POLS, FE and RE

[Giorgio Carlini]

Hello,
I'm estimating a TFP growth convergence regression type model in a panel with 20 NUTS-2 regions (reg) and N=17, from 1995 to 2012.
I've estimated the model with POLS (clustering by reg), FE (clustering by NUTS-1 geo1 area, i.e. North-west, North East, Centre and South), and RE.
My questions are:
1) I get similar results between POLS and RE. I've read previous posts on this issue, and I guess this is because my dependent variable is in difference. Or is it also due to limited between variability in the data. I've included an xtsum below, however some of the variable show high between variability. Please I'm not clear on this point.
2) The tests below also show that I should go for FE. Is this correct?
Thanks a lot.

These are my POLS FE and RE estimation codes and results:

#

xi: xtreg gtfp_100  lagltfp_rel_lev95    lagdettotoccrfl  lagnetai lagpsii  lagnetae     lagpsie ///
lagrspri_stock_va   lagtotroadskms   tau1995-tau2012  i.geo1 ///
if cod_reg<21 , vce(cluster reg)
i.geo1            _Igeo1_1-4          (_Igeo1_1 for geo1==Centro omitted)
note: tau1995 omitted because of collinearity
note: tau2012 omitted because of collinearity

Random-effects GLS regression                   Number of obs      =       340
Group variable: cod_reg                         Number of groups   =        20

R-sq:  within  = 0.6088                         Obs per group: min =        17
       between = 0.2743                                        avg =      17.0
       overall = 0.6005                                        max =        17

                                                Wald chi2(19)      =         .
corr(u_i, X)   = 0 (assumed)                    Prob > chi2        =         .

                                        (Std. Err. adjusted for 20 clusters in reg)
-----------------------------------------------------------------------------------
                  |               Robust
         gtfp_100 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
------------------+----------------------------------------------------------------
lagltfp_rel_lev95 |    -.04525   .0178226    -2.54   0.011    -.0801817   -.0103184
  lagdettotoccrfl |  -.0940007   .0516523    -1.82   0.069    -.1952374     .007236
         lagnetai |  -.0144664   .0082807    -1.75   0.081    -.0306964    .0017635
          lagpsii |    .011355   .0316344     0.36   0.720    -.0506473    .0733574
         lagnetae |   .0014869    .006694     0.22   0.824    -.0116331    .0146069
          lagpsie |  -.0566569   .0332526    -1.70   0.088    -.1218307    .0085169
lagrspri_stock_va |   .0000626   .0006108     0.10   0.918    -.0011346    .0012598
   lagtotroadskms |   .0053038   .0036947     1.44   0.151    -.0019376    .0125452
          tau1995 |          0  (omitted)
          tau1996 |   .0251443    .003433     7.32   0.000     .0184157    .0318728
          tau1997 |   .0282562   .0043658     6.47   0.000     .0196994     .036813
          tau1998 |   .0249205   .0036223     6.88   0.000      .017821      .03202
          tau1999 |   .0308059   .0043438     7.09   0.000     .0222922    .0393196
          tau2000 |   .0397295   .0053952     7.36   0.000      .029155    .0503039
          tau2001 |   .0232508    .005381     4.32   0.000     .0127043    .0337972
          tau2002 |   .0056954   .0059057     0.96   0.335    -.0058796    .0172705
          tau2003 |   .0090072   .0066685     1.35   0.177    -.0040628    .0220772
          tau2004 |   .0257581    .004193     6.14   0.000       .01754    .0339763
          tau2005 |   .0195458   .0042295     4.62   0.000     .0112561    .0278355
          tau2006 |   .0227237   .0039394     5.77   0.000     .0150026    .0304447
          tau2007 |   .0216878   .0037919     5.72   0.000     .0142557    .0291198
          tau2008 |   .0013876   .0042288     0.33   0.743    -.0069007     .009676
          tau2009 |   -.025776   .0030627    -8.42   0.000    -.0317788   -.0197732
          tau2010 |   .0301896   .0056313     5.36   0.000     .0191525    .0412267
          tau2011 |   .0166942   .0038948     4.29   0.000     .0090605    .0243279
          tau2012 |          0  (omitted)
         _Igeo1_2 |  -.0012142   .0019681    -0.62   0.537    -.0050716    .0026431
         _Igeo1_3 |  -.0031313   .0028951    -1.08   0.279    -.0088056     .002543
         _Igeo1_4 |  -.0062082   .0037191    -1.67   0.095    -.0134974    .0010811
            _cons |   .0355761   .0402259     0.88   0.376    -.0432652    .1144173
------------------+----------------------------------------------------------------
          sigma_u |          0
          sigma_e |  .01185219
              rho |          0   (fraction of variance due to u_i)
-----------------------------------------------------------------------------------




