Need Help for Panel Data Regression

[Hakan Self]

Hello dear Community,
I would like to do a regression for my research with panel data. More specifically i have firms revenue as my dependent variable and observations of many firms in a time period of 5 years.
As far as i know i have to -xtset- my data.

xtset company_id year

now i do not know which equation i should use, -xtreg- or -reghdfe-. I want to consider for company fixed effects. Also i am not sure if clustering would be useful in my case.

reghdfe revenue x1 x2, absorb(company_id)

also what would be the equivalent with -xtreg- of this equiation ?

[Carlo Lazzaro]

Hakan:
welcome to this forum.
The equivalent -xtreg- code is:

xtset company_id year
xtreg revenue x1 x2, fe

The -vce(cluster company_id)- standard errors make sense in case of heteroskedasticity and/or autocorrelation of the epsilon..

[Jared Greathouse]

I spoke about clustering in a post yesterday where I linked a paper on cluster SEs, so I'll do the same here.

[Carlo Lazzaro]

Hakan:
as an aside to Jared's helpful reference, another towering working paper on the very same topic is

http://cameron.econ.ucdavis.edu/research/Cameron_Miller_JHR_2015_February.pdf

[Jared Greathouse]

This paper was published in JHR, no? It isn't still a working paper is it? Carlo Lazzaro

[Carlo Lazzaro]

Jared:
you're right.
The working paper was published on JHR in 2015 (I cannot share the full reference at the moment).

[Carlo Lazzaro]

Cameron and Trivedi's Microeconometrics textbook (Cambridge,2005) also covers this topic in different chapters.

[Hakan Self]

Thanks for your answers. So to make things clear are these equivalent ?

xtset company_id year
xtreg revenue x1 x2, fe vce(cluster company_id)

xtset company_id year
reghdfe revenue x1 x2, absorb(company_id) vce(cluster company_id)

if not, what how would the correct -reghdfe- code look like ?

[Carlo Lazzaro]

Hakan:
they are almost equivalent (as you can see from the following toy-example):

use "https://www.stata-press.com/data/r17/nlswork.dta"
. xtreg ln_wage i.year c.age##c.age, fe vce(cluster idcode)

Fixed-effects (within) regression               Number of obs     =     28,510
Group variable: idcode                          Number of groups  =      4,710

R-squared:                                      Obs per group:
     Within  = 0.1162                                         min =          1
     Between = 0.1078                                         avg =        6.1
     Overall = 0.0932                                         max =         15

                                                F(16,4709)        =      79.11
corr(u_i, Xb) = 0.0613                          Prob > F          =     0.0000

                             (Std. err. adjusted for 4,710 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
        year |
         69  |   .0647054   .0155249     4.17   0.000     .0342693    .0951415
         70  |   .0284423   .0264639     1.07   0.283    -.0234395     .080324
         71  |   .0579959   .0384111     1.51   0.131    -.0173078    .1332996
         72  |   .0510671   .0502675     1.02   0.310    -.0474808     .149615
         73  |   .0424104   .0624924     0.68   0.497    -.0801038    .1649247
         75  |   .0151376    .086228     0.18   0.861    -.1539096    .1841848
         77  |   .0340933   .1106841     0.31   0.758    -.1828994     .251086
         78  |   .0537334   .1232232     0.44   0.663    -.1878417    .2953084
         80  |   .0369475   .1473725     0.25   0.802    -.2519716    .3258667
         82  |   .0391687   .1715621     0.23   0.819    -.2971733    .3755108
         83  |    .058766   .1836086     0.32   0.749    -.3011928    .4187249
         85  |   .1042758   .2080199     0.50   0.616    -.3035406    .5120922
         87  |   .1242272   .2327328     0.53   0.594    -.3320379    .5804922
         88  |   .1904977   .2486083     0.77   0.444    -.2968909    .6778863
             |
         age |   .0728746    .013687     5.32   0.000     .0460416    .0997075
             |
 c.age#c.age |  -.0010113   .0001076    -9.40   0.000    -.0012224   -.0008003
             |
       _cons |   .3937532   .2469015     1.59   0.111    -.0902893    .8777957
-------------+----------------------------------------------------------------
     sigma_u |  .40275174
     sigma_e |  .30127563
         rho |  .64120306   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. reghdfe ln_wage i.year c.age##c.age, abs(idcode) vce(cluster idcode)
(dropped 551 singleton observations)
(MWFE estimator converged in 1 iterations)

