- )()( bbbb ââââ= NNNNN ÃŽÃŽÃŽ In light of Rao (1973, p. 62) and Mittelhammer (1996, p. 254) ââ )()( NNN RR Ãà )())((min bbbb ââââ NNNNN ÃŽÃŽÃŽÃŽ and (C.17) )())(()( minmin bbbb âââââ NNNNN ÃŽÃŽÃŽ ÃŽÃŽ 2 * Ãà ââ NÃŽ for some 0* >ÃŽ by Assumption 5.
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- **** O=ââ ââ ZHHHZH QQQ . It follows as a special case of PÃtscher and Prucha (1997, Lemma F1) that )1()() Ë ( 1 ** 1 **** 1**1 pNN oN =ââââ ââââ ZHHHZH QQQZZ (( . (D.5) 109 It follows further that )1(** pNN o=â PP ( and )1(* ON =P with * NP defined in the Lemma. By arguments analoguous to the proof of Lemma 1 it follows that )1(*2/1 pNN ON =ââ ÃŽF , )1(**2/1 pNN ON =ââ ÃŽF , and also that )1(2/1 pNNN ON =â ÃŽHM and )1(])[( 1 1 ,, 2/1 pNNNN S s NmNmN ON =âââ â = â â ÃŽHMMMI à . As a consequence, )1() Ë ( *2/1*2/1 pNNNNN oNN +ââ=â â ÃŽFPÃŽÃŽ ( and )1(*2/1* pNNN ON =ââ â ÃŽFP , observing again that )1()( pNN o=âÃà ( . This completes the proof, recalling that *** NNN PFT = .
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- APPENDIX C I. Proof of Theorem 1 (Consistency of NÃ~ ) As a preliminary step, we now give a version of Lemma C.1 and Remark C.2 in Kelejian and Prucha (2008) that is applicable to the higher-order case.
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- ÃŽÃq ÃŽÃÃÃŽÃŽIÃà â â âââ=â ++ (C.37) In light of the discussion above the first term on the right-hand side is zero on Ã-sets of probability approaching 1 (compare PÃtscher and Prucha, 1997, p. 228ff.). This yields )1(),( ~),~(~ )~( 2/12/1 pNNNN NNN NNN oNN + â â â=â + ÃŽÃqÃŽ à ÎÃq ÃŽÃà . (C.38) Next observe that )1( ~),~(~ 1 pNNNNN NNN N o=âââ â â â+ ÃŽÃŽÃŽÃŽ à ÎÃq ÃŽ B , since (C.39) )1( ~ 1 pNN o=â â+ ÃŽÃŽ and )1( ),~( pNN NNN o=âââ â â ÃŽ à ÎÃq B . (C.40) We next consider the distribution of the vector ),(2/1 NNNN ÃŽÃq . In light of (C.29) and Lemma C.1 the elements of ),(2/1 NNNN ÃŽÃq can be expressed as ),(2/1 NNNN ÃŽÃq â â â â â â â â â â â â â â â â â â â â â â â â = â â â â NNSN NNSN NNN NNN N N N N
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- Corollary F4 in PÃtscher and Prucha (1997) Assume that NÃŽ and NÃŽ are sequences of random vectors in p R and q R respectively, and let NA be a sequence of bounded non-random qp à matrices. Suppose )1(pNNN o+= ÃŽAÃŽ and that ),(~ ÃŽÃŽÃŽÃŽ Nd N â with ÃŽ being positive definite. Define NNN ÃŽAÃŽ = and ),(~ NNNNNN N AÃŽAÃŽAÃŽAà â= . Let ,, ÃŽÃŽ NN FF and à NF be the cumulative distribution functions of ,, NN ÃŽÃŽ and Nà , respectively. ( )(xFN à is the cdf of a normal distribution with mean NNÃŽA and variance-covariance matrix NN AÃŽA â .) Assume further that 0)(inflim min >âââ NNN AAÃŽ holds. Then 0)()( ââ xFxF NN ÃŽÃŽ as N â â (i.e., the difference between the cdf of NÃŽ and NÃŽ converges to zero at all continuity points of the cdf of NÃŽ ), and 0)()( ââ xFxF NN ÃÃŽ as N â â. (i.e., the difference between the cdf of NÃŽ and Nà converges to zero at all continuity points of the cdf of Nà ).