xtreg gtfp_100  lagltfp_rel_lev95    lagdettotoccrfl  lagnetai lagpsii  lagnetae     lagpsie  ///
lagrspri_stock_va   lagtotroadskms  ///
tau1995-tau2012  ///
if cod_reg<21 ,  vce(cluster geo1) fe
note: tau1995 omitted because of collinearity
note: tau2012 omitted because of collinearity

Fixed-effects (within) regression               Number of obs      =       340
Group variable: cod_reg                         Number of groups   =        20

R-sq:  within  = 0.6611                         Obs per group: min =        17
       between = 0.0470                                        avg =      17.0
       overall = 0.1783                                        max =        17

                                                F(3,3)             =         .
corr(u_i, Xb)  = -0.8272                        Prob > F           =         .

                                        (Std. Err. adjusted for 4 clusters in geo1)
-----------------------------------------------------------------------------------
                  |               Robust
         gtfp_100 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
------------------+----------------------------------------------------------------
lagltfp_rel_lev95 |  -.3010526   .0451169    -6.67   0.007    -.4446346   -.1574707
  lagdettotoccrfl |  -.0210583   .1237399    -0.17   0.876     -.414854    .3727375
         lagnetai |   .0117222   .0296807     0.39   0.719    -.0827351    .1061794
          lagpsii |    .048451   .0132737     3.65   0.035     .0062082    .0906938
         lagnetae |   .0238814   .0094674     2.52   0.086    -.0062481    .0540109
          lagpsie |  -.0255698   .0479345    -0.53   0.631    -.1781187     .126979
lagrspri_stock_va |    .003575    .000634     5.64   0.011     .0015572    .0055928
   lagtotroadskms |   .0138893   .0075632     1.84   0.164    -.0101802    .0379588
          tau1995 |          0  (omitted)
          tau1996 |   .0602008    .008763     6.87   0.006     .0323128    .0880887
          tau1997 |   .0673868   .0051916    12.98   0.001     .0508649    .0839088
          tau1998 |   .0629318   .0064116     9.82   0.002     .0425271    .0833365
          tau1999 |   .0665205   .0052882    12.58   0.001      .049691    .0833499
          tau2000 |   .0746015   .0103181     7.23   0.005     .0417646    .1074384
          tau2001 |   .0585666   .0081672     7.17   0.006     .0325748    .0845584
          tau2002 |   .0425568   .0071275     5.97   0.009      .019874    .0652397
          tau2003 |   .0389972   .0106292     3.67   0.035     .0051705     .072824
          tau2004 |   .0509879   .0076617     6.65   0.007     .0266048    .0753709
          tau2005 |   .0442269    .007049     6.27   0.008     .0217939      .06666
          tau2006 |   .0460886   .0057566     8.01   0.004     .0277686    .0644086
          tau2007 |   .0434926   .0064391     6.75   0.007     .0230006    .0639846
          tau2008 |    .021933   .0079666     2.75   0.071    -.0034205    .0472864
          tau2009 |  -.0124006   .0024856    -4.99   0.015    -.0203108   -.0044903
          tau2010 |   .0305118   .0096626     3.16   0.051    -.0002389    .0612624
          tau2011 |   .0185516    .007333     2.53   0.085    -.0047853    .0418885
          tau2012 |          0  (omitted)
            _cons |  -.1294257   .0594517    -2.18   0.118    -.3186275    .0597761
------------------+----------------------------------------------------------------
          sigma_u |  .02448668
          sigma_e |  .01185219
              rho |  .81018832   (fraction of variance due to u_i)
-----------------------------------------------------------------------------------




