R-squared fixed effects

[Henk Janssen]

Dear all,

I am having some trouble understanding the true R-squared for my fixed effects model. For the below data I want to perform firm (gvkey) fixed effects regression.

* Example generated by -dataex-. For more info, type help dataex
clear
input float(logtobin_w ESGcompensation tenure) double femaledummy float(independent boardsize) double TOTAL_SEC float(stock_executive option_executive logsales leverage roa_w) double(ESGscore year) long gvkey
  .11830975 .       7.5                  0       .8 10          2477.4176   .2633955  .12278788  7.481996  .26053295  .11522503                  . 2010 1004
 -.18968147 .  7.727273                  0 .8181818 11 2631.8862000000004   .4328289  .07859492  7.637475    .360874   .1014245                  . 2011 1004
 -.06206616 .  7.727273                  0 .8181818 11 1706.0653333333335  .13900936   .1739342  7.681145   .3316019  .11474566                  . 2012 1004
-.017516488 .  8.727273                  0 .8181818 11 2211.0458333333336  .18035705   .2751986  7.618251  .28824732  .11639009                  . 2013 1004
  .12475272 .  9.727273                  0 .8181818 11 2767.3296000000005  .25445083          0   7.37419  .10165016  .05524752                  . 2014 1004
-.016281042 . 10.727273                  0 .8181818 11 2072.3328333333334   .2479061   .2186806  7.416138  .10269745    .094931                  . 2015 1004
   .1741487 0     10.75 .08333333333333333 .8333333 12 2719.3033333333333  .22130963  .13197964  7.477378  .10458081  .09853068              24.84 2016 1004
   .3389418 0  9.583333 .08333333333333333 .8333333 12 2596.2338333333328  .22687325   .1721262  7.466399  .11621959  .08296714              23.81 2017 1004
  .08878828 0 10.583333 .08333333333333333 .8333333 12 1613.2109999999996   .2315839   .1953064  7.626473  .09339573  .10117321              22.62 2018 1004
   .3576642 0  3.181818 .09090909090909091 .9090909 11          7240.1846   .6474589          0 10.621083   .4246824  .18364143              70.36 2015 1045
   .3279516 0  3.692308 .15384615384615385 .9230769 13  6626.606833333334   .6696367          0 10.601125   .4747825   .1527675              70.27 2016 1045
   .3400276 0 4.6923075 .15384615384615385 .9230769 13  6336.568400000003   .7499329          0  10.65034   .4876839  .13248113              72.02 2017 1045
   .2206872 0      5.75 .16666666666666666 .9166667 12  6059.041599999998   .7889238          0 10.704165     .56172   .0925388              69.36 2018 1045
   .1878603 0       6.4                 .2        0 10  6211.196799999999   .7659218          0  10.73134   .5574465  .10040837  71.34000000000002 2019 1045
  .06461501 1  9.636364  .2727272727272727 .9090909 11  3567.396833333333   .2376441          0    8.0906    .298824  .09208658  79.19000000000001 2010 1075
  .10431954 1  9.833333                .25 .9166667 12          4634.7656  .36076725          0  8.083755  .26668325   .0895096              72.13 2011 1075
  .11441843 1       9.8                 .2       .9 10  5922.154799999998   .3203236          0  8.102224   .2551711  .09388095              70.62 2012 1075
   .1143407 1      10.8                 .2       .9 10 3066.1481666666664   .4162558          0   8.14747  .25835332  .09342367  67.78999999999999 2013 1075
  .20094657 1  9.363636 .18181818181818182 .9090909 11  4414.744599999999   .2996751          0  8.158125  .24886835  .08583486              61.63 2014 1075
  .15798543 1      10.7                 .1       .9 10          4144.8194   .3282692          0  8.159215   .2541859  .08976582              68.98 2015 1075
    .217275 1      11.7                 .1       .9 10          5279.9712   .3267494          0  8.160142  .27017725  .08384103  62.47000000000001 2016 1075
   .2351784 1 11.636364 .18181818181818182 .9090909 11           4392.529  .27997512          0  8.179003   .2918555  .08628815  55.73000000000001 2017 1075
  .21915583 1      11.3                 .2       .9 10  4146.410833333334   .3376084          0  8.213719  .29520902  .07676775              63.25 2018 1075
