - Cov(X,Y) and b = Y-box Var(x) (c) Again suppose that our regression line criterion is to minimize the sum of squared errors. This criterion is being applied to the sample of 10 data points originally cited in this exercise (1) Looking at the above diagram, write out the total sum of squared errors for an arbitrary line using a's and b's. (ii) Determine the value of 'a' which minimizes the expression you found in part (c)(i). Likewise, determine the value of 'b' which minimizes the expression. Argue that this produces the exact same regression line you just found in part (b). Determine the total sum of squared errors using this regression line
