Pooled OLS with Clustered Robust S.E vs Random Effects (Robust)?

[Thomas Kurian]

I am working on a project with a time invariant independent variable (childhood nutrition at age 5). I have so far been working with a pooled OLS model with clustered robust standard errors at the panel (Household) level as my dataset is panel in nature-with clustered S.Es to correct for serial autocorrelation of the errors within a cluster (household) across time). I have 2000 clusters with 5 observations per cluster.

I cannot use a fixed effects due to the time invariant nature of my main independent variable. I was hoping to compare my results with pooled OLS with clustered robust standard errors with a random effects model. I was having some trouble understanding the relative benefits of doing so as from reading, my understanding is both models aim to correct for serial correlation and heteroscedasticity in errors across time. I have 2 specific questions:

1) What are the benefits of Random Effects over Pooled OLS with clustered robust standard errors ? Do they correct for different phenomena? (Any resources i.e. links you might be able to provide would be useful here
2) In STATA, what does the robust command do with random effects? Does not using robust with random effects correct for serial correlation anyway?

[Carlo Lazzaro]

Thomas:
without sharing what you typed and what Stata gave you back, it's difficult to get really helpful replies.
That said, if you have a T>N panel dataset, you may want to consider -xtgls- and -xtregar-.
As far as your questions are concerned:
1) I would start from -regress-, -xtreg-, -xtgls- and -xtregar- entries (and related references) in Stata .pdf manual;
2) Under regress-, the -robust- option takes heteroskedasticity only into account; under -xtreg- (with -fe- or -re- specification) both -robust- and -cluster- option do the very same job (ie, cluster-robust standard error, that takes heteroskedasticity and/or autocorrelation into account). -xt- commands for T>N panel datasets (like the one you're dealing with) have more sophisticated options to model the error term.

[Thomas Kurian]

Carlo- thank you for your reply. I will try to post my code in a separate reply. Prior to this:

Could you clarify what T>N refers to in this instance? To clarify, I have a panel survey with 5 rounds , with 2000 households. As I am clustering at the household (panel) level, does that mean that N in this instance is 2000? Or is that T?

Secondly, under -regress- with the add on (cluster panel) I understand this takes into account both heteroskedasticity and serial correlation across clusters (as opposed to the basic robust command, which as you mentioned only takes into account the former. As it appears the robust command in xtreg, re, also controls for both of these elements , is there any (general rule) regarding the relative benefits of either approach (pooled OLS with robust clustered standard errors at the panel level vs Random Effects with robust?) Or should I just run both approaches and compare them?

[Carlo Lazzaro]

Thomas:
you're correct. You have a N>T panel datasets; hence, among the -xt- commands, if your regressand is continuous, -xtreg- is the way to go.
Moreover, is right to cluster your standard error at the -panelid-.
If you have heteroskedasticity and serial correlation after -regress-, it's wise to invoke the -cluster- option; -robust- works fine if you have heteroskedasticity ony.
Under -xtreg- both options do the very same job.
Whenever I have a panel dataset with a continuous regressand, I consider .xtreg- only, as it can provide me with more details as far as the results are concerned (ie, -sigma_e- and -sigma_u- statistics).
Eventually, -hausman- (if you have default standard errors) or the community-contributed programm -xtoverid- can point you to -fe- or -re- specification.

[Thomas Kurian]

Thanks Carlo, that's very helpful. The one last question I have is that as my regressand is time invariant but continuous, I understand it would not be possible to do the Hausman test of fe vs re ?

[Carlo Lazzaro]

Thomas:
if your continuous regressand is time-invariant there's no way to make -xtreg,fe- works.

[Thomas Kurian]

Thanks Carlo. Actually one other question I had is with panel random effects, what is the difference in interpretation between not using robust and using robust (i.e. xtreg re Vs xtreg re robust)? Without adding a robust , does random effects not already correct for serial correlation ?

[Carlo Lazzaro]

Thomas:
without robust or cluster (ie, cluster-robust) standard error, -xtreg- does not take heteroskedasticity and/or autocorrelation of the systematic error into account.

[Thomas Kurian]

Should results using Pooled OLS with clustered standard errors (reg vce(cluster panelid) at the panel level differ from a random effects model with robust (xtreg re robust)? I can post my data which seems to be showing this is the case

[Carlo Lazzaro]

Thomas:
this is expected. As you can see from the following toy-example, the pooled OLS point estimates overlap those from -xtreg,fe- (but not those from -xtreg,re-) as fra as the shared coefficients are concerned:

. use "https://www.stata-press.com/data/r16/nlswork.dta"
(National Longitudinal Survey.  Young Women 14-26 years of age in 1968)

. reg ln_wage i.idcode i.year c.age##c.age if idcode<=4, vce(cluster idcode)

Linear regression                               Number of obs     =         50
                                                F(2, 3)           =          .
                                                Prob > F          =          .
                                                R-squared         =     0.6590
                                                Root MSE          =     .28764

