The first-differentiated Stone's index, ∆ logP* may be decomposed into three components:
n nn
(9) ∆ log P * = ∑ w. ∙∆ log pj + ∑ ∆ w ■ log pj - ∑ ∆ w ∙∆ log pj .
j =1j =1j =1
The second and third term are likely to be quite small since, in the context of time-series data,
shares usually do not change much from one observation to the next (Alston and Chalfant).
Substituting the first term of ∆ logP * from Equation (9) into the first-differentiated LA/AIDS in
Equation (8) yields
(10) ∆wi≈∑n γij∆logpj+βi[∆logX-∑n wj∆logpj].
j =1j =1
Equation (10) is similar to the Rotterdam model. Any difference is in the specification of the
income term. Theil and Clements (1987) refer to DQ in Equation (1) as a finite change version of
the Divisia volume index (Alston and Chalfant). It is approximately equal to
(11) DQ* =∆ log X -∆ log P ° ,
n
where ∆ log P ° = ∑ wj ■ ∆ log pj, which is similar to the first-differentiated Stone’s price index
j=1
in Equation (9). Substituting DQ* for DQ, the Rotterdam model is re-specified as follows
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