Springer Texts in Business and Economics

Simple Versus Multiple Regression Coefficients. This is based on Baltagi (1987)

The OLS residuals from Yi = у + 82v2i + 83v3i + wi, say tVi, satisfy the

following conditions:

n n n

wі = 0 ^]iViv2i = 0 ^^wiV 3i = 0

i=i i=i i=i

with Yi = у + 82V2i + 83V3i + vi.

n

Multiply this last equation by v2i and sum, we get P Yiv2i = °2 P v? i +

i=i

"ols = Y ^VEv.2 = E Yix^]xi2 = Y WExi2.

i=1 i=1 i=1 i=1 i=1 i=1

b. Regressing a constant 1 on Xi we get

n n n

b = 'Y/ Ъ/Y Xi2 with residuals wi = 1 — I nX^ ^ XiM Xi i=1 i=1 i=1

nn

so that, regressing Yi on wi yields a = wiYi/ w2.

i=1 i=1

_ n _ n

n _ _ n n nY EXi2-n^XiYi

But P wiYi = nY — nX £ XiYi/ P Xi2 = —i=^-5------------------------ ^------- and

i=1 i=1 i=1 P Xi2

i=1

_ n n 2 n

n 2 2 2nX X Xi n X Xi2-n2X2 n X Xi2

P w2 = n + ----------- ^ .

i=1 P Xi2 P Xi2 P Xi2 P Xi2

i = 1 i = 1 i = 1 i=1

xi2

i=1

n

X XiYi - nXY

a=4 = Y - " olsX

xi2

i=1

c. From part (a), var("ols) = a2/ ^2 = a2/ xi2 as it should be. From

i=1 i i=1

n

n a2 P X2

part (b), var (aols) = a2/ P w? = tt1 as it should be.