Springer Texts in Business and Economics

The Wald, LR, and LM Inequality. This is based on Baltagi (1994). The likelihood is given by Eq. (2.1) in the text

(1)

(3)
(4)
(5)
(6)
(7)
where I11 denotes the (1,1) element of the information matrix evaluated at the unrestricted maximum likelihood estimates. It is easy to show from (1) that n (Xi — 10)2
(8)
2a2

Hence, using (4) and (8), one gets

Hence, using (3) and (11), one gets

where the last equality follows from (10). L(f, 52) is the restricted max­imum; therefore, logL(ft, о2) < logL(ft, 52), from which we deduce that W > LR. Also, L(ft, &2) is the unrestricted maximum; therefore log L(ft, о2) > log L(ft, 52), from which we deduce that LR > LM.

An alternative derivation of this inequality shows first that LM W/n LR ( W

n 1 + (W/n) n n)

Then one uses the fact that y > log(1 + y) > y/(1 + y) for y = W/n.