    .225777 1 11.272727 .18181818181818182        0 11  4581.278166666668   .3218178          0  8.152258   .3145051 .068340935              62.58 2019 1075
   .6258997 0  9.833333 .08333333333333333 .9166667 12         12272.8692   .4724094  .05779778 10.467855   .3181561  .17932525  81.90999999999998 2010 1078
   .7224569 1       6.8                 .2       .9 10  9680.794599999997   .3559583 .072812445 10.567495  .25572947  .19975054  83.65999999999998 2011 1078
   .7601307 1  5.090909 .36363636363636365 .9090909 11 11890.849799999998   .3991205  .07130705 10.593477   .3045435   .1869076              80.87 2012 1078
   .5852639 1  6.090909 .36363636363636365 .9090909 11  7548.088600000002  .38093635   .2320279  9.991864  .15274835  .10227458              73.87 2013 1078
   .7529889 0  7.090909 .36363636363636365 .9090909 11          7549.2126   .2302896  .23029397  9.915762   .1900666  .10841914  68.14999999999999 2014 1078
    .736783 0  8.090909 .36363636363636365 .9090909 11  6784.863333333333   .2967345  .29628077  9.923535  .21822193  .11743885              75.64 2015 1078
   .5212999 0  9.090909 .36363636363636365 .9090909 11  8267.837400000002  .27706397  .27710316  9.945253   .4178407  .09446322              79.51 2016 1078
   .6417528 0  9.090909 .36363636363636365 .8181818 11  9900.922285714287  .19566767  .17468856 10.217934   .3662164  .09202623              77.27 2017 1078
   .8903543 0  9.333333  .3333333333333333 .8333333 12 10830.147599999998   .3510712   .3511061 10.328036  .29127774    .112575              72.39 2018 1078
   .6530657 0 4.5555553  .1111111111111111 .7777778  9          3883.2235  .28704077   .2051793  8.778634   .4877115   .1629734               67.7 2010 1161
   .3645646 0  5.111111  .1111111111111111 .8888889  9  5402.038714285714   .3292796   .3075194  8.789965   .4069439   .1495761  64.90999999999998 2011 1161
   .2571971 0       5.6                 .1       .8 10  3749.732000000001   .4543789  .22222343   8.59822      .5105       .056  65.69000000000001 2012 1161
   .4196974 0       4.6                 .1       .7 10  4492.095800000001  .58390486   .1801578 8.5752735  .47452155  .05579894  65.79999999999998 2013 1161
   .4057164 0  4.181818 .09090909090909091 .7272727 11  5496.783333333333  .59413034   .1839726  8.613594   .5872047 .073798776  63.76999999999999 2014 1161
   .6225287 0       3.4                 .2       .6 10          4663.4784   .6716943   .1503329  8.291797   .7275651  .04278551              80.48 2015 1161
  1.4030088 0         5  .2222222222222222 .6666667  9 5291.8571999999995   .6315684  .19050725  8.359838   .4320988  .04278551              71.73 2016 1161
  1.2907536 0       4.5                .25      .75  8          4795.2928  .56837773  .15451786  8.580919   .3940678  .06468926  69.35000000000001 2017 1161
  1.5674034 0  4.111111  .2222222222222222 .7777778  9  5194.050333333334   .5859488   .1699061  8.775703   .2743635  .12949955              68.12 2018 1161
   .6420469 1  7.272727 .18181818181818182 .9090909 11  5596.504399999999  .25204903   .1994436  9.107864   .3056664   .1707698  79.81999999999998 2010 1209
   .5409696 1  7.818182 .18181818181818182 .9090909 11  5197.334600000001   .2258526  .18627685  9.218507   .3192426   .1750719              84.98 2011 1209
   .5037185 1  7.545455  .2727272727272727 .9090909 11  4969.750800000001   .2349808  .18571424  9.170736   .3123576  .13755327  84.45000000000002 2012 1209
    .623949 1  8.545455  .2727272727272727 .9090909 11 3123.1486666666674  .23844536  .20301737   9.22822   .3514602  .13531017              75.51 2013 1209
    .765121 1  5.416667 .16666666666666666 .9166667 12  5653.788166666666  .24315766   .2003613 9.2533045     .34414  .14402303              78.76 2014 1209
   .7700632 1       6.4                 .2       .9 10           5296.816   .3667904  .07949234  9.199775  .33713534  .16095217  79.16999999999999 2015 1209
