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- Remark 4: Since it is also well known how to account for random weighting matrix [see Lemmas 3.4 and 3.5 of Pakes and Pollard (1989)], we abstract from it in all the proofs below and instead directly assume in the concerned propositions that the weighting matrix AN is possibly based on some preliminary consistent estimators of the concerned parameters such that AN P − → A where A is a positive definite matrix. Hence in what follows let b θN := b θLM N (A).
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- Under our maintained assumptions and (30) and (31), it can then be shown that kM(β0, γ0, b θN )k and hence kb θN − θ0k is OP (N−1/2). Details are available from the authors. Given this, and that our assumptions are essentially same as that in Theorem 3.5 of Pakes and Pollard (1989), the rest of the proof is also similar. Hence we only provide a sketch of the proof below, and highlight the differences that appear only to the end of the proof.
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