What is the meaning of the constant using "reg y i.id i.jd" and "xtreg y i.jd ,fe"?

Ruby Yao

Join Date: Apr 2023

Posts: 9
#1

What is the meaning of the constant using "reg y i.id i.jd" and "xtreg y i.jd ,fe"?

10 Apr 2023, 01:15

Hello everyone,

I am wondering what is the meaning of the constant using xtreg y i.jd ,fe and reg y i.id i.jd.

If there is only one group, e.g. group id1-3
Using reg y i.id, the constant 3.5 seems to be the average of y in id1

Using xtreg y, fe, the constant 3.433333 seems to be the grand average of y

If there are two groups, e.g. group id1-3, group jd1-5
What is the meaning of the constant 3.566667 using reg y i.id i.jd?

What is the meaning of the constant 3.5 using xtreg y i.jd, fe?

Below is the average of y in different combinations of id and jd. Thank you.

Last edited by Ruby Yao; 10 Apr 2023, 01:18.
Tags: constant, factor variable, fixed effects, xtreg
Carlo Lazzaro

Join Date: Apr 2014

Posts: 17689
#2

10 Apr 2023, 01:28

Ruby:
the very essential answer is that -_cons- in -regress- and -xtreg,fe- are totally different beasts.
See: https://stats.oarc.ucla.edu/stata/fa...d-by-xtreg-fe/.
In addition, skimming through your post, the reason why screenshots are deprecated by the FAQ becomes apparent.

Last edited by Carlo Lazzaro; 10 Apr 2023, 01:30.

Kind regards,
Carlo
(Stata 19.0)
Comment
Ruby Yao

Join Date: Apr 2023

Posts: 9
#3

10 Apr 2023, 08:05

Dear Carlo,

Thank you very much for your answer and your reference. Sorry for the screenshots. I did not check the FAQ carefully and will revise them.

I have read the page you recommend. I am still slightly confused about the -cons- in the two-groups scenario, e.g. -reg y i.id i.jd- and -xtreg y i.jd ,fe-. Is there any way to interpret the -_cons- in the two-groups scenario?
Comment
Ruby Yao

Join Date: Apr 2023

Posts: 9
#4

10 Apr 2023, 08:06

Dear Carlo,

Thank you very much for your answer and your reference. Sorry for the screenshots. I did not check the FAQ carefully and will revise them.

I have read the page you recommend. I am still slightly confused about the -cons- in the two-group scenario, e.g. -reg y i.id i.jd- and -xtreg y i.jd ,fe-. Is there any way to interpret the -_cons- in the two-group scenario?
Comment

Carlo Lazzaro

Join Date: Apr 2014
Posts: 17689

10 Apr 2023, 08:43

Ruby:
1) -regress- constant refers to id==1 and jd==1. You can see it when calculatiing -predict- by hand.
The following toy-example proves that fitted+ui from -xtreg,fe- equals fitted from -regress-.
Again, the _cons from -xtreg,fe- has no relevance.

Code:

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

. reg ln_wage i.idcode i.year if idcode<=3

      Source |       SS           df       MS      Number of obs   =        39
-------------+----------------------------------   F(16, 22)       =      2.84
       Model |  3.48635949        16  .217897468   Prob > F        =    0.0122
    Residual |  1.68937946        22  .076789976   R-squared       =    0.6736
-------------+----------------------------------   Adj R-squared   =    0.4362
       Total |  5.17573896        38  .136203657   Root MSE        =    .27711

