Book cover for Essays in Honor of Cheng Hsiao

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Advances in Econometrics

Essays in Honor of Cheng Hsiao

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Emerald Publishing Limited

Volume

DOI:

https://doi.org/10.1108/S0731-9053202041

ISBN electronic:

978-1-78973-957-2

ISBN print:

978-1-78973-958-9

Series ISSN:

0731-9053

Publication date:

2020

In the Appendix, we provide the detailed derivation of the Lemma 2.1 by showing that the JIVE for the Arellano–Bond GMM estimation when only one first differenced lag is used as instrument (Hsiao & Zhang, 2015), namely, in this case, the instrument matrix has the form

\underset{(T - 2) \times (T - 2)}{{\bar{W}}_{i}} = (\begin{matrix} Δ y_{i 1} & 0 & \dots & 0 \\ 0 & Δ y_{i 2} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & Δ y_{i, T - 2} \end{matrix}),

(A.1)

and let H be defined the same as (2.8) while the dimension changes to (T − 2) × (T − 2).

In this case, the Arellano–Bond type GMM for model (2.1) using (A.1) is given by

\begin{array}{l} {\hat{γ}}_{GMM} = [{(\sum_{i = 1}^{N} {\bar{W}}_{i} Δ y_{i, - 1})}^{'} {(\sum_{i = 1}^{N} {\bar{W}}_{i} H {\bar{W}}_{i}^{'})}^{- 1} {(\sum_{i = 1}^{N} {\bar{W}}_{i} Δ y_{i, - 1})}^{- 1}] \\ \times [{(\sum_{i = 1}^{N} {\bar{W}}_{i} Δ y_{i, - 1})}^{'} {(\sum_{i = 1}^{N} {\bar{W}}_{i} H {\bar{W}}_{i}^{'})}^{- 1} (\sum_{i = 1}^{N} {\bar{W}}_{i} Δ y_{i})], \end{array}

(A.2)

for this above Arellano–Bond GMM using only one instrument, it is shown by Hsiao and Zhang (2015, p. 314) that

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