HDFE Linear regression                            Number of obs   =     27,959
Absorbing 1 HDFE group                            F(  16,   4158) =      79.11
Statistics robust to heteroskedasticity           Prob > F        =     0.0000
                                                  R-squared       =     0.6593
                                                  Adj R-squared   =     0.5995
                                                  Within R-sq.    =     0.1162
Number of clusters (idcode)  =      4,159         Root MSE        =     0.3013

                             (Std. err. adjusted for 4,159 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
        year |
         69  |   .0647054   .0155252     4.17   0.000     .0342677    .0951432
         70  |   .0284423   .0264645     1.07   0.283    -.0234422    .0803268
         71  |   .0579959   .0384118     1.51   0.131    -.0173118    .1333037
         72  |   .0510671   .0502685     1.02   0.310    -.0474861    .1496203
         73  |   .0424104   .0624936     0.68   0.497    -.0801104    .1649313
         75  |   .0151376   .0862297     0.18   0.861    -.1539187    .1841939
         77  |   .0340933   .1106863     0.31   0.758    -.1829111    .2510976
         78  |   .0537334   .1232256     0.44   0.663    -.1878546    .2953214
         80  |   .0369475   .1473754     0.25   0.802    -.2519871    .3258822
         82  |   .0391687   .1715655     0.23   0.819    -.2971914    .3755288
         83  |    .058766   .1836122     0.32   0.749    -.3012121    .4187442
         85  |   .1042758    .208024     0.50   0.616    -.3035625     .512114
         87  |   .1242272   .2327373     0.53   0.594    -.3320624    .5805167
         88  |   .1904977   .2486132     0.77   0.444    -.2969171    .6779125
             |
         age |   .0728746   .0136873     5.32   0.000     .0460402    .0997089
             |
 c.age#c.age |  -.0010113   .0001076    -9.39   0.000    -.0012224   -.0008003
             |
       _cons |   .3956251   .2469216     1.60   0.109    -.0884733    .8797234
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
      idcode |      4159        4159           0    *|
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation

.

That said, in this case I would stick with -xtreg,fe-, as you have one fixed effect only.

[Hakan Self]

yes. but you did it with -i.year-. I just want to know that would be correct the correct -reghdfe- for my specific case ?

[Carlo Lazzaro]

Hakan:
yes, it is correct, but possibly misspecified.

[Hakan Self]

what do you mean with misspecified ? How should i make it then ?

[Carlo Lazzaro]

Hakan:
in the following toy-example:

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

. xtreg ln_wage c.age##c.age, fe vce(cluster idcode)

Fixed-effects (within) regression               Number of obs     =     28,510
Group variable: idcode                          Number of groups  =      4,710

R-squared:                                      Obs per group:
     Within  = 0.1087                                         min =          1
     Between = 0.1006                                         avg =        6.1
     Overall = 0.0865                                         max =         15

                                                F(2,4709)         =     507.42
corr(u_i, Xb) = 0.0440                          Prob > F          =     0.0000

                             (Std. err. adjusted for 4,710 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         age |   .0539076    .004307    12.52   0.000     .0454638    .0623515
             |
 c.age#c.age |  -.0005973    .000072    -8.30   0.000    -.0007384   -.0004562
             |
       _cons |    .639913   .0624195    10.25   0.000     .5175415    .7622845
-------------+----------------------------------------------------------------
     sigma_u |   .4039153
     sigma_e |  .30245467
         rho |  .64073314   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. predict fitted, xb
(24 missing values generated)

. g sq_fitted=fitted^2
(24 missing values generated)

. xtreg ln_wage fitted sq_fitted , fe vce(cluster idcode)

Fixed-effects (within) regression               Number of obs     =     28,510
Group variable: idcode                          Number of groups  =      4,710

R-squared:                                      Obs per group:
     Within  = 0.1092                                         min =          1
     Between = 0.1033                                         avg =        6.1
     Overall = 0.0881                                         max =         15

                                                F(2,4709)         =     523.09
corr(u_i, Xb) = 0.0467                          Prob > F          =     0.0000