------------------------------------------------------------------------------
     ln_wage | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
      idcode |
          2  |  -.3898423   .1155629    -3.37   0.003     -.629505   -.1501795
          3  |  -.4648596   .1120424    -4.15   0.000    -.6972212   -.2324979
             |
        year |
         69  |    .208967   .3918928     0.53   0.599    -.6037689    1.021703
         70  |  -.2747772   .3439816    -0.80   0.433    -.9881514    .4385969
         71  |  -.3613911    .326316    -1.11   0.280    -1.038129    .3153467
         72  |  -.2056973    .326316    -0.63   0.535    -.8824352    .4710406
         73  |  -.0310461    .326316    -0.10   0.925     -.707784    .6456917
         75  |   .0416271    .326316     0.13   0.900    -.6351107     .718365
         77  |   .0358937    .326316     0.11   0.913    -.6408441    .7126316
         78  |   .2433199    .326316     0.75   0.464    -.4334179    .9200578
         80  |   .2726139    .326316     0.84   0.412    -.4041239    .9493518
         82  |   .1747839   .3439816     0.51   0.616    -.5385903    .8881581
         83  |   .2924489    .326316     0.90   0.380    -.3842889    .9691868
         85  |   .3712589    .326316     1.14   0.267     -.305479    1.047997
         87  |   .2960361    .326316     0.91   0.374    -.3807017     .972774
         88  |   .3038639    .326316     0.93   0.362    -.3728739    .9806018
             |
       _cons |   1.958421   .2989038     6.55   0.000     1.338532    2.578309
------------------------------------------------------------------------------


. list fitted_reg if idcode==3 & year==68

       +----------+
       | fitted~g |
       |----------|
    1. | 1.493561 |
       +----------+

. . di 1.958421-.4648596
1.4935614


. xtreg ln_wage i.year 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  = 0.5446                                         min =         12
     Between = 0.2670                                         avg =       13.0
     Overall = 0.3678                                         max =         15

                                                F(14,22)          =       1.88
corr(u_i, Xb) = -0.0356                         Prob > F          =     0.0897

------------------------------------------------------------------------------
     ln_wage | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
        year |
         69  |    .208967   .3918928     0.53   0.599    -.6037689    1.021703
         70  |  -.2747772   .3439816    -0.80   0.433    -.9881514    .4385969
         71  |  -.3613911    .326316    -1.11   0.280    -1.038129    .3153467
         72  |  -.2056973    .326316    -0.63   0.535    -.8824352    .4710406
         73  |  -.0310461    .326316    -0.10   0.925     -.707784    .6456917
         75  |   .0416271    .326316     0.13   0.900    -.6351107     .718365
         77  |   .0358937    .326316     0.11   0.913    -.6408441    .7126316
         78  |   .2433199    .326316     0.75   0.464    -.4334179    .9200578
         80  |   .2726139    .326316     0.84   0.412    -.4041239    .9493518
         82  |   .1747839   .3439816     0.51   0.616    -.5385903    .8881581
         83  |   .2924489    .326316     0.90   0.380    -.3842889    .9691868
         85  |   .3712589    .326316     1.14   0.267     -.305479    1.047997
         87  |   .2960361    .326316     0.91   0.374    -.3807017     .972774
         88  |   .3038639    .326316     0.93   0.362    -.3728739    .9806018
             |
       _cons |   1.659677   .2833366     5.86   0.000     1.072073    2.247281
-------------+----------------------------------------------------------------
     sigma_u |  .24956596
     sigma_e |  .27711004
         rho |  .44784468   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(2, 22) = 9.64                       Prob > F = 0.0010

. predict fitted_xtreg, xb

. predict fe_xtreg, u

. list fitted_reg fitted_xtreg fe_xtreg if idcode==3 & year==68

       +--------------------------------+
       | fit~_reg   fit~treg   fe_xtreg |
       |--------------------------------|
   25. | 1.493561   1.659677   -.166116 |
       +--------------------------------+

. di 1.659677 -.166116
1.493561

.

Last edited by Carlo Lazzaro; 10 Apr 2023, 08:52.

Kind regards,
Carlo
(Stata 19.0)

Comment

Ruby Yao

Join Date: Apr 2023
Posts: 9

11 Apr 2023, 06:06

Dear Carlo,

Thanks a lot for your help.

Does it mean that the _cons from -reg y i.id i.jd- and -xtreg y i.jd , fe- both refer to the average fitted value of y when id==1 and jd==1?