   .8828155 1     4.375                .25     .875  8          5156.0642   .3697537          0 9.1616125   .3447852  .16982825              73.57 2016 1209
   .8072674 1     5.375                .25     .875  8  6279.959199999998   .4150184          0  9.010376  .21458586  .13681555              79.04 2017 1209
   .8526819 1     6.375                .25     .875  8          6434.1898   .4485046          0  9.097194   .1987976  .14923117  84.77999999999999 2018 1209
  .17025746 .  11.88889  .2222222222222222 .8888889  9 1961.2088333333336   .3100396  .11475078  8.251221   .3058247  .14258662                  . 2010 1230
    .252262 .      10.8                 .3       .8 10          1907.7008   .2622375  .13603291 8.3705015  .25156882  .14142445                  . 2011 1230
  .25676554 .      11.8                 .3       .8 10  2261.820833333334  .28720525  .08877523  8.446127  .18746594  .14459582                  . 2012 1230
  .41635615 . 11.727273  .2727272727272727 .8181818 11 2087.1042000000007    .374419  .15566844  8.509967  .14919493  .15690304                  . 2013 1230
   .6560078 .      10.3                 .4       .9 10 1872.3498571428568  .27144995  .14502862  8.588211  .12991425   .1983498                  . 2014 1230
   .7763644 0  7.454545 .45454545454545453 .9090909 11 2361.6336000000006  .42100295  .10166698  8.630165  .10500536   .2525639  55.67999999999999 2015 1230
   .5901425 0  8.454545 .45454545454545453 .9090909 11 2692.6487999999995   .4581262  .08749834  8.687948  .29753062  .18359767              64.86 2016 1230
   .4026812 0  9.555555  .4444444444444444 .8888889  9 3092.7891666666665   .5221846  .09153342  8.978786  .23919925  .16294228  60.86999999999999 2017 1230
   .2950791 0  7.363636 .45454545454545453 .9090909 11  2582.232833333334   .3301577  .11376386  9.019664   .1927236  .10749634              54.26 2018 1230
   .2051185 . 11.666667  .1111111111111111 .7777778  9 1088.4212000000005   .2609991   .1322854  7.451241   .2197327  .08765724                  . 2011 1254
   .5058599 .      11.5                 .1       .8 10  1201.115142857143  .28250012   .1015951  7.352441  .27173635  .14945073                  . 2012 1254
   .4850742 . 10.857142 .14285714285714285 .8571429  7 1420.2306666666666   .5040103          0  7.400743   .2291917  .13658576                  . 2013 1254
   .5901712 . 11.857142 .14285714285714285 .8571429  7           1798.992    .326683          0  7.446702  .26651448  .14374375                  . 2014 1254
   .6101473 0 12.857142 .14285714285714285 .8571429  7 1884.6286000000002  .36298385          0   7.54163  .25745597  .17541023  34.88999999999999 2015 1254
   .4182765 0 13.857142 .14285714285714285 .8571429  7 1594.1343333333334  .50208896          0  7.571268   .3666088  .11684445               39.5 2016 1254
  .23310487 0        11 .14285714285714285 .7142857  7 2080.5836000000004   .4044619          0  7.624082   .3813571  .09757508  37.67000000000001 2017 1254
   .2245757 0  9.857142 .14285714285714285 .8571429  7          2124.7818   .4033992          0  7.706523     .35237  .08998518 39.300000000000004 2018 1254
  .28666762 0  6.714286  .2857142857142857        0  7 2073.7504000000004   .4096306          0  7.697621   .4298165  .07229212              50.32 2019 1254
   .5978834 1       6.5                 .1       .9 10 10855.913199999997   .3528117   .2853227 10.415413  .17560925   .1248348              49.18 2010 1300
   .5801171 1       6.3                 .1       .9 10          15717.509          0   .2849802  10.50586  .18978597   .1071895              60.96 2011 1300
   .6297287 1       7.3                 .1       .9 10 13103.977199999998  .28917408   .2767683 10.536487  .17910305  .13205744              63.25 2012 1300
   .7847168 1         7                .25 .9166667 12 13788.763600000004 .027408276   .2786295 10.572726  .19432156   .1574777  66.94000000000001 2013 1300
   .8463459 1         8                .25 .9166667 12 11117.501333333334  .21651234  .29472128 10.604256   .1910409   .1617786  53.02999999999998 2014 1300
