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From |
vwiggins@stata.com (Vince Wiggins, StataCorp) |

To |
statalist@hsphsun2.harvard.edu |

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

Date |
Wed, 26 Jun 2002 15:45:05 -0500 |

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/

**Follow-Ups**:**Re: st: Cross-Sectional Time Series***From:*anirban basu <abasu@midway.uchicago.edu>

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