xtreg -fe vs xtpcse, what is the difference between them?

[SangWon Choi]

Dear All,

Hello, I am a Korean student with a question on panel analysis.
I understand the meaning of fixed effect coefficient (within effect), but I was wondering what Panel FGLS coefficient (comes from xtpcse, Prais-Winston) means.
I have looked at stata manual sheets for solutions, but I couldn't solve it.
So, if there's anyone who can tell me the difference of coefficient between xtreg -fe and xtpcse, I'd much appreciate it.
(Like, fixed effect coefficient =within effect, Panel FGLS coefficient = ???)

Thank you so much in advance.

[Carlo Lazzaro]

SangWon:
-xtreg- is for N>T panel datasets. By default, it assumes that -epsilon- is neither heteroskedastic, nor autocorrelated within and/or across panels;
-xtpcse- is mainlty for T>N panel datasets. Quoting from its helpfile,

...xtpcse assumes that the disturbances are, by default, heteroskedastic and contemporaneously correlated across panels.

[SangWon Choi]

Thank you for your post Carlo,

Yes, actually, I have seen many of your posts to study panel analysis and these are so helpful. Thank you again.
Maybe I would make my question much more precise.
Then how about results analysis each of them?

for example like below,

fixed effects model
age = 3.6 + 1.6*exercise_hours + residuals

Panel FGLS model
age = 3.6 + 1.6*exercise_hours + residuals

in each model, the variable 'exercise' coefficients are the same. In fixed effect, I would say 'increasing exercise hours by 1 will increase 1.6 age
Then is it right to say in the same way about coefficient of panel FGLS?

[Carlo Lazzaro]

SangWon:
yes. As you can see in the following toy-example, the predicted values are the same:

. use https://www.stata-press.com/data/r17/grunfeld.dta

. xtreg mvalue c.time##c.time, fe

Fixed-effects (within) regression               Number of obs     =        200
Group variable: company                         Number of groups  =         10

R-squared:                                      Obs per group:
     Within  = 0.1025                                         min =         20
     Between =      .                                         avg =       20.0
     Overall = 0.0069                                         max =         20

                                                F(2,188)          =      10.74
corr(u_i, Xb) = -0.0000                         Prob > F          =     0.0000

-------------------------------------------------------------------------------
       mvalue | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
         time |   -29.4019   17.12643    -1.72   0.088    -63.18657     4.38277
              |
c.time#c.time |    2.13131   .7921775     2.69   0.008     .5686106    3.694009
              |
        _cons |   1084.558   78.09038    13.89   0.000     930.5121    1238.604
--------------+----------------------------------------------------------------
      sigma_u |  1334.9167
      sigma_e |  331.92143
          rho |  .94177501   (fraction of variance due to u_i)
-------------------------------------------------------------------------------
F test that all u_i=0: F(9, 188) = 323.50                    Prob > F = 0.0000

. predict fitted, xb

. xtpcse mvalue c.time##c.time

Linear regression, correlated panels corrected standard errors (PCSEs)

Group variable:   company                       Number of obs     =        200
Time variable:    year                          Number of groups  =         10
Panels:           correlated (balanced)         Obs per group:
Autocorrelation:  no autocorrelation                          min =         20
                                                              avg =         20
                                                              max =         20
Estimated covariances      =        55          R-squared         =     0.0069
Estimated autocorrelations =         0          Wald chi2(2)      =       8.94
Estimated coefficients     =         3          Prob > chi2       =     0.0114

-------------------------------------------------------------------------------
              |           Panel-corrected
       mvalue | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
--------------+----------------------------------------------------------------
         time |   -29.4019   26.53782    -1.11   0.268    -81.41507    22.61127
              |
c.time#c.time |    2.13131   1.227498     1.74   0.083    -.2745428    4.537162
              |
        _cons |   1084.558   121.0029     8.96   0.000     847.3967     1321.72
-------------------------------------------------------------------------------

. predict fitted2, xb

. list company time mvalue fitted fitted2 fitted2 if company==1

     +----------------------------------------------------------+
     | company   time   mvalue     fitted    fitted2    fitted2 |
     |----------------------------------------------------------|
  1. |       1      1   3078.5   1057.288   1057.288   1057.288 |
  2. |       1      2   4661.7    1034.28    1034.28    1034.28 |
  3. |       1      3   5387.1   1015.534   1015.534   1015.534 |
  4. |       1      4   2792.2   1001.051   1001.051   1001.051 |
  5. |       1      5   4313.2   990.8314   990.8314   990.8314 |
     |----------------------------------------------------------|
  6. |       1      6   4643.9   984.8739   984.8739   984.8739 |
  7. |       1      7   4551.2    983.179    983.179    983.179 |
  8. |       1      8   3244.1   985.7468   985.7468   985.7468 |
  9. |       1      9   4053.7   992.5771   992.5771   992.5771 |
 10. |       1     10   4379.3    1003.67    1003.67    1003.67 |
     |----------------------------------------------------------|
 11. |       1     11   4840.9   1019.026   1019.026   1019.026 |
 12. |       1     12   4900.9   1038.644   1038.644   1038.644 |
 13. |       1     13   3526.5   1062.525   1062.525   1062.525 |
 14. |       1     14   3254.7   1090.668   1090.668   1090.668 |
 15. |       1     15   3700.2   1123.074   1123.074   1123.074 |
     |----------------------------------------------------------|
 16. |       1     16   3755.6   1159.743   1159.743   1159.743 |
 17. |       1     17     4833   1200.674   1200.674   1200.674 |
 18. |       1     18   4924.9   1245.868   1245.868   1245.868 |
 19. |       1     19   6241.7   1295.325   1295.325   1295.325 |
 20. |       1     20   5593.6   1349.044   1349.044   1349.044 |
     +----------------------------------------------------------+

Tha said, I would not consider these two codes as interchangeable, as their underlying assumptions are pretty different.

[SangWon Choi]

Thank you Carlo, Then, the important things are underlying assumptions. Now I understand!
It is helpful to me. Thank you again, and I wish you have good Christmas day!

Regards,
Choi, Sangwon.

[Carlo Lazzaro]

Choi, Sangwon:
happy with reading that it was of some help.
Thanks for the XMas wishes: I do reciprocate the very same to you and your dears.