Getting residuals from reghdfe - wrong or right?

[Alberto Alvarez]

Dear Stata Users,
Please help me to understand whether I am wrong. I am using the following regression to estimate residuals:

use http://www.stata-press.com/data/r15/nlswork.dta
 
sort idcode year

reghdfe ln_wage hours, absorb(idcode year) vce(cluster idcode) residuals
(dropped 550 singleton observations)
(MWFE estimator converged in 7 iterations)

HDFE Linear regression                            Number of obs   =     27,917
Absorbing 2 HDFE groups                           F(   1,   4159) =       2.35
Statistics robust to heteroskedasticity           Prob > F        =     0.1257
                                                  R-squared       =     0.6555
                                                  Adj R-squared   =     0.5949
                                                  Within R-sq.    =     0.0004
Number of clusters (idcode)  =      4,160         Root MSE        =     0.3030

                             (Std. Err. adjusted for 4,160 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       hours |    .000749    .000489     1.53   0.126    -.0002097    .0017076
       _cons |   1.650541   .0178933    92.24   0.000     1.615461    1.685621
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
      idcode |      4160        4160           0    *|
        year |        15           0          15     |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation

The value of residuals for idcode == 1 and year == 70 is -0.40762286. However, if I plug in the estimates that I got: Y_hat = 0,000749*20 + 1,650541 = 1,665521. Thus, residuals should equal: Y – Y_hat = 1,451214 - 1,665521 = -0,214307. This is different from what the program is giving. Am I wrong in my calculations or there is smth wrong with the code?

[Carlo Lazzaro]

Alberto:
the issue here is to use -xbd- instead of -xb- to predict the fitted values:

. reghdfe ln_wage hours, absorb(idcode year) vce(cluster idcode) residuals
(dropped 550 singleton observations)
(MWFE estimator converged in 7 iterations)

HDFE Linear regression                            Number of obs   =     27,917
Absorbing 2 HDFE groups                           F(   1,   4159) =       2.35
Statistics robust to heteroskedasticity           Prob > F        =     0.1257
                                                  R-squared       =     0.6555
                                                  Adj R-squared   =     0.5949
                                                  Within R-sq.    =     0.0004
Number of clusters (idcode)  =      4,160         Root MSE        =     0.3030

                             (Std. err. adjusted for 4,160 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       hours |    .000749    .000489     1.53   0.126    -.0002097    .0017076
       _cons |   1.650541   .0178933    92.24   0.000     1.615461    1.685621
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
      idcode |      4160        4160           0    *|
        year |        15           0          15     |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation

. predict fitted, xb

. predict fitted_abs, xbd


. list idcode year ln_wage hours _reghdfe_resid fitted fitted_abs in 1

     +---------------------------------------------------------------------+
     | idcode   year    ln_wage   hours   _reghdfe~d    fitted   fitted_~s |
     |---------------------------------------------------------------------|
  1. |      1     70   1.451214      20   -.40762286   1.66552   1.8588368 |
     +---------------------------------------------------------------------+

. di -1.8588368+1.451214
-.4076228

.

[Alberto Alvarez]

Thank you for your reply. The correct way to estimate residuals then is using "predict fitted, xb" and getting residuals as the difference betwen y - fitted. Am I right?

[Carlo Lazzaro]

Alberto:
that is the usual way with OLS.
The community-contributed module -reghdfe- allows two options for calculatind predicted values (from its helpfile):

xb                    xb fitted values; the default
           xbd                   xb + d_absorbvars

If you go with the latter, in your code, you'll obtain the right residual value.

[Alberto Alvarez]

Thank you!

[Carlo Lazzaro]

Alberto:
as you can see the very same issue creeps up whe we compare the fitted values obtained via the two possible appraoches to run a fixed effect regression on a panel dataset:

. use "https://www.stata-press.com/data/r17/nlswork.dta"
(National Longitudinal Survey of Young Women, 14-24 years old in 1968)

. regress ln_wage i.idcode i.year age if idcode<=3

      Source |       SS           df       MS      Number of obs   =        39
-------------+----------------------------------   F(17, 21)       =      2.68
       Model |  3.54194923        17  .208349955   Prob > F        =    0.0171
    Residual |  1.63378973        21  .077799511   R-squared       =    0.6843
-------------+----------------------------------   Adj R-squared   =    0.4288
       Total |  5.17573896        38  .136203657   Root MSE        =    .27893

