## Regression Diagnostics and Specification Tests

8.1 Since H = PX is idempotent, it is positive semi-definite with b0H b > 0 for any arbitrary vector b. Specifically, for b0 = (1,0,.., 0/ we get hn > 0. Also, H2 = H. Hence,

n

hii =J2 hb2 > h2i > 0.

j=i

From this inequality, we deduce that hjy — h11 < 0 or that h11(h11 — 1/ < 0. But h11 > 0, hence 0 < h11 < 1. There is nothing particular about our choice of h11. The same proof holds for h22 or h33 or in general hii. Hence, 0 < hii < 1 for i = 1,2,.., n.

8.2 A Simple Regression With No Intercept. Consider yi = xi" + ui for i = 1,2,.., n

a. H = Px = x(x0x)_1x0 = xx0/x0x since x0x is a scalar. Therefore, hii =

n

x2/ x2 for i = 1,2,.., n. Note that the xi’s are not in deviation form as

i=1

in the case of a simple regression with an intercept. In this case tr(H/ =

n

tr(Px/ = tr(xx0//x0x ...

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