st: RE: RE: RE: how many d.f. in the vcv for the within estimator?

["Schaffer, Mark E" <M.E.Schaffer@hw.ac.uk>]

Gregorio,

One way to think about it is via the underlying asymptotics in a
large-sample setting.

If you regard the number of individual effects N as fixed and
T->infnity, then you would be right to say that (ignoring the constant)
you could indeed use (N*T-k) = T*[N-k/T] as the degrees of freedom.  If
it's T that is going off to infinity, then asymptotically there's no
difference between this and the textbook N*(T-1)-k = T*[N-1/T-k/T].  In
both cases the 1/T terms disappear as T gets large, and you're left with
N*T.  Or, put another way, the ratio (N*T-k)/[N*(T-1)-k] goes to 1 as
T->infinity.

But if it's N that is going off to infinity, then the number of
individual effects is also going off to infinity.  In that case the two
expressions are not equivalent asymptotically, because N*(T-1)-k will
always be lagging behind N*T-k, even asymptotically.  That is, the ratio
(N*T-k)/[N*(T-1)-k] goes to T/(T-1) as N->infinity.  If T is small, this
ratio will be very different from 1.  So you need to incorporate this in
how the error variance is calculated.

I hope this makes sense.  I get uneasy when I make these hand-wavey
asymptotic arguments....

