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st: panel within transformation cause serial correlation?

From   Eddy <>
Subject   st: panel within transformation cause serial correlation?
Date   Fri, 12 Dec 2003 06:53:06 -0800 (PST)

Dear listers,

In a typical panel data model with individual fixed effect, we have

   y_{it} = a_i + B*X + e_{it}, --- (1)

where a_i is individual effect. Assume e_{it} is iid distributed for
i and t. A standard estimation procedure is to first do the "within"
transformation to get rid of the potentially large number of the a_i
dummies. The transformation essentially subtracts the group means
from the variables:

  y_{it} -y_{i.} = B*(X_{it}-X_{i.}) + (e_{it}-e_{i.}),  -- (2)

where e_{i.} = (1/T) *(e_{i1} + e_{i2} + ... + e_{iT}).

It can be shown numerically that OLS estimations on models (1) and
(2) give you exactly the same results.

My question is: In model (2), the transformed error term
(e_{it}-e_{i.}) seems to be serially correlated within any given
individual (i.e., for any i), but the OLS estimation assumes no
correlation. Thus, how come the serial correlation can be ignored
in estimating (2), and the results are still the same as (1)?

To be more clear, consider the transformed error terms of individual
i in period t and t-1. They are

 (e_{it}-e{i.})   = e_{it}   - (1/T) *(e_{i1} + e_{i2} + ... +
 (e_{it-1}-e{i.}) = e_{it-1} - (1/T) *(e_{i1} + e_{i2} + ... +

. I think they are correlated because of the common term on the RHS
of the expressions.

Any insight will be appreciated.


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