Now if we let denote the betahat value after the

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Now, if we let * denote the betahat value after the transformation, * 1 1 2 2 (10 (10 ))( ) ( ))( ) ˆ ˆ (10 (10 )) ( ) i i i i i i X X Y Y X X Y Y X X X X β β = = = The slope is equal but of opposite sign as the original. Makes sense since all we did was turn the X variable on its head, so the relationship should be the same but backwards. For this problem, that means the slope is .000177. The intercept will be * * 0 1 1 1 1 0 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ (10 ) (10 ) 10 10 Y X Y X Y X β β β β β β β = = + = + = + , which here is -9780.89. This change is a little tricky to decipher. Essentially the intercept term is shifting to account for the flip in the sign of the regression line. The direction of the shift will depend on whether beta1hat is positive or negative and its size. b) If instead, corruption was the “Y” variable, the effect on the slope is exactly the same as above. * 1 1 2 2 ( ))(10 (10 )) ( ))( ) ˆ ˆ ( )) ( ) i i i i i i X X Y Y X X Y Y X X X X β β = = = , or .000177. The intercept also has to adjust. * * 0 1 1 0 ˆ ˆ ˆ ˆ (10 ) 10 10 Y X Y X β β β β = = + = , or 3.153. The shift of the intercept is exactly the same as the transformation of C. I confirmed all these results by rerunning the regressions with the transformed values of corruption.
Either version of the regression is fine Version a, in which we regress corrup on gdp_cap country corrup gdp_cap corrup_d^2 gdpcap_d^2 corrup_d*gdpcap_d Yhat Y-Yhat^2 Argentina 8 8580 8.064 905,430.247 -2702.15 5.328615 7.136299382 Australia 1.3 20930.21 14.901 129,929,655.301 -44001.61 3.143264 3.397620913 Austria 2.5 26577.11 7.077 290,551,423.027 -45345.32 2.144053 0.126698116 Belarus 7.3 2112 4.579 55,049,588.436 -15876.03 6.473118 0.683734568 Belgium 4.6 25006.05 0.314 239,460,429.239 -8669.45 2.42205 4.743466586 Zimbabwe 7.5 653 5.474 78,828,490.032 -20773.65 6.731285 0.590922117 Avgs/sums 5.160240964 9531.541 490.039 11,050,880,223.600 -1955436.60 144.027 Beta1hat -0.000176948 Beta0hat 6.846832778 SSR 144.027 SST 490.039 SSE 346.01 R^2 0.706090138 VarB1Hat 0.0000000001609 SDB1Hat 0.000012685 Test Statistic -13.94972039 p-value 3.03703E-23 Version b, in which we regress gdp_cap on corrup country corrup gdp_cap corrup_d^2 gdpcap_d^2 corrup_d*gdpcap_d Yhat Y-Yhat^2 Argentina 8 8580 8.064 905,430.247 -2702.15 -1800.15 107747538 Australia 1.3 20930.21 14.901 129,929,655.301 -44001.61 24935.33 16041023.29 Austria 2.5 26577.11 7.077 290,551,423.027 -45345.32 20146.89 41347736.78 Belarus 7.3 2112 4.579 55,049,588.436 -15876.03 993.1086 1251918.068 Belgium 4.6 25006.05 0.314 239,460,429.239 -8669.45 11767.11 175269524.7 Zimbabwe 7.5 653 5.474 78,828,490.032 -20773.65 195.0344 209732.5353 Avgs/sums 5.160240964 9531.541 490.039 11,050,880,223.600 -1955436.60 3,247,962,684.987 Regress gdp_cap on corrup Beta1hat -3990.37 Beta0hat 30122.81695 SSR 3,247,962,684.987 SST 11,050,880,223.600 SSE 7802917538.61 R^2 0.706090138 VarB1Hat 81,826.80 SDB1Hat 286.05 Test Statistic -13.94972039 p-value 3.03703E-23
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