xi: xtreg gtfp_100  lagltfp_rel_lev95    lagdettotoccrfl  lagnetai lagpsii  lagnetae     lagpsie  ///
lagrspri_stock_va   lagtotroadskms  /// 
tau1995-tau2012  i.geo1  ///
if cod_reg<21,  robust re
i.geo1            _Igeo1_1-4          (_Igeo1_1 for geo1==Centro omitted)
note: tau1995 omitted because of collinearity
note: tau2012 omitted because of collinearity

Random-effects GLS regression                   Number of obs      =       340
Group variable: cod_reg                         Number of groups   =        20

R-sq:  within  = 0.6088                         Obs per group: min =        17
       between = 0.2743                                        avg =      17.0
       overall = 0.6005                                        max =        17

                                                Wald chi2(19)      =         .
corr(u_i, X)   = 0 (assumed)                    Prob > chi2        =         .

                                    (Std. Err. adjusted for 20 clusters in cod_reg)
-----------------------------------------------------------------------------------
                  |               Robust
         gtfp_100 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
------------------+----------------------------------------------------------------
lagltfp_rel_lev95 |    -.04525   .0178226    -2.54   0.011    -.0801817   -.0103184
  lagdettotoccrfl |  -.0940007   .0516523    -1.82   0.069    -.1952374     .007236
         lagnetai |  -.0144664   .0082807    -1.75   0.081    -.0306964    .0017635
          lagpsii |    .011355   .0316344     0.36   0.720    -.0506473    .0733574
         lagnetae |   .0014869    .006694     0.22   0.824    -.0116331    .0146069
          lagpsie |  -.0566569   .0332526    -1.70   0.088    -.1218307    .0085169
lagrspri_stock_va |   .0000626   .0006108     0.10   0.918    -.0011346    .0012598
   lagtotroadskms |   .0053038   .0036947     1.44   0.151    -.0019376    .0125452
          tau1995 |          0  (omitted)
          tau1996 |   .0251443    .003433     7.32   0.000     .0184157    .0318728
          tau1997 |   .0282562   .0043658     6.47   0.000     .0196994     .036813
          tau1998 |   .0249205   .0036223     6.88   0.000      .017821      .03202
          tau1999 |   .0308059   .0043438     7.09   0.000     .0222922    .0393196
          tau2000 |   .0397295   .0053952     7.36   0.000      .029155    .0503039
          tau2001 |   .0232508    .005381     4.32   0.000     .0127043    .0337972
          tau2002 |   .0056954   .0059057     0.96   0.335    -.0058796    .0172705
          tau2003 |   .0090072   .0066685     1.35   0.177    -.0040628    .0220772
          tau2004 |   .0257581    .004193     6.14   0.000       .01754    .0339763
          tau2005 |   .0195458   .0042295     4.62   0.000     .0112561    .0278355
          tau2006 |   .0227237   .0039394     5.77   0.000     .0150026    .0304447
          tau2007 |   .0216878   .0037919     5.72   0.000     .0142557    .0291198
          tau2008 |   .0013876   .0042288     0.33   0.743    -.0069007     .009676
          tau2009 |   -.025776   .0030627    -8.42   0.000    -.0317788   -.0197732
          tau2010 |   .0301896   .0056313     5.36   0.000     .0191525    .0412267
          tau2011 |   .0166942   .0038948     4.29   0.000     .0090605    .0243279
          tau2012 |          0  (omitted)
         _Igeo1_2 |  -.0012142   .0019681    -0.62   0.537    -.0050716    .0026431
         _Igeo1_3 |  -.0031313   .0028951    -1.08   0.279    -.0088056     .002543
         _Igeo1_4 |  -.0062082   .0037191    -1.67   0.095    -.0134974    .0010811
            _cons |   .0355761   .0402259     0.88   0.376    -.0432652    .1144173
------------------+----------------------------------------------------------------
          sigma_u |          0
          sigma_e |  .01185219
              rho |          0   (fraction of variance due to u_i)
-----------------------------------------------------------------------------------

