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From |
anirban basu <abasu@midway.uchicago.edu> |

To |
statalist@hsphsun2.harvard.edu |

Subject |
Re: st: Cross-Sectional Time Series |

Date |
Wed, 26 Jun 2002 16:45:30 -0500 (CDT) |

Hi David and Vince, Thanks for your insights and helpful comments. This was a good learning experience.. Anirban ______________________________________ ANIRBAN BASU Doctoral Student Harris School of Public Policy Studies University of Chicago (312) 563 0907 (H) ________________________________________________________________ On Wed, 26 Jun 2002, Vince Wiggins, StataCorp wrote: > I have one additional comment in the continuing thread comparing the results > of -regress-, -xtreg, fe-, and -xtreg , re-. > > While I agree with the comparisons between the models presented by Mark > Schaffer <M.E.Schaffer@hw.ac.uk> and David Drukker <ddrukker@stata.com>, there > is a more mundane reason why the example presented by Anirban Basu > <abasu@midway.uchicago.edu> elicits virtually identical estimates from > -regress-, -xtreg, fe-, and -xtreg, re-. The short answer is they have to be > identical, at least to machine precision of the computations. > > Anirban Basu asks us to generate data in the following manner, > > . mat C= (1, 0.6, 0.6, 0.6 \ 0.6, 1, 0.6, 0.6 \ 0.6, 0.6, 1, 0.6 \ /* > */ 0.6, 0.6, 0.6, 1) > . drawnorm y1 y2 y3 y4, n(1000) means(1 3 4 7) corr(C) > . gen id=_n > . reshape long y , i(id) j(time) > > Anirban is using -drawnorm- to create 4 correlated variables and then > -reshape- to turn these into a panel data with 4 values for a single y. This > is a fine way to create data with a random effect. Here are the first three > panels: > > . list in 1/12 > > id time y > 1. 1 1 -.0939699 > 2. 1 2 2.265574 > 3. 1 3 2.323656 > 4. 1 4 6.053069 > 5. 2 1 1.367081 > 6. 2 2 3.062155 > 7. 2 3 4.830178 > 8. 2 4 7.105754 > 9. 3 1 1.145398 > 10. 3 2 4.087784 > 11. 3 3 3.99791 > 12. 3 4 6.942679 > > > Anirban, the asks us to try the OLS, fixed-effects, and random-effects > estimators on this data by typing, > > . regress y time > > . xtreg y time , i(id) fe > and, > . xtreg y time , i(id) re > > What is unusual about this model is that we are including -time- as a > regressor. Note that we have perfectly balanced panels of 4 observations > each, and that the variable -time- exactly repeats itself -- counting 1, 2, 3, > 4 in each panel. > > What does this mean for the fixed-effects (FE) transformation? The FE > transformation just subtracts the panel mean for each variable (dependent and > independent) from each value. The panel mean for time is 2.5 in every panel. > This means the the FE transformation just subtracts a constant value from > -time-. Subtracting a constant from a regressor does not have any effect on > its estimated coefficient. > > But wait, we also subtracted the panel means from the dependent variable y and > those means were not the same for each panel. As it turns out, when panels > are balanced, the FE transformation of any variable produces a variable that > has a regression coefficient of exactly 1 when regressed against the > untransformed variable. Thus, the relationship with a variable that has not > been transformed (like -time-, that had only a constant subtracted) remains > exactly the same. > > So, with only a single independent variable that repeats exactly in each > balanced panel, OLS and fixed-effects regression will produce the same > estimate of the coefficient on the regressor (within machine tolerance of the > different computations performed). > > Side-note: While I was aware of the behaviour of variables that repeat within > panel for balanced panels, I hadn't previously considered why the FE > transformation of the dependent variable has no effect. A little scribbling > on the white board from Bobby Gutierrez <rgutierrez@stata.com> shows that when > the FE transformation is expressed in matrix form it is idempotent for balanced > panels. That causes the transformation to essentially fall out of regression > of y on y-transformed leaving a coefficient of 1. > > What about the random-effects (RE) estimator? The GLS random-effects > estimator is just a matrix-weighted combination of the FE estimator and the > between-effects (BE) estimator. The BE estimator is a regression of the > panel-level mean of each variable (again, dependent and independent). As we > saw above, the panel-level mean for -time- is a constant 2.5 in every panel > and thus is collinear with the constant. This means that the between > estimator cannot estimate B_time and provides no additional information for > this coefficient. It has no contribution to the RE estimator. So, the RE > estimator must be identical to the FE estimator in a model with a single > covariate that repeats exactly within each balanced panel. > > > -- Vince > vwiggins@stata.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/ > * * 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/

**Follow-Ups**:**Re: st: Cross-Sectional Time Series***From:*Mark Schaffer <M.E.Schaffer@hw.ac.uk>

**References**:**Re: st: Cross-Sectional Time Series***From:*vwiggins@stata.com (Vince Wiggins, StataCorp)

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