sigma_u=0 in random effects model

[Fabian Wagner]

Hi all,
this is my first post so please let me know if anything can be improved upon!

I am conducting an analysis in which I explain the logged levelized-cost of electrity of onshore wind energy (log_lcoe_on) by cumulative capacity (log_cc_on) and a knowledge stock (log_ks_rd_aug), which consists of accumulated past expenditures in R&D. Additionally, I use feed-in tariffs measured in USD per kilowatt hour (this is not my full specification but the issue also occurs in this more parsimonious specification). The data consists of annual country-level observations (6 countries with 20 years each).

To this I added country-level FE. While conducting postestimation analyses I noticed that "sigma_u" after running random effects is 0. Previous Statalist entries suggested that this implies an absence of panel-wise effects and that pooled OLS should be used (which in this case yields the same coefficients as RE) (https://www.statalist.org/forums/for...plier-equals-1). However, including FE did change results by quite a bit and also the repoted "rho" and "F-values" suggest the FE model to be suitable. I could identify that this issue only occurs when feed-in tariffs are included, but I don't know why it is like that - it may be because of the zero values. Another issue is that, even without feed-in tarifs, the Hausman test returns a negative chi2 which I understood is critical, too.

Since I don't understand why these issues occur I also don't know how to solve them. Any help on this would be highly appreciated!

Please find my outputs and a plot of the feed-in tariff below.

.         ********* POLS *********
.         reg log_lcoe_on log_cc_on log_ks_rd_aug fd_in_trff, vce(robust)

Linear regression                               Number of obs     =        120
                                                F(3, 116)         =      27.37
                                                Prob > F          =     0.0000
                                                R-squared         =     0.3781
                                                Root MSE          =     .24315

-------------------------------------------------------------------------------
              |               Robust
  log_lcoe_on | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
    log_cc_on |  -.0460654   .0085744    -5.37   0.000    -.0630481   -.0290827
log_ks_rd_aug |  -.0631807   .0138594    -4.56   0.000    -.0906309   -.0357305
   fd_in_trff |   1.499136   .4131649     3.63   0.000     .6808104    2.317461
        _cons |  -1.437684   .1321524   -10.88   0.000    -1.699428   -1.175939
-------------------------------------------------------------------------------

.        
.        
.         ********* FE *********
.         xtreg log_lcoe_on log_cc_on log_ks_rd_aug fd_in_trff, fe

Fixed-effects (within) regression               Number of obs     =        120
Group variable: cntry                           Number of groups  =          6

R-squared:                                      Obs per group:
     Within  = 0.7476                                         min =         20
     Between = 0.1010                                         avg =       20.0
     Overall = 0.3001                                         max =         20

                                                F(3,111)          =     109.60
corr(u_i, Xb) = -0.7116                         Prob > F          =     0.0000

-------------------------------------------------------------------------------
  log_lcoe_on | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
    log_cc_on |  -.0031357   .0130257    -0.24   0.810    -.0289471    .0226757
log_ks_rd_aug |  -.2184333   .0176014   -12.41   0.000    -.2533116   -.1835551
   fd_in_trff |   .7140114   .3078071     2.32   0.022      .104071    1.323952
        _cons |  -.1522363   .1348406    -1.13   0.261     -.419432    .1149595
--------------+----------------------------------------------------------------
      sigma_u |  .31892833
      sigma_e |  .15586972
          rho |  .80719591   (fraction of variance due to u_i)
-------------------------------------------------------------------------------
F test that all u_i=0: F(5, 111) = 34.26                     Prob > F = 0.0000

.        
.        
.         ********* RE *********
.         xtreg log_lcoe_on log_cc_on log_ks_rd_aug fd_in_trff, re

Random-effects GLS regression                   Number of obs     =        120
Group variable: cntry                           Number of groups  =          6

R-squared:                                      Obs per group:
     Within  = 0.5856                                         min =         20
     Between = 0.0398                                         avg =       20.0
     Overall = 0.3781                                         max =         20

                                                Wald chi2(3)      =      70.53
corr(u_i, X) = 0 (assumed)                      Prob > chi2       =     0.0000