I've also used the same estimation methods without robust in order to get the results to choose the proprer one. These are the results:

#

1)
F test that all u_i=0:     F(19, 296) =     3.35             Prob > F = 0.0000

2)
xttest0

Breusch and Pagan Lagrangian multiplier test for random effects

        gtfp_100[cod_reg,t] = Xb + u[cod_reg] + e[cod_reg,t]

        Estimated results:
                         |       Var     sd = sqrt(Var)
                ---------+-----------------------------
                gtfp_100 |   .0003688       .0192045
                       e |   .0001405       .0118522
                       u |          0              0

        Test:   Var(u) = 0
                             chibar2(01) =     0.00
                          Prob > chibar2 =   1.0000

3)
hausman FE RE , sigmamore

Note: the rank of the differenced variance matrix (8) does not equal the number of coefficients being tested (24); be sure this is what you expect, or there may be
        problems computing the test.  Examine the output of your estimators for anything unexpected and possibly consider scaling your variables so that the
        coefficients are on a similar scale.

                 ---- Coefficients ----
             |      (b)          (B)            (b-B)     sqrt(diag(V_b-V_B))
             |       FE           RE         Difference          S.E.
-------------+----------------------------------------------------------------
lagltfp_r~95 |   -.3010526      -.04525       -.2558026        .0405769
lagdettoto~l |   -.0210583    -.0940007        .0729425        .0689508
    lagnetai |    .0117222    -.0144664        .0261886        .0158818
     lagpsii |     .048451      .011355         .037096        .0283857
    lagnetae |    .0238814     .0014869        .0223945        .0222148
     lagpsie |   -.0255698    -.0566569         .031087        .0484602
lagrspri_s~a |     .003575     .0000626        .0035124        .0023246
lagtotroad~s |    .0138893     .0053038        .0085855        .0118428
     tau1996 |    .0602008     .0251443        .0350565        .0069425
     tau1997 |    .0673868     .0282562        .0391306        .0061284
     tau1998 |    .0629318     .0249205        .0380113        .0060186
     tau1999 |    .0665205     .0308059        .0357146        .0058911
     tau2000 |    .0746015     .0397295         .034872        .0060373
     tau2001 |    .0585666     .0232508        .0353158        .0066595
     tau2002 |    .0425568     .0056954        .0368614        .0059096
     tau2003 |    .0389972     .0090072        .0299901        .0051444
     tau2004 |    .0509879     .0257581        .0252297        .0043123
     tau2005 |    .0442269     .0195458        .0246811        .0044523
     tau2006 |    .0460886     .0227237        .0233649        .0039436
     tau2007 |    .0434926     .0216878        .0218048        .0037612
     tau2008 |     .021933     .0013876        .0205453        .0035564
     tau2009 |   -.0124006     -.025776        .0133755        .0025573
     tau2010 |    .0305118     .0301896        .0003222         .000566
     tau2011 |    .0185516     .0166942        .0018574        .0005124
------------------------------------------------------------------------------
                           b = consistent under Ho and Ha; obtained from xtreg
            B = inconsistent under Ha, efficient under Ho; obtained from xtreg

    Test:  Ho:  difference in coefficients not systematic

                  chi2(8) = (b-B)'[(V_b-V_B)^(-1)](b-B)
                          =       46.11
                Prob>chi2 =      0.0000
                (V_b-V_B is not positive definite)

This is the xtsum:

#

xtsum gtfp_100  lagltfp_rel_lev95    lagdettotoccrfl  lagnetai lagpsii  lagnetae     lagpsie   ///
lagrspri_stock_va   lagtotroadskms if cod_reg<21

