Advanced Econometrics Takeshi Amemiya

Concentrated Likelihood Function

We often encounter in practice the situation where the parameter vector 0O can be naturally partitioned into two subvectors Oq and fi0 as 0O = (a£, fi'0)'. The regression model is one such example; in this model the parameters consist of the regression coefficients and the error variance. First, partition the maximum likelihood estimator as 0 = (o', fi')'. Then, the limit distribution of •Jr{& — Oo) can easily be derived from statement (4.2.23). Partition the in­verse of the asymptotic variance-covariance matrix of statement (4.2.23) con­formably with the partition of the parameter vector as

(4.2.39)

Then, by Theorem 13 of Appendix 1, we have v'TXd - Oo) — N[0, (A - BC-'BT1].

Let the likelihood function be L(a, fi). Sometimes it is easier to maximize L in two steps (first, maximize it with respect to fi, insert the maximizing value of fi back into L; second, maximize L with respect to a) than to maximize L simultaneously for a and fi. More precisely, define

L*(a) = L[a, fi(a)],

where fi(a) is defined as a root of

(4.2.42)
(4.2.43)
We call L*(a) the concentrated likelihoodjunction of a. In this subsection we shall pose and answer affirmatively the question: If we treat L*(a) as if it were a proper likelihood function and obtain the limit distribution by (4.2.23), do we get the same result as (4.2.40)? From Theorem 4.1.3 and its proof we have
where в+ lies between [o£, ІкріоУ] and в0. But we have

Next, differentiating both sides of the identity

3 log L*{a) _ 3 log L[a, fla)] да да

with respect to a yields

Combining (4.2.51) and (4.2.52) yields

which implies

Finally, we have proved that (4.2.44), (4.2.49), and (4.2.54) lead precisely to the conclusion (4.2.40) as desired.