Multiple regression analysis y-Bo+Bx+ Bx2+... Bkxk+ Estimation Economics 20- Prof anderson
Economics 20 - Prof. Anderson 1 Multiple Regression Analysis y = b0 + b1 x1 + b2 x2 + . . . bk xk + u 1. Estimation
Parallels with Simple regression ◆ Bo is still the intercept oB to Pk all called slope parameters uis still the error term(or disturbance Still need to make a zero conditional mean assumption, so now assume that ◆B(lx1x2,…,x)=0 o Still minimizing the sum of squared residuals. so have k+l first order conditions Economics 20- Prof anderson
Economics 20 - Prof. Anderson 2 Parallels with Simple Regression b0 is still the intercept b1 to bk all called slope parameters u is still the error term (or disturbance) Still need to make a zero conditional mean assumption, so now assume that E(u|x1 ,x2 , …,xk ) = 0 Still minimizing the sum of squared residuals, so have k+1 first order conditions
Interpreting Multiple regression y=Bo+B,,+ B2x2+.+Bkxk, So △y=△1x1+△B2x2+…+△kxk so holding x, ,,xk fixed implies that △y=△Bx1, that is each B has a ceteris paribus interpreta tion Economics 20- Prof anderson
Economics 20 - Prof. Anderson 3 Interpreting Multiple Regression a interpreta tion , that is each has ˆ ˆ so holding ,..., fixed implies that , ˆ ... ˆ ˆ ˆ ,so ˆ ... ˆ ˆ ˆ ˆ 1 1 2 1 1 2 2 0 1 1 2 2 ceteris paribus y x x x y x x x y x x x k k k k k b b b b b b b b b = = + + + = + + + +
A“ Partialling Out” Interpretation Consider t he case where k=2 . ie y=Bo+B,x,+B,x,, then 1,where rl are the residuals from the estimated regression x=ro+y2I Economics 20- Prof anderson 4
Economics 20 - Prof. Anderson 4 A “Partialling Out” Interpretation ( ) 1 0 2 2 1 2 1 1 1 0 1 1 2 2 regression ˆ ˆ ˆ ˆ the residuals from the estimated ˆ ˆ , where ˆ are ˆ , then ˆ ˆ ˆ ˆ Consider t he case where 2, i.e. x x r y r r y x x k i i i i b b b b = + = = + + =
Partialling out continued e Previous equation implies that regressing y on x, and x, gives same effect of x, as regressing y on residuals from a regression ofx, on x 2 o This means only the part of xlt that is uncorrelated with xi2 are being related to y so we re estimating the effect ofx, on y after x, has been"partialled out Economics 20- Prof anderson 5
Economics 20 - Prof. Anderson 5 “Partialling Out” continued Previous equation implies that regressing y on x1 and x2 gives same effect of x1 as regressing y on residuals from a regression of x1 on x2 This means only the part of xi1 that is uncorrelated with xi2 are being related to yi so we’re estimating the effect of x1 on y after x2 has been “partialled out