   .8096887 1      7.75                .25 .9166667 12         13847.0564          0   .3161889 10.560515   .2447076  .16743045              62.37 2015 1300
   .8198145 1      8.75                .25 .9166667 12 11173.043399999997   .3011529   .3437505  10.57903  .29134193  .15441585  66.64999999999999 2016 1300
   .9738825 1  9.076923 .23076923076923078 .8461539 13 11548.070999999998  .21835007  .25763392 10.609897   .3011097   .1555391                 67 2017 1300
   .8560433 1  8.416667                .25 .9166667 12  8241.814571428575   .4841225   .1834174   10.6407  .28065014  .15354924  72.66999999999999 2018 1300
  1.0402052 1  9.416667                .25        0 12  9906.926833333333   .3646107  .19597636 10.510777  .28471854  .14460029  74.06999999999998 2019 1300
   .9323655 0  8.555555                  0 .7777778  9           2140.735   .3200472  .22701335   6.97714  .04778805  .16074465  52.63999999999999 2010 1327
   .6521157 0  9.555555                  0 .7777778  9  2749.407166666667    .427952  .23105904  7.257653 .013800863   .2039374              46.99 2011 1327
     .80136 0 10.555555                  0 .7777778  9 1911.8248333333333   .3996799  .26350126  7.357927          0   .1783866 48.730000000000004 2012 1327
   .7419222 0    11.625                  0      .75  8 2461.6748000000002   .3658626   .2300524  7.491087          0  .19587673              47.07 2013 1327
   1.345916 0    10.375               .125     .875  8               3245   .4701273  .15088746  7.736962          0   .2333378              43.57 2014 1327
  1.4949583 0    11.375               .125     .875  8 3519.5010000000007   .3934705  .20700005  8.088991          0  .29371822              47.31 2015 1327
  1.3172176 0    12.375               .125     .875  8 3090.5408749999992   .4749785   .3120078  8.098339          0  .29371822              41.54 2016 1327
  1.4328263 0        12  .1111111111111111 .7777778  9          4661.9418   .5601915   .1291313  8.202866          0  .29371822  55.72000000000001 2017 1327
   1.248179 0        13  .1111111111111111 .7777778  9  5194.166166666666   .6930379          0  8.260493          0  .29371822              55.49 2018 1327
  1.0762497 0        12  .2222222222222222        0  9           5433.453   .7233093          0  8.124683          0  .29211092  55.11000000000001 2019 1327
  .23008296 1        11 .07692307692307693 .7692308 13 7736.2581999999975   .2331519   .2332289  10.43005   .1577297  .17052774  68.46000000000001 2010 1380
 .020101056 1        12 .15384615384615385 .7692308 13  7501.111999999998   .2473889    .250891  10.55753    .154768   .1661897  73.16999999999999 2011 1380
 -.07162579 1        13 .15384615384615385 .7692308 13  6373.796600000001  .55704385          0 10.537177    .186713  .16677792  69.93000000000004 2012 1380
  .05197859 1  5.894737 .10526315789473684 .9473684 19  6965.755199999999  .53402585          0 10.011624  .13561304   .1438228              69.98 2013 1380
-.029052453 1  7.142857 .14285714285714285 .9285714 14          9405.2334   .3771886  .09429847  9.281451  .15519208   .1422054  74.15999999999998 2014 1380
 -.17600115 1  8.142858 .14285714285714285 .9285714 14  6649.987200000002   .4433318  .11083187  8.800264    .193888  .05588536              76.76 2015 1380
   .1664449 1  9.272727 .18181818181818182 .9090909 11          6101.3272    .432174  .11034811  8.468423  .23779742  .04278551  70.58000000000001 2016 1380
   .1561371 1      8.75 .16666666666666666 .9166667 12 7350.4857999999995   .3766926   .1088413  8.606302   .3018778  .04278551              77.38 2017 1380
  .09665885 1      9.75 .16666666666666666 .9166667 12  6510.368000000001  .39493415  .12856022  8.751949   .3112957  .11864881  76.59000000000002 2018 1380
   .4281915 1 11.181818 .18181818181818182        0 11  8077.180200000002   .3147502  .10647754  8.778788   .3641539  .12744468              78.64 2019 1380
  .25403136 .  10.88889  .2222222222222222 .7777778  9 2314.3152000000005   .3474218  .21813995  8.159303    .090723  .10021309                  . 2010 1410
end
format %ty year