HTH,
Mark

> -----Original Message-----
> From: owner-statalist@hsphsun2.harvard.edu 
> [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of 
> Impavido, Gregorio
> Sent: Wednesday, December 12, 2012 1:49 PM
> To: statalist@hsphsun2.harvard.edu
> Subject: st: RE: RE: how many d.f. in the vcv for the within 
> estimator?
> 
> Mark, thank you.  I do not have your text at hand but so does 
> Baltagi "Econometric Analysis of Panel Data" fourth edition 
> page 16. I am not disputing the point.  I was trying to 
> understand the rationale. I would have thought that the Q 
> transform is not simply a computational expedient but a way 
> to improve efficiency re LSDV.
> 
> -----Original Message-----
> From: owner-statalist@hsphsun2.harvard.edu 
> [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of 
> Schaffer, Mark E
> Sent: Tuesday, December 11, 2012 7:55 PM
> To: statalist@hsphsun2.harvard.edu
> Subject: st: RE: how many d.f. in the vcv for the within estimator?
> 
> Gregorio,
> 
> The classical VCV does indeed need to incorporate the dof 
> adjustment associated with the N individual effects.  See 
> e.g. Hayashi's Econometrics p. 334 (the textbook I have at 
> hand - you can find it in other texts as well).
> 
> HTH,
> Mark
> 
> > -----Original Message-----
> > From: owner-statalist@hsphsun2.harvard.edu [mailto:owner- 
> > statalist@hsphsun2.harvard.edu] On Behalf Of Impavido, Gregorio
> > Sent: 12 December 2012 00:41
> > To: statalist@hsphsun2.harvard.edu
> > Subject: st: how many d.f. in the vcv for the within estimator?
> > 
> > Dear all,
> > 
> > I am trying to replicate the result of -xtreg, fe- in mata. 
> Given that
> the
> > individual effects are not estimated after the within transform,
> shouldn't the
> > degrees of freedom used for the estimate of the variance of the
> residuals be
> > (N*T-k-1) instead of N*(T-1)-k (where N=number of panels, 
> > T=periods,k=number of regressors)?  I have added the intercept = the
> meant
> > of the dep variable as in STATA manual.
> > 
> > I paste the code used below which reproduces the results after if
> force df_r=
> > N*(T-1)-k.  Any suggestion would be welcome.
> > 
> > With kind regards
> > Gregorio
> > ==================
> > * this do file reproduces the results of -xtreg, fe- using mata
> > * it works only with balanced panels
> > * For corrections and suggestions, Gregorio Impavido
> (gimpavido@imf.org)
> > **********************************************************
> > *********************
> > ************************************ START
> > ************************************
> > **********************************************************
> > *********************
> > use "http://www.stata-press.com/data/r9/grunfeld.dta ", 
> clear rename 
> > invest I rename mvalue F rename kstock C sort company time
> > gen touse=(I!=. & F!=. & C!=.)      // ignore eventual missing obs
> > 
> > * example of panel within estimator
> > mata:
> > mata clear                          // clear the workspace
> > 
> > T = 20                              // Number of observations per
> groups
> > N = 10                              // Number of groups
> > 
> > Z = st_data(.,("F","C"),"touse")    // (NTxk) matrix of regressors
> > Y = st_data(.,("I"),"touse")        // (NTx1) vector of dep var
> > i = J(rows(Z),1,1)                  // (NTx1) vector of 
> ones, declare
> a
> > X = Z,i
> > it = J(T,1,1)                       // (Tx1) vector of 
> ones, declare a
> > in = J(N,1,1)                       // (Tx1) vector of 
> ones, declare a
> > B = pinv(T)*I(N)#(it*it')           // (NTxNT) between-individual
> operator
> > Bbar = pinv(N)*(in*in')#I(T)        // (NTxNT) between-individual
> operator
> > Q = I(N*T)-B                        // (NTxNT) Q within transform
> > * Qbar = I(N*T)-Bbar                  // (NTxNT) Qbar within period
> operator
> > Ybar = B*Bbar*Y                     // (NTx1) vector of mean dep var
> > Zbar = B*Bbar*Z                     // (NTxk) matrix of 
> mean regressor
> var
> > Ytilda = Q*Y + Ybar                 // added overall mean
> > Ztilda = Q*Z + Zbar                 // added overall mean
> > Xtilda = Ztilda,i                   // (NTxk+1) add column of 1s
> (intercept)
> > b_fe = pinv(Xtilda'Xtilda)*Xtilda'*Ytilda	// (k+1x1) vector of
> beta hat
> > k = cols(Z)                         // (1x1) No of regressors
> > u = (Ytilda-Xtilda*b_fe)            // (NTx1) uhat, fitted residuals
> > df_r = (N*(T-1)-k)                  // (1x1) residual d.f. 
> (shouldn't
> be NT-k-1??)
> > rss = (u'*u)                        // (1x1) unrestricted 
> residual sum
> of squares
> > mse = rss/df_r                      // (1x1) mean squared error
> > vcv = mse*pinv(Xtilda'Xtilda)       // (NTxk+1) VCOV matrix
> > se = sqrt(diagonal(vcv))            // (k+1x1) vector of s.e. of the
> beta hat
> > t = b_fe:/se                        // (k+1x1) vector of t 
> statistics
> > pt = 2*ttail(df_r,abs(t))           // (k+1x1) vector of pvalues
> > crit = invttail(df_r,0.025)         // (k+1x1) bhat~T(df_r)(b,V(b))
> > cil = b_fe-crit*se                  // (k+1x1) vector of low CI
> > cih = b_fe+crit*se                  // (k+1x1) vector of high CI
> > 
> > rss, df_r, mse
> > b_fe, se, t, pt, cil, cih
> > 
> > end
> > xtset company time
> > xtreg I F C, fe                     // to cross check
> > **********************************************************
> > *********************
> > ************************************* END
> > *************************************
> > **********************************************************
> > *********************
> > 
> > *
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> 
> 
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----- 
Sunday Times Scottish University of the Year 2011-2013
Top in the UK for student experience
Fourth university in the UK and top in Scotland (National Student Survey 2012)


We invite research leaders and ambitious early career researchers to 
join us in leading and driving research in key inter-disciplinary themes. 
Please see www.hw.ac.uk/researchleaders for further information and how
to apply.

Heriot-Watt University is a Scottish charity
registered under charity number SC000278.


*
*   For searches and help try:
*   http://www.stata.com/help.cgi?search
*   http://www.stata.com/support/faqs/resources/statalist-faq/
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