Exapl e t hae Tl hSan hl reo l epfl Sor pxl hsi BRAFR {a0 TLGE02-ECINEEFMly by s2-Al5FRGvHy, wRmG AIR TL-E-EGETCIMEE5FN5 5.5 Asymptotic distribution of a function of b LE5 f(o bRGvEIF'II J CIN5-MAIAECNd CI5-MALAAy d-ffFHFN5-CbIRIAMG-INEIIb-WR wC 5TfiNd 5LRl-m-5-Mg d-EFbAfIMII f(b)By ELRTGIIFRRNEIM af(6 f(6)=f()+0m(1)+ ingate afB2-EGm GFx II SLRIEm afn( I→→ aBn ask TLRIRnGMFFSFFm bECTnFEMgl-g-bIR-Ib a B-TLAE (f(b)1f()a∈(or(a2Qm→ TLG-Ef(b) GETLGEGMEnd d-E5FbA5-IM-M5LRIl-m-5CHAPTER 5 LARGE—SAMPLE PROPERTIES OF THE LSE 6 Because ε ′ ε n = 1 n n i=1 ε 2 i P→ σ 2 ε ′X n = 1 n Xiεi P→ 0 X′X n = 1 n XiX ′ i → Q and n n − K → 1, s 2 P→ σ 2 . That is, σ 2 is consistently by s 2 . Alternatively, we may use σˆ 2 = 1 n ˆε ′ ˆε. This is also consistent. 5.5 Asymptotic distribution of a function of b. Let f (b) be a vector of J continuous and continuously differentiable functions of b. We want to find the limiting distribution of f (b). By the Taylor expansion f (b) = f (β) + ∂f (β) ∂β′ (b − β) + remainder. ∂f(β) ∂β′ is a matrix of the form ∂f1(β) ∂β1 · · · ∂f1(β) ∂βk . . . ∂fJ(β) ∂β1 · · · ∂fJ (β) ∂βk = Γ. The remainder term becomes negligible if b P→ β. Thus √ n (f (b) − f (β)) d→ N  0, Γ  σ 2Q −1  Γ ′  . That is, f (b) also has a normal distribution in the limit