Going:

- reg logtobin_w c.ESGcompensation##c.tenure femaledummy independent boardsize TOTAL_SEC stock_executive option_executive logsales leverage roa_w ESGscore i.year i.gvkey

- areg logtobin_w c.ESGcompensation##c.tenure femaledummy independent boardsize TOTAL_SEC stock_executive option_executive logsales leverage roa_w ESGscore i.year, absorb(gvkey)

yields the same estimates in coefficients and P-values. But more importantly, the R-sq. values are the same. However, if I use:

- xtreg logtobin_w c.ESGcompensation##c.tenure femaledummy independent boardsize TOTAL_SEC stock_executive option_executive logsales leverage roa_w ESGscore i.year, fe vce(cluster gvkey)

the R-sq. is significantly lower. It is even lower than other regressions with industry fixed effects. Those have fewer regressors (as #firms > #industries), so per definition the R-sq. for firm fixed effects should be higher.

Does anyone know what is causing this? Thank you in advance.

[Andrew Musau]

The statistic is defined differently in regress/areg in comparison to xtreg. In fact, the latter reports 3 different R-squared statistics. See the PDF manual entry of xtreg for definitions and formulas.

help xtreg

#9 in the following thread illustrates the calculation of the within R-squared statistic: https://www.statalist.org/forums/for...ted-as-missing

[Henk Janssen]

- Andrew

Thank you for the advice. Would this mean that going -regress/-areg makes more sense with regard to the estimation of R-sq.? Logically, the R-sq. should be higher in the regression including firm fixed effects than only industry fixed effects right? With -xtreg and firm fixed effects it is significantly lower.

[Henk Janssen]

"Would this mean that going -regress/-areg makes more sense with regard to the estimation of R-sq."

With this, I mean in order to compare the R-sq. to my other estimates with -reg and only i.year or i.year and i.industry dummies.

[Carlo Lazzaro]

Henk:
as Andrew already pointed out, those R-sqs are not equivalent.
When it comes to -xtreg,fe- you should consider the within R_sq.

[Andrew Musau]

With this, I mean in order to compare the R-sq. to my other estimates with -reg and only i.year or i.year and i.industry dummies.

You want to compare like for like. So yes, if some of your regressions report the Least Squares Dummy Variables (LSDV) R-squared, then you should report the same even for the model estimated using xtreg.

Logically, the R-sq. should be higher in the regression including firm fixed effects than only industry fixed effects right?

Yes. If you have too many dummies, it may be inefficient to estimate the model using regress/ areg. You can install reghdfe from SSC which reports the LSDV R-squared and allows you to absorb multiple dummies. On the other hand, as Carlo states, report the within R-squared consistently (across all models).

[Henk Janssen]

- Carlo

Thank you. I understand that I should consider the within. However, this still is lower than my other R-squared using -reg.

I think for now I will just go -reg i.year i.firm to be able to 'compare'.

