HTJA n 2 2A ^1)^1 h h H tRr TA + n 2A l)R R th
mm m J m m m m / /
1 1 ∖ 1
A nn -AA "1 R
∖ m m f f
(γ ʌ
Γ
2
V 2)
Y1
→ΓT RqTrC(1)∑ XkXkTC(1)2Rι
k= 1
q-r
)
Γ - Γ22V Γ,,
1 2 r,m2
where the latter limit matrix is of full rank s. Therefore,
Theorem 5. If the null hypothesis (29) is false, then the s1 ordered largest solutions of the
generalized eigenvalue problem (31) converge in distribution to the ordered solutions of the
generalized eigenvalue problem
m
det ΓTRq2rC(1)∑ XkXk2C(1) 2RrrΓ,
k= 1
(
(
, AL
λ λ I. RqTrC(1)£XkXk2C(1)2R1
∖1
Vl
(34)
k= 1
q~r
√
γ1 + rTVm,n γ2
: ɪ0,
)-
whereas the remaining s - s1 solutions of (31) converge in probability to zero, where Γ1, Γ2
and s1 are defined in (33).
6.2. 2he lambda-max and trace tests for linear restrictions
Theorems 4 and 5 suggest to use the maximum solution, or the sum ,Tm(H), say, of all
solutions, of eigenvalue problem (31) as a basis for a test of the null hypothesis (29). We
only discuss the trace test in detail, as the asymptotic properties of the lambda-max tests can
be derived along similar lines as for the trace test.
It follows straightforwardly from Theorems 4 and 5 that under the null hypothesis
(29),
20