Fixed effects panel regression with robust standard error- which code to use?

[Titir Bhattacharya]

Hi,

I'm using the following code to perform fixed effects panel regression

areg y x1 x2 i.year,a(HHID)

I want to obtain robust standard errors and tried

areg y x1 x2 i.year,a(HHID) robust

But on looking further, I found another option

vce(robust)

and now I'm confused as to what is the difference between the two.

I would greatly appreciate it if someone could explain the difference between these two options and which one would be appropriate for my purpose.

Thanks,

[Carlo Lazzaro]

Titir:
there's no difference, as you can easily test yourself:

. use http://www.stata-press.com/data/r16/auto2.dta
. areg mpg weight gear_ratio, absorb(rep78) robust

Linear regression, absorbing indicators         Number of obs     =         69
Absorbed variable: rep78                        No. of categories =          5
                                                F(   2,     62)   =      58.56
                                                Prob > F          =     0.0000
                                                R-squared         =     0.6734
                                                Adj R-squared     =     0.6418
                                                Root MSE          =     3.5109

------------------------------------------------------------------------------
             |               Robust
         mpg |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      weight |  -.0051031   .0007151    -7.14   0.000    -.0065326   -.0036736
  gear_ratio |    .901478   1.502386     0.60   0.551    -2.101748    3.904704
       _cons |   34.05889   6.167363     5.52   0.000     21.73051    46.38727
------------------------------------------------------------------------------

. areg mpg weight gear_ratio, absorb(rep78) vce(robust)

Linear regression, absorbing indicators         Number of obs     =         69
Absorbed variable: rep78                        No. of categories =          5
                                                F(   2,     62)   =      58.56
                                                Prob > F          =     0.0000
                                                R-squared         =     0.6734
                                                Adj R-squared     =     0.6418
                                                Root MSE          =     3.5109

------------------------------------------------------------------------------
             |               Robust
         mpg |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      weight |  -.0051031   .0007151    -7.14   0.000    -.0065326   -.0036736
  gear_ratio |    .901478   1.502386     0.60   0.551    -2.101748    3.904704
       _cons |   34.05889   6.167363     5.52   0.000     21.73051    46.38727
------------------------------------------------------------------------------

[Titir Bhattacharya]

Titir:
there's no difference, as you can easily test yourself:

. use http://www.stata-press.com/data/r16/auto2.dta
. areg mpg weight gear_ratio, absorb(rep78) robust

Linear regression, absorbing indicators Number of obs = 69
Absorbed variable: rep78 No. of categories = 5
F( 2, 62) = 58.56
Prob > F = 0.0000
R-squared = 0.6734
Adj R-squared = 0.6418
Root MSE = 3.5109

------------------------------------------------------------------------------
| Robust
mpg | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
weight | -.0051031 .0007151 -7.14 0.000 -.0065326 -.0036736
gear_ratio | .901478 1.502386 0.60 0.551 -2.101748 3.904704
_cons | 34.05889 6.167363 5.52 0.000 21.73051 46.38727
------------------------------------------------------------------------------

. areg mpg weight gear_ratio, absorb(rep78) vce(robust)

Linear regression, absorbing indicators Number of obs = 69
Absorbed variable: rep78 No. of categories = 5
F( 2, 62) = 58.56
Prob > F = 0.0000
R-squared = 0.6734
Adj R-squared = 0.6418
Root MSE = 3.5109

------------------------------------------------------------------------------
| Robust
mpg | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
weight | -.0051031 .0007151 -7.14 0.000 -.0065326 -.0036736
gear_ratio | .901478 1.502386 0.60 0.551 -2.101748 3.904704
_cons | 34.05889 6.167363 5.52 0.000 21.73051 46.38727
------------------------------------------------------------------------------

thanks Carlo for your reply.

[Jeff Wooldridge]

You should be clustering your standard errors at the HHID level. Serial correlation is prevalent in panel data applications. Using "robust" with areg isn't doing that. In fact, if T is not pretty large, those standard errors aren't correct even for heteroskedasticity.

xtset HHID year
xtreg y x1 ... xK i.year, fe vce(cluster HHID)

[Titir Bhattacharya]

You should be clustering your standard errors at the HHID level. Serial correlation is prevalent in panel data applications. Using "robust" with areg isn't doing that. In fact, if T is not pretty large, those standard errors aren't correct even for heteroskedasticity.

xtset HHID year
xtreg y x1 ... xK i.year, fe vce(cluster HHID)

Thank you Prof. Wooldridge. I'll do that.

[Joro Kolev]

Professor Wooldridge might be alluding to this paper
Stock, James H., and Mark W. Watson. "Heteroskedasticity‐robust standard errors for fixed effects panel data regression." Econometrica 76, no. 1 (2008): 155-174
which shows that robust only standard errors are not consistent in fixed effects panel data applications.

Because of this, and as a matter of Stata syntax, once you switch to -xtreg- as Professor Wooldridge showed above, you do not even have the choice. Stata takes ownership of the whole situation, and computes cluster-robust standard errors (with clustering done at the panel level) whether you specify the robust only option, or the robust cluster(panel id) option.

