By Lemma 2,
nN ∑i Dx,T XiXiDχ,τ
=Op(1),
nN Pi Dχ,τzizi) ( nN Pi zizi) -1 (nN Pi ziziDχ,T)
and by definition,
Gx,T Dx-,1T = Jx-,T1 = O(1) ,
as (N, T →∞) . Thus,
θTGχ,T ©A3 — CB-1C0} Gχ,T = Op (θT) = Op (1). (89)
Next, consider
√NTGχ,T {θT [(A4 + A5) - CB3-1 (B4 + B5)] }
θ2 √tg d-√ *d Dχ,τxi(ui1+zi) Λ
T x,τ x,Tj— (NPiDχ,τχizi)(N Pi ziz0) √NP Pizi (ui+zi))ʃ
By Lemma 2, under the local alternatives to random effects (Assumption 11),
√1N Pi Dx,Txi(ui + vi)
nN Pi Dx,T xi¾i)( nN Pi ziz0) N √1N
P ´ = Op(1).
∑i zi (Ui + Vi) ) J
By definition,
Thus,
θT √ΤGχ,T D-,T = O
√ntgxt ©θT £(A4 + A5)-CB-1 (B4 + B5)]}=Op{√0 = op(1). (90)
Substituting (89) and (90) into (88), we have
√nt(bg - β) = [Gx,TA1Gx,T + op(1)] 1[√NTGx,Ta2 + op(1)]
= √NT (bw - β) + Op(1).
The last equality results from Lemma 2(a), (b) and Theorem 5. ¥
Part (b)
Similarly to Part (a), we can easily show that under the assumptions given
in Part (b), the denominator in (88) is
NT X X √ Gx,T (xit xi) (xit xi) Gx,T + op (1) . (91)
Consider the second term of the numerator of (88):
θ2√TGχ,T I
√1N Pi xi(zi + zi)
— (nN ∑i Xizi)(NN ∑i zizi)
1( √1N Pi zi (zi + zi
)´ . (92)
56
More intriguing information
1. Strengthening civil society from the outside? Donor driven consultation and participation processes in Poverty Reduction Strategies (PRSP): the Bolivian case2. The name is absent
3. SME'S SUPPORT AND REGIONAL POLICY IN EU - THE NORTE-LITORAL PORTUGUESE EXPERIENCE
4. An institutional analysis of sasi laut in Maluku, Indonesia
5. The name is absent
6. Optimal Private and Public Harvesting under Spatial and Temporal Interdependence
7. Synthesis and biological activity of α-galactosyl ceramide KRN7000 and galactosyl (α1→2) galactosyl ceramide
8. Demand Potential for Goat Meat in Southern States: Empirical Evidence from a Multi-State Goat Meat Consumer Survey
9. The name is absent
10. The name is absent