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From |
"Enrico Pellizzoni" <Enrico.Pellizzoni@borsaitalia.it> |

To |
statalist@hsphsun2.harvard.edu |

Subject |
Re: st: panel within transformation cause serial correlation? |

Date |
Tue, 16 Dec 2003 09:32:01 +0100 |

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. ______________________________ Enrico Pellizzoni Borsa Italiana Spa Research & Development Piazza Affari, 6 - 20123 Milano Tel: 02 72426 304 Fax: 02 86464323 E-mail: enrico.pellizzoni@borsaitalia.it Eddy <eddy_05831@yahoo.com> To: statalist@hsphsun2.harvard.edu 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. __________________________________ Do you Yahoo!? New Yahoo! Photos - easier uploading and sharing. http://photos.yahoo.com/ * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/ ---------------------------------------------------------------------------- L'utilizzo non autorizzato del presente messaggio è vietato e potrebbe costituire reato. Se il presente messaggio non e' a Lei indirizzato, Le saremmo grati se, via e-mail, ne comunicasse l'errata ricezione. Il contenuto del presente messaggio non deve essere considerato come trasmesso o autorizzato da Borsa Italiana. Borsa Italiana non si assume alcuna responsabilità per eventuali intercettazioni, modifiche o danneggiamenti del presente messaggio e-mail. The unauthorized use of this e-mail is prohibited and could constitute an offence. Please notify Borsa Italiana immediately by reply e-mail if you are not the intended recepient. The contents of this message shall be understood as neither given nor endorsed by Borsa Italiana. Borsa Italiana does not accept liability for corruption, interception or amendment, if any, or the consequences thereof. ---------------------------------------------------------------------------- (Embedded image moved to file: pic07579.pcx)

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