Variable         |      Mean   Std. Dev.       Min        Max |    Observations
-----------------+--------------------------------------------+----------------
gtfp_100 overall | -.0090632   .0192045  -.0827193   .0452053 |     N =     340
         between |              .002689    -.01442  -.0049467 |     n =      20
         within  |             .0190242  -.0802943   .0410888 |     T =      17
                 |                                            |
laglt~95 overall | -.0750882   .0964169  -.3628312   .1190492 |     N =     340
         between |             .0834291  -.2464719    .080493 |     n =      20
         within  |             .0516171  -.2183106   .0482665 |     T =      17
                 |                                            |
lagdet~l overall |  .0978757   .0232197   .0589524   .1836198 |     N =     340
         between |             .0211351   .0666674    .149588 |     n =      20
         within  |             .0106559   .0645574   .1319075 |     T =      17
                 |                                            |
lagnetai overall |  .4892074   .1627707    .233325    1.23396 |     N =     340
         between |             .1571984   .2725485    .928585 |     n =      20
         within  |             .0543073   .2613606   .7945828 |     T =      17
                 |                                            |
lagpsii  overall |  1.089251    .032663    1.00995   1.179449 |     N =     340
         between |             .0264269   1.045245   1.145038 |     n =      20
         within  |             .0200364   1.018452   1.167375 |     T =      17
                 |                                            |
lagnetae overall |  .4741275   .1603552   .2287767   .9504244 |     N =     340
         between |             .1582435   .2668897    .790624 |     n =      20
         within  |             .0430657   .3174492   .6424334 |     T =      17
                 |                                            |
lagpsie  overall |  1.101113   .0261305   1.034469    1.17758 |     N =     340
         between |             .0217327   1.068396   1.148636 |     n =      20
         within  |             .0152572   1.060608   1.155469 |     T =      17
                 |                                            |
lagrspr~ overall |   2.16662   2.037671   .0784046   10.12346 |     N =     340
         between |              2.04059   .1297701   8.967184 |     n =      20
         within  |             .4296822   .9524957   3.921881 |     T =      17
                 |                                            |
lagtot~s overall |   .599534   .1451635   .2292234    .992353 |     N =     340
         between |             .1359085   .2333248   .8243992 |     n =      20
         within  |             .0589331   .4515679   .9143746 |     T =      17

[Carlo Lazzaro]

Giorgio (as reminded by the FAQ, please note the preference on this forum for real and complete given and family names. Thanks)
welcome to this forum.
Some remarks about your query:
- it is not clear why you started out your inference session with pooled OLS. As you have panel data, your first choice should have been -xtreg-;
- it is also not clear why you tested so many regression approaches:
- please note that, unless you're working with a pretty old Stata release, the -xi:- prefix is redundant with built-in Stata commands. If you plan to create categorical variables and/or interactions, use -fvvralist- notation instead;
- you cannot play with default and robust standard error at your convenience, because they work on different assumptions;
- if you suspect heteroskedasticity in the idiosyncratic error and/or autocorrelation in your panel dataset and wisely impose robust/clustered standard errors, you cannot compare -fe- vs -re- specification via -hausman-, but you should switch to the community-contributed -xtoverid- programme (just type -search xtoverid- from within Stata to spot and install it). Being glorious but a but old-fashioned, -xtoverid- does not support -fvvarlist- notation and you have to go back to -xi:- prefix or create interactions by hand.

[Giorgio Carlini]

Thanks a lot Carlo for the reply,
yes probably I should first check which estimation method is more appropriate.
I've now run RE and FE with clustered standard errors and used xtoverid programme.
The message I get is:
xtoverid
Error - saved RE estimates are degenerate (sigma_u=0) and equivalent to pooled OLS
From previous posts I guess this is because the xtreg re is identical to normal regression.
Therefore I should go for FEin my case, given that the F test is in favour of the FE?
Thanks

[Carlo Lazzaro]

Giorgio (the kind remnder about (re-)registering with real and complete given and family names still holds. Thanks):
- you were probably led astray by alternating default and non-default standard errors in your regression models.
I think -xtoverid- oucome provide the most reliable guidance: your dataset does not support the evidence of a group-wise effect; hebce, you should go pooled OLS.