And I am wondering why is the _cons the average fitted value of y when id==1 and jd==1 rather than the average true value of y when id==1 and jd==1.

I find that there is no observation when idcode==1 & year==68 in the nlswork.dta above. So I post my data here to illustrate my question.

The dataset is as below.

Code:

* Example generated by -dataex-. To install: ssc install dataex
clear
input float(id jd y)
1 1 3
1 1 4
1 2 4
1 2 3
1 3 6
1 3 5
1 4 3
1 4 2
1 5 2
1 5 3
2 1 5
2 1 4
2 2 3
2 2 4
2 3 2
2 3 3
2 4 5
2 4 3
2 5 4
2 5 3
3 1 2
3 1 3
3 2 4
3 2 3
3 3 3
3 3 4
3 4 2
3 4 6
3 5 3
3 5 2
end

The average true value of y when id==1 and jd==1 is 3.5.

Code:

sum y if id==1&jd==1

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
           y |          2         3.5    .7071068          3          4

The average fitted value of y when id==1 and jd==1 from -reg y i.id i.jd- is 3.566667, different from the average true value. Why is the _cons from reg the average fitted value of y when id==1 and jd==1 rather than the average true value of y when id==1 and jd==1?

Code:

reg y i.id i.jd

      Source |       SS           df       MS      Number of obs   =        30
-------------+----------------------------------   F(6, 23)        =      0.47
       Model |  4.06666667         6  .677777778   Prob > F        =    0.8247
    Residual |        33.3        23  1.44782609   R-squared       =    0.1088
-------------+----------------------------------   Adj R-squared   =   -0.1236
       Total |  37.3666667        29  1.28850575   Root MSE        =    1.2033

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
          id |
          2  |         .1   .5381126     0.19   0.854    -1.013171    1.213171
          3  |        -.3   .5381126    -0.56   0.583    -1.413171    .8131708
             |
          jd |
          2  |   9.45e-16   .6947004     0.00   1.000    -1.437097    1.437097
          3  |   .3333333   .6947004     0.48   0.636    -1.103764    1.770431
          4  |   9.44e-16   .6947004     0.00   1.000    -1.437097    1.437097
          5  |  -.6666667   .6947004    -0.96   0.347    -2.103764    .7704307
             |
       _cons |   3.566667   .5812281     6.14   0.000     2.364305    4.769029
------------------------------------------------------------------------------

predict fitted_reg,xb

sum fitted_reg if id==1&jd==1

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
  fitted_reg |          2    3.566667           0   3.566667   3.566667

The average fitted value of y when id==1 and jd==1 from -xtreg y i.jd, fe- is 3.5, the same as the average true value. Does it mean that the _cons from xtreg is not only the average fitted value but also the average true value of y when id==1 and jd==1?

Code:

xtset id
       panel variable:  id (balanced)
xtreg y i.jd,fe

Fixed-effects (within) regression               Number of obs     =         30
Group variable: id                              Number of groups  =          3

R-sq:                                           Obs per group:
     within  = 0.0877                                         min =         10
     between = 0.0000                                         avg =       10.0
     overall = 0.0856                                         max =         10

                                                F(4,23)           =       0.55
corr(u_i, Xb)  = 0.0000                         Prob > F          =     0.6991

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
          jd |
          2  |  -4.28e-16   .6947004    -0.00   1.000    -1.437097    1.437097
          3  |   .3333333   .6947004     0.48   0.636    -1.103764    1.770431
          4  |  -4.44e-16   .6947004    -0.00   1.000    -1.437097    1.437097
          5  |  -.6666667   .6947004    -0.96   0.347    -2.103764    .7704307
             |
       _cons |        3.5   .4912274     7.13   0.000     2.483819    4.516181
-------------+----------------------------------------------------------------
     sigma_u |   .2081666
     sigma_e |  1.2032565
         rho |  .02906016   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(2, 23) = 0.30                       Prob > F = 0.7442

predict fitted_xtreg,xb

sum fitted_xtreg if id==1&jd==1

Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
fitted_xtreg |          2         3.5           0        3.5        3.5

Thank you.

