Constant in Fixed Effect

[keiyou Wong]

I created a dataset using following codes

clear
set obs  100
gen id = _n
expand 10
bysort id : gen year = 2000 + _n -1
xtset id year

gen z = rnormal()
gen u = rnormal()
gen x = 3 * z + u +rnormal()
gen y = 6 + 3*x  + u

And I regressed the model with "reghdfe" command according to the following code:

reghdfe y x z, a(i.id i.year)  // model1
est sto r1

reghdfe y x z, a(i.id i.year) noconstant  // model2
est sto r2

I found the estimate results are completely the same whether I added constant or not.

while I regressed the model with "reg" command according to:

reg y x z i.id i.year  // model3
est sto reg1

reg y x z i.id i.year ,noconstant  // model4
est sto reg2

the estimate results are different.

I plot the coefficient like this:



I am so confused about the difference where only regression command are changed. I compared the result 1 with result 3, and found the estimation are the same. So theoretically result 2 and 4 should be the same, but it's NOT.

I was wondering what exactly the difference between "reghdfe" and "reg" and "noconstant" option is meaningful in reghdfe ?

Thanks for Answering
Best regards!

[Andrew Musau]

You cannot separate the constant from the fixed effects, so the regress results with option -nocons- will only affect estimates of the unit dummies if you allow all dummies to be included (or otherwise stated, will not affect estimates of coefficients on the time-varying variables).

reg y x z i.id i.year
reg y x z ibn.id i.year ,noconstant

The default with factor variables is to omit the lowest category. So option -nocons- with no changes to the default behavior estimates a different model where more than the required indicators are omitted. In any case, the estimate of the constant term is meaningless in FE models. More discussion of this here: https://www.stata.com/support/faqs/s...effects-model/.

[keiyou Wong]

You cannot separate the constant from the fixed effects, so the regress results with option -nocons- will only affect estimates of the unit dummies if you allow all dummies to be included (or otherwise stated, will not affect estimates of coefficients on the time-varying variables).

reg y x z i.id i.year
reg y x z ibn.id i.year ,noconstant

The default with factor variables is to omit the lowest category. So option -nocons- with no changes to the default behavior estimates a different model where more than the required indicators are omitted. In any case, the estimate of the constant term is meaningless in FE models. More discussion of this here: https://www.stata.com/support/faqs/s...effects-model/.

Oh I get it !

 Source |       SS           df       MS      Number of obs   =     1,000
-------------+----------------------------------   F(110, 889)     =   1906.00
       Model |  103970.988       110  945.190801   Prob > F        =    0.0000
    Residual |   440.85695       889  .495902081   R-squared       =    0.9958
-------------+----------------------------------   Adj R-squared   =    0.9953
       Total |  104411.845       999  104.516361   Root MSE        =     .7042

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |    3.47117   .0166212   208.84   0.000     3.438549    3.503792
           z |   -1.44663   .0558097   -25.92   0.000    -1.556164   -1.337096
             |
          id |
          2  |  -.1935297   .3149917    -0.61   0.539    -.8117438    .4246843
          3  |  -.1796347   .3153945    -0.57   0.569    -.7986393    .4393698
          4  |  -.7138173   .3150467    -2.27   0.024    -1.332139   -.0954952
          5  |  -.3636884   .3150227    -1.15   0.249    -.9819632    .2545864
          6  |  -.5128189   .3150383    -1.63   0.104    -1.131125    .1054867
          7  |  -.4384328   .3150544    -1.39   0.164     -1.05677    .1799044
          8  |  -.1211999   .3149669    -0.38   0.700    -.7393652    .4969654
          9  |   .1020693   .3151303     0.32   0.746    -.5164168    .7205554
         10  |  -.1151052   .3150153    -0.37   0.715    -.7333657    .5031552

I checked the previous regress results with option -nocons- and found it did omit the first id. The results based on your code are consistent.

Thanks for answering!