[Giorgio Carlini]

Thanks again Carlo,
about re-registering I've just sent an email asking how to add a complete family name. I was not able to modify in my account profile. Sorry about that.
Sorry also for coming back to the question of estimation method choice. I'm confused here. You suggested that the -xtoverid- outcome points to pooled OLS.
My interpretation was that this is a test that indicates FE vs RE. The F test instead is for FE vs pooled OLS. So whlie -xtoverid- suggests FE, F test reject (H0: choose pooled OLS).
What am I missing here?
Thanks

[Carlo Lazzaro]

Giorgio:
as far as re-registering is concerned, see https://www.statalist.org/forums/help#realnames (I guess you have already followed the instructions, but just in case...).
Your previous -hausman- test was run with default standard errors, while it seems that you imposed cluster robust standard errors in your regression models.
This is the main issue: if you impose non-default standard errors you cannot test -fe- vs -re- specification with -hausman- pretending that you stick with default standard errors.
Since you imposed clustered robust standard errors, I suggested to test which specification is appropriate for your data via the user-written programma -xtoverid-, which points you out to pooled OLS.
As an aside, when -fe- is the way to go, -xtoverid- returns a statistical significant p-value.

[Giorgio Carlini]

Thanks for the reply Carlo.
I've replied to the contact office for the re-registering issue, saying to change the family name.
I've a last question now. Why FE is not appropriate here as the F test suggest? Should I only view at the -xtoverid- result?
Thanks

[Carlo Lazzaro]

Giorgio:
the reason is that you ran you regression models with clustered robust standard errors (and the F-test you mention is not reported whe you invoke non-default standard errors under -xtreg,fe-), then you test -fe- vs -re- specification via -hausman- (that supports default standard errors only). This way, -hausman- outcome is unreliable, whereas the -xtoverid- outcome is the right one to consider.

[Giorgio Carlini]

Thanks Carlo,
I see your point now about default and clustered standard errors.
So basically the -xtoverid- outcome, which is the correct one in my case, i.e.

xtoverid
Error - saved RE estimates are degenerate (sigma_u=0) and equivalent to pooled OLS

is telling me that my time invariant regional effects do not vary between individuals, so pooling OLS is fine?

[Carlo Lazzaro]

Giorgio:
thanks for providing the list with your full family name, too.
Yes, your interpretation is correct. Your panel data does not show evidence of a panel-wise effect. Hence, you should switch to pooled OLS and cluster your standard errors on -panelid- (beware that the options -robust- and -cluster- under -regress- are not as interchangeable as their -xtreg- cousins).

[Giorgio Carlini]

Thanks Carlo.
so if the time invariant regional effects are relatively unimportant, since the variance is zero, we are also assuming that they are not correlated with all the explanatory variables?

[Carlo Lazzaro]

Giorgio:
your panel dataset does not seem to show any evidence of a panel-wise effect.
Hence, the additional orthogonality condition concerning the (assumed) absence of correlation of the panel-wise effect (ui) with the vector of regressors seems off target here, simply because you do not have a panel-wise effect to worry about.

[Giorgio Carlini]

Thanks Carlo,
if I'm going with pooled OLS here, I presume I should use IV given that in my model I've a lagged dependent variable on the rhs. The exogeneity assumption is violeted, am I correct?

[Carlo Lazzaro]

Giorgio:
if you have a lagged dependent variable as a predictor, -xtreg- will give you back biased results.
You should switch to the dynamic panel regression approach (see -xtabond-), instead.
As you're surely aware of, it's realy tricky.

[Giorgio Carlini]

Carlo,
the point is that I've T=17 and N is too small (N=20), so with dynamic panel there is a problem with instruments proliferation (even after using the collapse option and short lags for the instrumental variables). So I need to use alternative estimation methods