Comment

Carlo Lazzaro

Join Date: Apr 2014
Posts: 17689

11 Apr 2023, 08:16

Ruby:
you're seemingly ignoring the role of residuals:

Code:

. reg y i.id i.jd

      Source |       SS           df       MS      Number of obs   =        30
-------------+----------------------------------   F(6, 23)        =      0.47
       Model |  4.06666667         6  .677777778   Prob > F        =    0.8247
    Residual |        33.3        23  1.44782609   R-squared       =    0.1088
-------------+----------------------------------   Adj R-squared   =   -0.1236
       Total |  37.3666667        29  1.28850575   Root MSE        =    1.2033

------------------------------------------------------------------------------
           y | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
          id |
          2  |         .1   .5381126     0.19   0.854    -1.013171    1.213171
          3  |        -.3   .5381126    -0.56   0.583    -1.413171    .8131708
             |
          jd |
          2  |   9.45e-16   .6947004     0.00   1.000    -1.437097    1.437097
          3  |   .3333333   .6947004     0.48   0.636    -1.103764    1.770431
          4  |   9.44e-16   .6947004     0.00   1.000    -1.437097    1.437097
          5  |  -.6666667   .6947004    -0.96   0.347    -2.103764    .7704307
             |
       _cons |   3.566667   .5812281     6.14   0.000     2.364305    4.769029
------------------------------------------------------------------------------

. predict fitted, xb

. predict residual, res

. list id jd y fitted residual if id==1&jd==1

     +------------------------------------+
     | id   jd   y     fitted    residual |
     |------------------------------------|
  1. |  1    1   3   3.566667   -.5666667 |
  2. |  1    1   4   3.566667    .4333333 |
     +------------------------------------+

. xtset id

Panel variable: id (balanced)

. 
.        panel variable:  id (balanced)
command panel is unrecognized
r(199);

. 
. xtreg y i.jd,fe

Fixed-effects (within) regression               Number of obs     =         30
Group variable: id                              Number of groups  =          3

R-squared:                                      Obs per group:
     Within  = 0.0877                                         min =         10
     Between = 0.0000                                         avg =       10.0
     Overall = 0.0856                                         max =         10

                                                F(4,23)           =       0.55
corr(u_i, Xb) = 0.0000                          Prob > F          =     0.6991

------------------------------------------------------------------------------
           y | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
          jd |
          2  |  -4.28e-16   .6947004    -0.00   1.000    -1.437097    1.437097
          3  |   .3333333   .6947004     0.48   0.636    -1.103764    1.770431
          4  |  -4.44e-16   .6947004    -0.00   1.000    -1.437097    1.437097
          5  |  -.6666667   .6947004    -0.96   0.347    -2.103764    .7704307
             |
       _cons |        3.5   .4912274     7.13   0.000     2.483819    4.516181
-------------+----------------------------------------------------------------
     sigma_u |   .2081666
     sigma_e |  1.2032565
         rho |  .02906016   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(2, 23) = 0.30                       Prob > F = 0.7442

. predict fitted_xtreg, xb

. predict u_error_xtreg, u

. predict e_error_xtreg, e


. list id jd y fitted_xtreg u_error_xtreg e_error_xtreg if id==1&jd==1

     +-----------------------------------------------+
     | id   jd   y   fitted~g   u_erro~g   e_error~g |
     |-----------------------------------------------|
  1. |  1    1   3        3.5   .0666667   -.5666667 |
  2. |  1    1   4        3.5   .0666667    .4333333 |
     +-----------------------------------------------+

.

If you take a look at observed, fitted and residuals, after a bit of algebra, the results of -regress- with -i.id-among the predictors, and -xtreg,fe- overlap (as expected).

Kind regards,
Carlo
(Stata 19.0)

Announcement