                                 (Std. Err. adjusted for 4 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      idcode |
          2  |   -.394374   .0296125   -13.32   0.001    -.4886143   -.3001337
          3  |   .1010589   1.385597     0.07   0.946     -4.30853    4.510648
          4  |   .5522662   1.411066     0.39   0.722    -3.938377    5.042909
             |
        year |
         69  |    .224994   .2362011     0.95   0.411    -.5267035    .9766914
         70  |   .1649135   .4084011     0.40   0.713    -1.134801    1.464628
         71  |   .1712431   .5615442     0.30   0.780    -1.615841    1.958327
         72  |   .3136266   .8258606     0.38   0.729     -2.31463    2.941883
         73  |   .4386164   1.160727     0.38   0.731    -3.255333    4.132566
         75  |    .575746   1.642352     0.35   0.749     -4.65095    5.802442
         77  |    .651904   2.091338     0.31   0.776    -6.003666    7.307474
         78  |   .9314592   2.431102     0.38   0.727    -6.805391     8.66831
         80  |   .9700889   3.035563     0.32   0.770    -8.690428    10.63061
         82  |   1.063147   3.462289     0.31   0.779    -9.955403     12.0817
         83  |   1.379563   3.652919     0.38   0.731    -10.24566    13.00478
         85  |   1.854813   3.994786     0.46   0.674    -10.85838      14.568
         87  |   2.153058   4.439545     0.48   0.661    -11.97556    16.28167
         88  |   2.571633   4.744555     0.54   0.625    -12.52766    17.67092
             |
         age |   .2364322   .2603253     0.91   0.431     -.592039    1.064903
             |
 c.age#c.age |  -.0056102   .0021351    -2.63   0.078    -.0124052    .0011848
             |
       _cons |  -1.093667   4.089099    -0.27   0.806    -14.10701    11.91967
------------------------------------------------------------------------------


. xtreg ln_wage i.year c.age##c.age if idcode<=4, fe vce(cluster idcode)

Fixed-effects (within) regression               Number of obs     =         50
Group variable: idcode                          Number of groups  =          4

R-sq:                                           Obs per group:
     within  = 0.5342                                         min =         11
     between = 0.0151                                         avg =       12.5
     overall = 0.2227                                         max =         15

                                                F(3,3)            =          .
corr(u_i, Xb)  = -0.6249                        Prob > F          =          .

                                 (Std. Err. adjusted for 4 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        year |
         69  |    .224994   .2252089     1.00   0.391    -.4917214    .9417093
         70  |   .1649135   .3893952     0.42   0.700    -1.074316    1.404143
         71  |   .1712431   .5354114     0.32   0.770    -1.532675    1.875161
         72  |   .3136266   .7874271     0.40   0.717    -2.192318    2.819571
         73  |   .4386164   1.106709     0.40   0.718    -3.083427    3.960659
         75  |    .575746   1.565921     0.37   0.738    -4.407713    5.559205
         77  |    .651904   1.994012     0.33   0.765    -5.693933    6.997741
         78  |   .9314592   2.317964     0.40   0.715    -6.445338    8.308257
         80  |   .9700889   2.894296     0.34   0.760    -8.240852    10.18103
         82  |   1.063147   3.301163     0.32   0.769    -9.442628    11.56892
         83  |   1.379563   3.482922     0.40   0.719    -9.704648    12.46377
         85  |   1.854813   3.808879     0.49   0.660    -10.26674    13.97637
         87  |   2.153058    4.23294     0.51   0.646    -11.31805    15.62416
         88  |   2.571633   4.523756     0.57   0.609    -11.82498    16.96824
             |
         age |   .2364322   .2482104     0.95   0.411    -.5534841    1.026349
             |
 c.age#c.age |  -.0056102   .0020358    -2.76   0.070     -.012089    .0008685
             |
       _cons |  -1.036501   4.526557    -0.23   0.834    -15.44202    13.36902
-------------+----------------------------------------------------------------
     sigma_u |  .38900632
     sigma_e |  .28764391
         rho |  .64651254   (fraction of variance due to u_i)
------------------------------------------------------------------------------

.

. xtreg ln_wage i.year c.age##c.age if idcode<=4, vce(cluster idcode)

Random-effects GLS regression                   Number of obs     =         50
Group variable: idcode                          Number of groups  =          4

R-sq:                                           Obs per group:
     within  = 0.5077                                         min =         11
     between = 0.0107                                         avg =       12.5
     overall = 0.3789                                         max =         15

                                                Wald chi2(3)      =          .
corr(u_i, X)   = 0 (assumed)                    Prob > chi2       =          .

                                 (Std. Err. adjusted for 4 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        year |
         69  |   .1486743   .0549624     2.71   0.007     .0409499    .2563986
         70  |    .277369   .2887149     0.96   0.337    -.2885019    .8432399
         71  |   .1301287   .2621698     0.50   0.620    -.3837146     .643972
         72  |   .1961925   .2197171     0.89   0.372     -.234445    .6268301
         73  |   .2437975   .1144552     2.13   0.033     .0194694    .4681255
         75  |   .2229619   .0470414     4.74   0.000     .1307625    .3151613
         77  |   .0565625   .2272149     0.25   0.803    -.3887705    .5018956
         78  |   .2544722   .4275434     0.60   0.552    -.5834974    1.092442
         80  |   .2037513    .589187     0.35   0.729    -.9510341    1.358537
         82  |   .0580069   .6546599     0.09   0.929    -1.225103    1.341117
         83  |   .3523115   .6776646     0.52   0.603    -.9758868     1.68051
         85  |   .6482934   .7762536     0.84   0.404    -.8731357    2.169722
         87  |   .7630092   .8329341     0.92   0.360    -.8695117     2.39553
         88  |     1.0182    .886308     1.15   0.251    -.7189321    2.755331
             |
         age |   .2887857    .133158     2.17   0.030     .0278008    .5497707
             |
 c.age#c.age |  -.0050776   .0021492    -2.36   0.018      -.00929   -.0008652
             |
       _cons |  -2.402156   1.992439    -1.21   0.228    -6.307264    1.502953
-------------+----------------------------------------------------------------
     sigma_u |          0
     sigma_e |  .28764391
         rho |          0   (fraction of variance due to u_i)
------------------------------------------------------------------------------