Here it is:

. sysuse auto, clear
(1978 Automobile Data)

. xtset rep
       panel variable:  rep78 (unbalanced)

. xtreg price mpg headroom, fe robust

Fixed-effects (within) regression               Number of obs     =         69
Group variable: rep78                           Number of groups  =          5

R-sq:                                           Obs per group:
     within  = 0.2549                                         min =          2
     between = 0.0274                                         avg =       13.8
     overall = 0.2139                                         max =         30

                                                F(2,4)            =       4.02
corr(u_i, Xb)  = -0.4159                        Prob > F          =     0.1105

                                  (Std. Err. adjusted for 5 clusters in rep78)
------------------------------------------------------------------------------
             |               Robust
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |  -297.1542   104.8356    -2.83   0.047    -588.2246   -6.083836
    headroom |  -334.6746   249.9078    -1.34   0.252     -1028.53    359.1808
       _cons |   13476.44   2675.177     5.04   0.007     6048.955    20903.92
-------------+----------------------------------------------------------------
     sigma_u |  1266.7868
     sigma_e |  2613.7617
         rho |  .19021503   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. xtreg price mpg headroom, fe robust cluster(rep)

Fixed-effects (within) regression               Number of obs     =         69
Group variable: rep78                           Number of groups  =          5

R-sq:                                           Obs per group:
     within  = 0.2549                                         min =          2
     between = 0.0274                                         avg =       13.8
     overall = 0.2139                                         max =         30

                                                F(2,4)            =       4.02
corr(u_i, Xb)  = -0.4159                        Prob > F          =     0.1105

                                  (Std. Err. adjusted for 5 clusters in rep78)
------------------------------------------------------------------------------
             |               Robust
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |  -297.1542   104.8356    -2.83   0.047    -588.2246   -6.083836
    headroom |  -334.6746   249.9078    -1.34   0.252     -1028.53    359.1808
       _cons |   13476.44   2675.177     5.04   0.007     6048.955    20903.92
-------------+----------------------------------------------------------------
     sigma_u |  1266.7868
     sigma_e |  2613.7617
         rho |  .19021503   (fraction of variance due to u_i)
------------------------------------------------------------------------------

.

as you see the standard errors are numerically equivalent whether I specify robust only, or robust cluster(panel id). This is because Stata itself automatically switches to cluster robust standard errors once you declare your data as a panel.

[Titir Bhattacharya]

Professor Wooldridge might be alluding to this paper
Stock, James H., and Mark W. Watson. "Heteroskedasticity‐robust standard errors for fixed effects panel data regression." Econometrica 76, no. 1 (2008): 155-174
which shows that robust only standard errors are not consistent in fixed effects panel data applications.

Because of this, and as a matter of Stata syntax, once you switch to -xtreg- as Professor Wooldridge showed above, you do not even have the choice. Stata takes ownership of the whole situation, and computes cluster-robust standard errors (with clustering done at the panel level) whether you specify the robust only option, or the robust cluster(panel id) option.

Here it is:

. sysuse auto, clear
(1978 Automobile Data)

. xtset rep
panel variable: rep78 (unbalanced)

. xtreg price mpg headroom, fe robust

Fixed-effects (within) regression Number of obs = 69
Group variable: rep78 Number of groups = 5

R-sq: Obs per group:
within = 0.2549 min = 2
between = 0.0274 avg = 13.8
overall = 0.2139 max = 30

F(2,4) = 4.02
corr(u_i, Xb) = -0.4159 Prob > F = 0.1105

(Std. Err. adjusted for 5 clusters in rep78)
------------------------------------------------------------------------------
| Robust
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
mpg | -297.1542 104.8356 -2.83 0.047 -588.2246 -6.083836
headroom | -334.6746 249.9078 -1.34 0.252 -1028.53 359.1808
_cons | 13476.44 2675.177 5.04 0.007 6048.955 20903.92
-------------+----------------------------------------------------------------
sigma_u | 1266.7868
sigma_e | 2613.7617
rho | .19021503 (fraction of variance due to u_i)
------------------------------------------------------------------------------

. xtreg price mpg headroom, fe robust cluster(rep)

Fixed-effects (within) regression Number of obs = 69
Group variable: rep78 Number of groups = 5

R-sq: Obs per group:
within = 0.2549 min = 2
between = 0.0274 avg = 13.8
overall = 0.2139 max = 30

F(2,4) = 4.02
corr(u_i, Xb) = -0.4159 Prob > F = 0.1105

(Std. Err. adjusted for 5 clusters in rep78)
------------------------------------------------------------------------------
| Robust
price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
mpg | -297.1542 104.8356 -2.83 0.047 -588.2246 -6.083836
headroom | -334.6746 249.9078 -1.34 0.252 -1028.53 359.1808
_cons | 13476.44 2675.177 5.04 0.007 6048.955 20903.92
-------------+----------------------------------------------------------------
sigma_u | 1266.7868
sigma_e | 2613.7617
rho | .19021503 (fraction of variance due to u_i)
------------------------------------------------------------------------------

.

as you see the standard errors are numerically equivalent whether I specify robust only, or robust cluster(panel id). This is because Stata itself automatically switches to cluster robust standard errors once you declare your data as a panel.

Thank you Joro for your reply