Probably now I got your point.
Let's write your model as y_(it)=D*a_i+B*X+e_(it) (1)
where D is simply the matrix of dummy variables
It is true that:
the OLS estimation of this model (in Stata reg y x a_1 a_2 ...)
the within estimation of the model (in Stata xtreg y x )
give exactly the same result.
The fact is that the Within estimation in Stata doesn't perform you trasformation (2), but simply uses a partitioned OLS
estimation of (1).
The estimator for B, in a simple partitioned regression, is:
B=(X*Md*X')-1 * Md*X*y where Md= I-D*(D'*D)-1*D'. Md has the property that: Md*Md'=Md
This is exaclty the estiamtor you get from an OLS estimation of y_{it} -y_{i.} = B*(X_{it}-X_{i.}) + e_{it}
that is your transformation number 2, but there is not the correction in the error term.
This is the important point.
For this reason there is no serial correlation in the error and the only difference between (1) and (2) is in the
standard errors because in (1) the number of degrees of freedom is NT-N-K, while in (2) is NT-K.
(K is the number of varables in X, N is the nuber of individuals).
Anyway,the within estimation of the model in Stata ( xtreg y x ) automatically adjust for the number of degrees of
freedom.
That's why you have two exactly identical results.
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Enrico Pellizzoni
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Eddy
<[email protected]> To: [email protected]
Sent by: cc:
owner-statalist@hsphsun2. Subject: Re: st: panel within transformation cause serial correlation?
harvard.edu
12/12/2003 16.28
Please respond to
statalist
Eddy wrote:
> 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})
> (e_{it-1}-e{i.}) = e_{it-1} - (1/T) *(e_{i1} + e_{i2} + ... +
> e_{iT})
>
> . I think they are correlated because of the common term on the RHS
> of the expressions.
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