https://stats.stackexchange.com/questions/46185/qu…Multipl

Multipl.
Multiple regression can be obtained by sequential matchingReturning to the setting of the question, we have one target yy and two matchers x1x1 and x2x2. We seek numbers b1b1 and b2b2 for which yy is approximated as closely as possible by b1x1+b2x2b1x1+b2x2, again in the least-distance sense. Arbitrarily beginning with x1x1, Mosteller & Tukey match the remaining variables x2x2 and yy to x1x1. Write the residuals for these matches as x2 1 2 1 and y 1y 1, respectively: the 1 1indicates that x1x1 has been taken out of the variable.We can writey= 1 1+y 1 and x2= 2 1+x2 1.y= 1 1+y 1 and x2= 2 1+x2 1.Having taken x1x1 out of x2x2 and yy, we proceed to match the target residuals y 1y 1 to the matcher residuals x2 1 2 1. The final residuals are y 12y 12. Algebraically, we have writteny 1y= 3 2 1+y 12; whence= 1 1+y 1= 1 1+ 3 2 1+y 12= 1 1+ 3(x2 2 1)+y 12=( 1 3 2)x1+ 3 2+y 12.y 1= 3 2 1+y 12; whencey= 1 1+y 1= 1 1+ 3 2 1+y 12= 1 1+ 3(x2 2 1)+y 12=( 1 3 2)x1+ 3 2+y 12.This shows that the 3 3 in the last step is the coefficient of x2x2 in a matching of x1x1 and x2x2 to yy.We could just as well have proceeded by first taking x2x2 out of x1x1 and yy, producing x1 2 1 2 and y 2y 2, and then taking x1 2 1 2 out of y 2y 2, yielding a different set of residuals y 21y 21. This time, the coefficient of x1x1 found in the last step let s call it 3 3 is the coefficient of x1x1 in a matching of x1x1 and x2x2 to yy.Finally, for comparison, we might run a multiple (ordinary least squares regression) of yy against x1x1and x2x2. Let those residuals be y lmy lm. It turns out that the coefficients in this multiple regression are precisely the coefficients 3 3 and 3 3 found previously and that all three sets of residuals, y 12y 12, y 21y 21, and y lmy lm, are identical.
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