------------------------------------------------------------------------------
     ln_wage | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
      idcode |
          2  |  -.3898423     .11632    -3.35   0.003     -.631743   -.1479415
          3  |  -2.247118   2.111457    -1.06   0.299    -6.638133    2.143897
             |
        year |
         69  |  -.0920902   .5314565    -0.17   0.864    -1.197315    1.013134
         70  |  -.8648493    .779214    -1.11   0.280    -2.485314    .7556149
         71  |  -1.248506    1.09967    -1.14   0.269    -3.535396    1.038383
         72  |   -1.39387   1.443494    -0.97   0.345    -4.395779     1.60804
         73  |  -1.520276    1.79214    -0.85   0.406    -5.247236    2.206684
         75  |  -2.049717   2.495803    -0.82   0.421    -7.240024     3.14059
         77  |  -2.657565   3.203292    -0.83   0.416    -9.319175    4.004045
         78  |  -2.751196   3.557758    -0.77   0.448    -10.14996    4.647567
         80  |  -3.324016   4.267534    -0.78   0.445    -12.19884    5.550808
         82  |  -4.027975   4.983977    -0.81   0.428    -14.39272    6.336774
         83  |  -4.207353   5.333467    -0.79   0.439     -15.2989    6.884199
         85  |  -4.730657   6.044586    -0.78   0.443    -17.30106    7.839747
         87  |  -5.407995   6.755956    -0.80   0.432    -19.45777    8.641785
         88  |  -5.901929   7.348904    -0.80   0.431    -21.18481    9.380954
             |
         age |   .3010572   .3561559     0.85   0.407    -.4396095    1.041724
       _cons |  -2.882579   5.734884    -0.50   0.620    -14.80892    9.043766
------------------------------------------------------------------------------

. predict fitted, xb

. list ln_wage idcode year age fitted if _n==1

       +-------------------------------------------+
       |  ln_wage   idcode   year   age     fitted |
       |-------------------------------------------|
    1. | 1.451214        1     70    18   1.671601 |
       +-------------------------------------------+

. xtset idcode year

Panel variable: idcode (unbalanced)
 Time variable: year, 68 to 88, but with gaps
         Delta: 1 unit

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

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

R-squared:                                      Obs per group:
     Within  = 0.5596                                         min =         12
     Between = 0.4744                                         avg =       13.0
     Overall = 0.0413                                         max =         15

                                                F(15,21)          =       1.78
corr(u_i, Xb) = -0.9573                         Prob > F          =     0.1102

------------------------------------------------------------------------------
     ln_wage | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
        year |
         69  |  -.0920902   .5314565    -0.17   0.864    -1.197315    1.013134
         70  |  -.8648493    .779214    -1.11   0.280    -2.485314    .7556149
         71  |  -1.248506    1.09967    -1.14   0.269    -3.535396    1.038383
         72  |   -1.39387   1.443494    -0.97   0.345    -4.395779     1.60804
         73  |  -1.520276    1.79214    -0.85   0.406    -5.247236    2.206684
         75  |  -2.049717   2.495803    -0.82   0.421    -7.240024     3.14059
         77  |  -2.657565   3.203292    -0.83   0.416    -9.319175    4.004045
         78  |  -2.751196   3.557758    -0.77   0.448    -10.14996    4.647567
         80  |  -3.324016   4.267534    -0.78   0.445    -12.19884    5.550808
         82  |  -4.027975   4.983977    -0.81   0.428    -14.39272    6.336774
         83  |  -4.207353   5.333467    -0.79   0.439     -15.2989    6.884199
         85  |  -4.730657   6.044586    -0.78   0.443    -17.30106    7.839747
         87  |  -5.407995   6.755956    -0.80   0.432    -19.45777    8.641785
         88  |  -5.901929   7.348904    -0.80   0.431    -21.18481    9.380954
             |
         age |   .3010572   .3561559     0.85   0.407    -.4396095    1.041724
       _cons |  -3.866807   6.544144    -0.59   0.561     -17.4761    9.742485
-------------+----------------------------------------------------------------
     sigma_u |  1.2007631
     sigma_e |  .27892564
         rho |   .9488037   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(2, 21) = 6.09                       Prob > F = 0.0082

. predict fitted_fe, xb

. predict fitted_fe_ue, xbu

. g u= fitted_fe_ue- fitted_fe

. list ln_wage idcode year age fitted_fe fitted_fe_ue u if _n==1

       +-----------------------------------------------------------------+
       |  ln_wage   idcode   year   age   fitte~fe   fitte~ue          u |
       |-----------------------------------------------------------------|
    1. | 1.451214        1     70    18   .6873738   1.671601   .9842277 |
       +-----------------------------------------------------------------+

.

The obtain the same fitted values retrieved from -regress- after -xtreg,fe-, we have to sum up the fitted value+the panel-wise term of the composite error (i.e., -u-).