least square --coefficient correlation

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Economics for Management Studies
Victor Yu

CHAPTER 442 24 CORRELATION AND RIn chapter 3 we learned how to regress the output of wheat against the amount of fertilizer per acre Does it make any difference whether we regress wheat against fertilizer or fertilizer against wheatAnd if so how do we decide which variable is on the lefthand side of the equation and which is on the righthand side of the equationAnd if we cant determine which variable belongs on which side what do we do then Finally is there a measure of how well the equation line fits the data In this chapter we will answer these questionsA INDEPENDENT vs DEPENDENT VARIABLESIn ordinary mathematics the following five equations are identical 1YABX Y103X2YABX Y 103X YA 3 Y3103X XBBAY 4X103 13Y XBB A1 5XABY whereA BBB However this is not true in regression as the least squares technique will give differentY X answers depending on whether we have ABX or ABY This can be readily seen by looking back at Figure 36 in the previous chapter When we regressed Y against X we 2 Y Yminimized thethat is we minimized the sum of the squared vertical distances ii Y between the line and the observation 36B The least squares equation was 0X 2 X XHowever when we regress X against Y we minimizewhich is equivalent to ii minimizing the sum of the squared horizontal distances between the line and the observationsX 36D The least squares equation is 14 14YIf we solve this equation in terms of YX Y we get Y64 which is definitely not the same as 0X42CHAPTER 443All of this can be understood in terms of the leastsquares formula22XYXY XYXY XY i i i ii i22 0602 621602 462 64 22 0 602 62802 46216 22 6 662 62 1662 16 62 1622XYXYX66Y X Y24X 24 Y96iii ii i YXYiXiB24241 2XXiYBXA21222 0Y 0XThis regression can be seen in Figure 36B If we regress X against Y we substitute X for Y and Y for X in the least squares formula as X and Y change places X ABY N XYiXiYi1B2496 14 2YYiXYAB2142 2 1232X X 32 14Yor Y64 This can be seen in Figure 36D Hence when we regress Y against X we get a different line than when we regress X against Y When Y is regressed against X the least squares line minimizes the sum of squared 43
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