[Carlo Lazzaro]

Henk:
please note that under -regression.and -xtreg,re- (overall) R_sq are computed differently (please note that standard errors are intentionally not clustered due to the scant number of panels):

use "https://www.stata-press.com/data/r16/nlswork.dta"
. reg ln_wage c.age##c.age i.idcode i.year if idcode<=3

      Source |       SS           df       MS      Number of obs   =        39
-------------+----------------------------------   F(18, 20)       =      4.86
       Model |  4.21278813        18  .234043785   Prob > F        =    0.0005
    Residual |  .962950828        20  .048147541   R-squared       =    0.8139
-------------+----------------------------------   Adj R-squared   =    0.6465
       Total |  5.17573896        38  .136203657   Root MSE        =    .21943

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .0773019   .2865219     0.27   0.790    -.5203723    .6749761
             |
 c.age#c.age |  -.0045583   .0012212    -3.73   0.001    -.0071057    -.002011
             |
      idcode |
          2  |  -.4183815   .0918256    -4.56   0.000    -.6099263   -.2268366
          3  |   .6579353   1.834332     0.36   0.724    -3.168414    4.484284
             |
        year |
         69  |   .3367906   .4335876     0.78   0.446    -.5676572    1.241238
         70  |   .2089384   .6771373     0.31   0.761    -1.203545    1.621422
         71  |   .3144116   .9610926     0.33   0.747    -1.690392    2.319216
         72  |   .5888124   1.253657     0.47   0.644     -2.02627    3.203894
         73  |   .8912873   1.550825     0.57   0.572    -2.343676    4.126251
         75  |   1.246958   2.152898     0.58   0.569    -3.243908    5.737823
         77  |   1.560689   2.761762     0.57   0.578    -4.200247    7.321624
         78  |   1.941522   3.068213     0.63   0.534    -4.458659    8.341703
         80  |    2.34498   3.684737     0.64   0.532    -5.341247    10.03121
         82  |   2.698954   4.315145     0.63   0.539     -6.30228    11.70019
         83  |   2.994437   4.618087     0.65   0.524    -6.638723     12.6276
         85  |   3.538578   5.245889     0.67   0.508    -7.404154    14.48131
         87  |   3.965153   5.878139     0.67   0.508    -8.296429    16.22674
         88  |    4.40786   6.407149     0.69   0.499    -8.957218    17.77294
             |
       _cons |   1.341224   4.651269     0.29   0.776    -8.361153     11.0436
------------------------------------------------------------------------------

. xtreg ln_wage c.age##c.age i.year if idcode<=3, fe

Fixed-effects (within) regression               Number of obs     =         39
Group variable: idcode                          Number of groups  =          3

R-sq:                                           Obs per group:
     within  = 0.7404                                         min =         12
     between = 0.4068                                         avg =       13.0
     overall = 0.4014                                         max =         15

                                                F(16,20)          =       3.57
corr(u_i, Xb)  = -0.8560                        Prob > F          =     0.0042

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .0773019   .2865219     0.27   0.790    -.5203723    .6749761
             |
 c.age#c.age |  -.0045583   .0012212    -3.73   0.001    -.0071057    -.002011
             |
        year |
         69  |   .3367906   .4335876     0.78   0.446    -.5676572    1.241238
         70  |   .2089384   .6771373     0.31   0.761    -1.203545    1.621422
         71  |   .3144116   .9610926     0.33   0.747    -1.690392    2.319216
         72  |   .5888124   1.253657     0.47   0.644     -2.02627    3.203894
         73  |   .8912873   1.550825     0.57   0.572    -2.343676    4.126251
         75  |   1.246958   2.152898     0.58   0.569    -3.243908    5.737823
         77  |   1.560689   2.761762     0.57   0.578    -4.200247    7.321624
         78  |   1.941522   3.068213     0.63   0.534    -4.458659    8.341703
         80  |    2.34498   3.684737     0.64   0.532    -5.341247    10.03121
         82  |   2.698954   4.315145     0.63   0.539     -6.30228    11.70019
         83  |   2.994437   4.618087     0.65   0.524    -6.638723     12.6276
         85  |   3.538578   5.245889     0.67   0.508    -7.404154    14.48131
         87  |   3.965153   5.878139     0.67   0.508    -8.296429    16.22674
         88  |    4.40786   6.407149     0.69   0.499    -8.957218    17.77294
             |
       _cons |   1.465543   5.342682     0.27   0.787    -9.679096    12.61018
-------------+----------------------------------------------------------------
     sigma_u |  .54258328
     sigma_e |  .21942548
         rho |  .85944136   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(2, 20) = 10.43                      Prob > F = 0.0008

.

A more substantive issue relates to the correct specification of the functional form of the regerssand in your model (see -linktest-).