On Spatial Processes and Asymptotic Inference under Near$Epoch ...

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hence w.l.g. we can take di;n = 1. Furthermore, by Assumption 6(b) the input process " is -mixing, and the -mixing coe¢ cients satisfy Assumption 4(b). Consequently (C.2) follows directly from Theorem 2 applied to qi;n(Zi;n; ). Next observe that by Proposition 1 of Jenish and Prucha (2009), Assumption 6(c) implies that qi;n is L0 stochastically equicontinuous on , i.e., for every " > 0 1 lim sup n!1 jDnj X P i2Dn sup ( ; ) jqi;n(Zi;n; ) qi;n(Zi;n; )j > " ! ! 0 as ! 0: Furthermore, in light of Assumption 6(a) the qi;n(Zi;n; ) clearly satisfy the domination condition postulated by the ULLN in Jenish and Prucha (2009), stated as Theorem 2 in that paper. Given that we have already veri…ed the pointwise LLN in (C.2) it now follows directly from that theorem that 1 P sup jRn( ) ERn( )j 2 p ! 0 (C.3) with Rn( ) = jDnj i2Dn qi;n(Zi;n; ), and that the ERn( ) are uniformly equicontinuous on in the sense that lim sup sup n!1 2 sup ( ; ) To prove (C.1) observe that sup 2 sup 2 sup 2 jERn( ) ERn( )j ! 0 as ! 0: Qn( ) Q n( ) (C.4) jRn( ) 0 P Rn( ) ERn( )P ERn( )j + sup jRn( ) 2 0 (Pn P )Rn( )j jRn( ) 0 P Rn( ) ERn( )P ERn( )j + 2 sup jRn( )j 2 2 jPn P j : Furthermore observe that Assumption 6(a) we have E [sup 2 jqi;n(Zi;n; )j] K and E [sup 2 jqi;n(Zi;n; )j] 2 K for some …nite constant K. Thus and sup 2 E jRn( )j E sup jRn( )j jDnj 2 E sup jRn( )j 2 2 jDnj jDnj 2 X i;j2Dn 2 X i;j2Dn E sup 2 " 1 X i2Dn E sup jqi;n(Zi;n; )j K (C.5) 2 jqi;n(Zi;n; )j sup jqj;n(Zj;n; )j (C.6) 2 E sup jqi;n(Zi;n; )j 2 2 # 1=2 " E sup jqj;n(Zj;n; )j 2 Now consider the …rst terms on the r.h.s. of the last inequality of (C.4). From (C.5) we see that E jRn( )j takes on its values in a compact set. Given (C.3) it 32 2 # 1=2 K:
now follows immediately from part (a.I) of Lemma 3.3 of Pötscher and Prucha (1997) that sup jRn( ) 2 0 P Rn( ) ERn( )P ERn( )j p ! 0: (C.7) Next we show that also the second term on the r.h.s. of the last inequality of (C.4) converges in probability to zero. To see that this is indeed the case observe that sup 2 jRn( )j 2 = Op(1) in light of (C.6) and jPn P j p ! 0 by assumption. This completes the proof of (C.1). Having established that ERn( ) are uniformly equicontinuous on , the uniform equicontinuity of Q n( ) on follows immediately from Lemma 3.3(b) of Pötscher and Prucha (1997). Q.E.D. Proof of Theorem 4: Clearly by Theorem 3 we have b o n n = op(1). Step 1. The estimators b n corresponding to the objective function (23) satisfy the following …rst order conditions: h jDnj 1=2 Rn( b i n) = op(1): (C.8) r Rn( b n) 0 Pn The op(1) term on the r.h.s. re‡ects that the …rst order conditions may not hold if b n falls onto the boundary of , and that the probability of that event goes to zero as n ! 1, since the o n are uniformly in the in the interior of by Assumption 7(a). If b n is in the interior of , then the l.h.s. of (C.8) is zero. Taking the mean value expansion of Rn( b n) about Rn( b n) = Rn( o n) + r Rn( e n)( b n o n yields o n) (C.9) where e n 2 is between b n and o n (component-by-component). Let bAn = r Rn( b n) 0 Pnr Rn( e n) and b Bn = r Rn( b n) 0 Pn then combining (C.8) and (C.9) gives: = = jDnj 1=2 b n h I h I i A b+ b n An i A b+ b n An o n jDnj 1=2 b n jDnj 1=2 b n where b A + n denotes the generalized inverse of b An. o n o n h jDnj 1 i1=2 n ; (C.10) bA + n r Rn( b n) 0 h Pn jDnj 1=2 Rn( o i n) + b A + n op(1) bA + n b h 1=2 Bn n jDnj Rn( o i n) + b A + n op(1); Step 2. By Assumptions 7(c) the qi;n(Zi;n; o n) are uniformly L2-NED and uniformly L2+ -integrable with ci;n = 1. Given Assumptions 7(d),(g) it is now readily seen that the process fqi;n(Zi;n; o n); i 2 Dng satis…es all assumptions of the CLT for vector-valued NED processes, given as Corollary 1 in the text, 33
Page 1 and 2: On Spatial Processes and Asymptotic
Page 3 and 4: 1 Introduction 1 Models with cross-
Page 5 and 6: tion of the NED property under tran
Page 7 and 8: tion error declines “su¢ ciently
Page 9 and 10: Proposition 3 (Doukhan, 1994) Under
Page 11 and 12: and sup n sup 1 i n We have the fol
Page 13 and 14: Dobrushin (1968) showed that weak d
Page 15 and 16: R k be a zero-mean random …eld th
Page 17 and 18: Gallant and White (1988) or Pötsch
Page 19 and 20: uniformly to Q n. A su¢ cient cond
Page 21 and 22: Furthermore, for the r > 2 as speci
Page 23 and 24: with kYi E[YijFi(s)]k 2 fi("i+j; Z
Page 25 and 26: where (u; v; r) are the mixing coe
Page 27 and 28: The above CLT is in essence a varia
Page 29 and 30: Using part (a) of Lemma B.3 with Xi
Page 31 and 32: Next observe that Vns = n;s n " 1 n
Page 33: The above shows that indeed, for ea
Page 37 and 38: Recalling that supn h max jDnj 1 i
Page 39 and 40: [29] Jenish, N., and I.R. Prucha (2

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