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- Denote the expectation of Nx as )( NN E xÎ = and its variance-covariance matrix as )( NNEN xxx â=Î , which can be derived using Lemma A.1 in Kelejian and Prucha (2008). It then follows under Assumptions A.1-A.3, and provided that 0)(min 1 >ââ cN Nx ÎÎ holds, that ),0()(2/1 M d
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- In the following let â<K be a common bound for the row and column sums of the absolute elements of NB , NÃŽ , and NNNN ÃŽBÃŽB and of their respective elements. Then, using Lemma A.1 in Kelejian and Prucha (2008), we have ââ= = â = N i N j NjNiNijN bNEE 1 1 ,,, 1 ÃŽÃŽÃ (C.3) ââ= = â â N i N j NjNiNij EbN 1 1 ,,, 1 ÃŽÃŽ ââ= = â â N i N j NjNiNijbN 1 1 ,,, 1 Ãà 3 Kâ , 17 We use the fact that 2/)( NNNNNNNNNN ÃŽAAÃŽÃŽAÃŽÃŽAÃŽ â+â=ââ=â , which is a quadratic form in the symmetric matrix 2/)( NN AA â+ . 79 where we used HÃlderâs inequality in the last step. This proves that NEà is O(1).
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- It now follows from (C.38) and (C.39) and (C.43) that )1()()~( 2/12/112/1 pNNNNNNNN oN +ââ=â ââ vÃŽÃŽÃŽJÃŽÃà . (C.48) Since all nonstochastic terms on the right hand side from (C.48) are )1(O it follows that )~(2/1 NNN Ãà â is )1(pO . To derive the asymptotic distribution of )~(2/1 NNN Ãà â , we invoke (part of) Corollary F4 (together with the Assumptions stated in Corollary F3) in PÃtscher and Prucha (1997) (see Appendix B). In the present context we have ),(~ 2 2/1 S d NNN N I0ÃŽvÃŽÃŽ ââ= â , and )1()~(2/1 pNNNN oN +=â ÃŽAÃà , where 2/11 NNNNN ÃŽÃŽJÃŽA â= â .
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- Let A and B be symmetric, positive semidefinite matrices of dimension NN Ã . Then )()()()()( BAABBA TrTrTr LS ÎÎ ââ , where LÎ and SÎ denote the largest and smallest eigenvalue of A, respectively (Mittelhammer, 1996, p. 254).
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- n NN IÎxÎ xx âââ as ââN . 77 Lemma F1 in PÃtscher and Prucha (1997) Let NA and NB be real square random matrices. Let NB be non-singular with probability approaching 1. Let 0BA p NN ââ as N â â and let the sequences NB and + NB be bounded normwise in probability. Then the sequences NA and + NA are bounded normwise in probability, NA is non-singular with probability approaching 1, and 0BA p NN ââ ++ as N â â.
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- Nij ca ââ=1 , for 1>q (Kelejian and Prucha, 2008, Remark C.1).
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- NmNmNmNNmNmN uMuMI â â= = âââ= 1 1 ,,,,, ])~([)( ÃÃà 22 Compare Kelejian and Prucha (2008, p. 41ff.). 93 NN S m S m NmNmNmNNNmNmN ÃŽDMÃŽDMI â â= = âââ+ 1 1 ,,,,, ])~([)( ÃÃà NN ÃŽÃŽ += , where â â â= = â = ââ+â= S m S m N S m NmNmNNmNmNmNNNmNmNN 1 1 1 1 ,,,,,,, ])()~([)( ÃŽMIMÃŽDMIÃŽ ÃÃÃà NN S m NmNmNm ÃŽDMâ= â+ 1 ,,, ])~([ Ãà . (C.60) This can also be written as NNN gRÃŽ = , (C.61) where ],,[ ,3,2,1 NNNN RRRR = with N,1R â= â= S m NNmNmN 1 ,, ,)( DMI à ])(,...,)([ 1 1 ,,, 1 1 ,,1,2 N S m NmNmNNSN S m NmmNNN ÃŽMIMÃŽMIMR â = â = ââ ââ= Ãà , ],...,[ ,,1,3 NNSNNN DMDMR = , and â â â â â â â â â â ââ â= NNN NN N N ÃŽÃà Ãà Î )~( )~(g .
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- NN NN . (A.5) As can be seen from (A.5), Assumption 5 in the higher-order case requires that the assumption made by Kelejian and Prucha (2008) for the first-order case is fulfilled for at least one subset of moment conditions associated with one of the weights matrices. Note, however, that all weighting matrices enter the elements of each Ns,ÃŽ , Ss ,...,1= . If two weights matrices are collinear, for example, none of the matrices Ns,ÃŽ would have a smallest eigenvalue that is strictly positive and Assumption 5 would be hurt. APPENDIX B. For the convenience of the reader, Appendix B lists some Lemmata and Theorems as used in the subsequent proofs.
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- The following results will be used repeatedly in the proofs: If NA and NB are (sequences of) NN Ã matrices, whose row and column sums are bounded uniformly in absolute value (say by Ac and Bc ), then so are the row and column sums of NN BA and NN BA + by BAcc and BA cc + , respectively (Kelejian and Prucha, 1999, p. 526).
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