                             (Std. err. adjusted for 4,710 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
      fitted |   2.569185   .7085064     3.63   0.000     1.180181    3.958189
   sq_fitted |    -.47432   .2153021    -2.20   0.028    -.8964128   -.0522272
       _cons |  -1.290258    .580562    -2.22   0.026    -2.428431   -.1520844
-------------+----------------------------------------------------------------
     sigma_u |    .403403
     sigma_e |  .30238578
         rho |  .64025357   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. test sq_fitted=0

 ( 1)  sq_fitted = 0

       F(  1,  4709) =    4.85
            Prob > F =    0.0276

. xtreg ln_wage c.age##c.age i.year, fe vce(cluster idcode)

Fixed-effects (within) regression               Number of obs     =     28,510
Group variable: idcode                          Number of groups  =      4,710

R-squared:                                      Obs per group:
     Within  = 0.1162                                         min =          1
     Between = 0.1078                                         avg =        6.1
     Overall = 0.0932                                         max =         15

                                                F(16,4709)        =      79.11
corr(u_i, Xb) = 0.0613                          Prob > F          =     0.0000

                             (Std. err. adjusted for 4,710 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         age |   .0728746    .013687     5.32   0.000     .0460416    .0997075
             |
 c.age#c.age |  -.0010113   .0001076    -9.40   0.000    -.0012224   -.0008003
             |
        year |
         69  |   .0647054   .0155249     4.17   0.000     .0342693    .0951415
         70  |   .0284423   .0264639     1.07   0.283    -.0234395     .080324
         71  |   .0579959   .0384111     1.51   0.131    -.0173078    .1332996
         72  |   .0510671   .0502675     1.02   0.310    -.0474808     .149615
         73  |   .0424104   .0624924     0.68   0.497    -.0801038    .1649247
         75  |   .0151376    .086228     0.18   0.861    -.1539096    .1841848
         77  |   .0340933   .1106841     0.31   0.758    -.1828994     .251086
         78  |   .0537334   .1232232     0.44   0.663    -.1878417    .2953084
         80  |   .0369475   .1473725     0.25   0.802    -.2519716    .3258667
         82  |   .0391687   .1715621     0.23   0.819    -.2971733    .3755108
         83  |    .058766   .1836086     0.32   0.749    -.3011928    .4187249
         85  |   .1042758   .2080199     0.50   0.616    -.3035406    .5120922
         87  |   .1242272   .2327328     0.53   0.594    -.3320379    .5804922
         88  |   .1904977   .2486083     0.77   0.444    -.2968909    .6778863
             |
       _cons |   .3937532   .2469015     1.59   0.111    -.0902893    .8777957
-------------+----------------------------------------------------------------
     sigma_u |  .40275174
     sigma_e |  .30127563
         rho |  .64120306   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. predict fitted2, xb
(24 missing values generated)

. g sq_fitted2=fitted2^2
(24 missing values generated)

. xtreg ln_wage fitted2 sq_fitted2 , fe vce(cluster idcode)

Fixed-effects (within) regression               Number of obs     =     28,510
Group variable: idcode                          Number of groups  =      4,710

R-squared:                                      Obs per group:
     Within  = 0.1164                                         min =          1
     Between = 0.1094                                         avg =        6.1
     Overall = 0.0941                                         max =         15

                                                F(2,4709)         =     586.29
corr(u_i, Xb) = 0.0619                          Prob > F          =     0.0000

                             (Std. err. adjusted for 4,710 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
     fitted2 |   2.012332   .5365254     3.75   0.000     .9604909    3.064172
  sq_fitted2 |  -.3040363   .1616996    -1.88   0.060    -.6210431    .0129706
       _cons |  -.8379964    .443929    -1.89   0.059    -1.708305    .0323122
-------------+----------------------------------------------------------------
     sigma_u |  .40239556
     sigma_e |  .30114591
         rho |  .64099409   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. test sq_fitted=0

 ( 1)  sq_fitted2 = 0

       F(  1,  4709) =    3.54
            Prob > F =    0.0601

.

The first regression model is misspecified (sq_fitted test reaches statistical significance), whereas the second one does not support evidence of misspecification (sq_fitted test does not reach statistical significance).
The only difference btween the two codes rests on the regressor -i.year- plugged in the right-hand side of the regression equation in the second code.