-------------------------------------------------------------------------------
  log_lcoe_on | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
--------------+----------------------------------------------------------------
    log_cc_on |  -.0460654    .014093    -3.27   0.001    -.0736872   -.0184436
log_ks_rd_aug |  -.0631807   .0160545    -3.94   0.000    -.0946469   -.0317144
   fd_in_trff |   1.499136   .3981048     3.77   0.000     .7188644    2.279407
        _cons |  -1.437684   .1434132   -10.02   0.000    -1.718769   -1.156599
--------------+----------------------------------------------------------------
      sigma_u |          0
      sigma_e |  .15586972
          rho |          0   (fraction of variance due to u_i)
-------------------------------------------------------------------------------

.        
.        
.         ********* RE without fd_in_trff *********
.         xtreg log_lcoe_on log_cc_on log_ks_rd_aug, re  

Random-effects GLS regression                   Number of obs     =        120
Group variable: cntry                           Number of groups  =          6

R-squared:                                      Obs per group:
     Within  = 0.7179                                         min =         20
     Between = 0.2219                                         avg =       20.0
     Overall = 0.2984                                         max =         20

                                                Wald chi2(2)      =     116.87
corr(u_i, X) = 0 (assumed)                      Prob > chi2       =     0.0000

-------------------------------------------------------------------------------
  log_lcoe_on | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
--------------+----------------------------------------------------------------
    log_cc_on |  -.0292927   .0146369    -2.00   0.045    -.0579806   -.0006048
log_ks_rd_aug |  -.1317308   .0183369    -7.18   0.000    -.1676705   -.0957911
        _cons |  -.7754473   .1523597    -5.09   0.000    -1.074067   -.4768278
--------------+----------------------------------------------------------------
      sigma_u |  .04758568
      sigma_e |  .15888889
          rho |  .08231155   (fraction of variance due to u_i)
-------------------------------------------------------------------------------


.         ********* Hasuman test *********

.         qui: xtreg log_lcoe_on log_cc_on log_ks_rd_aug, fe      

.         est sto fixed

.         qui: xtreg log_lcoe_on log_cc_on log_ks_rd_aug, re      

.         est sto random

.         hausman fixed random

                 ---- Coefficients ----
             |      (b)          (B)            (b-B)     sqrt(diag(V_b-V_B))
             |     fixed        random       Difference       Std. err.
-------------+----------------------------------------------------------------
   log_cc_on |   -.0046799    -.0292927        .0246128               .
log_ks_rd_~g |   -.2204806    -.1317308       -.0887498               .
------------------------------------------------------------------------------
                          b = Consistent under H0 and Ha; obtained from xtreg.
           B = Inconsistent under Ha, efficient under H0; obtained from xtreg.

Test of H0: Difference in coefficients not systematic

chi2(2) = (b-B)'[(V_b-V_B)^(-1)](b-B)
        = -566.18

Warning: chi2 < 0 ==> model fitted on these data
         fails to meet the asymptotic assumptions
         of the Hausman test; see suest for a
         generalized test.

Thank you in advance!

Fabian

[Carlo Lazzaro]

Fabian:
1) your pooled OLS needs -vce(cluster cntry)- standard errors (under -regress- the -robust- option takes care of heteroskedasticity only);
2) without knowing your data, it is difficult to say why -re- reported sigma_u=0. However, the between R_sq is dramatically low and this warns you about the fact that -re- specificatiin is not good for your data.
3) even without -tariffs- the -rho- in -re- is dramatically low;
4) as -hausman- test failed to meet its asymptotic assumptions, check the -re- specification via the community-contributed module -xtoverid- (H₀: -re- is the way to go).

[Fabian Wagner]

Thank you for your reply, Carlo!
For the specification with feed-in tariffs -xtoverid- confirms that "RE estimates are degenerate (sigma_u=0) and equivalent to pooled OLS", whereas without feed-in tariffs a p-value of 0.000 is reported.
Am I interpreting it right that, due to a lack of between-country variation, the test results suggest to use pooled OLS (with proper SE - my bad) instead of FE or RE?

\Fabian

[Carlo Lazzaro]

Fabian:
however, your -xtreg,fe- showed evidence of a panel-wise effect (and the within R-sq is good too).
In addition, the rejected null of -xtoverid- points you toward fe.
Provided that the model is correctly specified, I would go -fe-.

[Jackson Monroe]

The correlation between u_i and X is quite high, so I'd recommend to FE specification to account for the unobserved heterogeneity (if u_i and y_i are correlated using POLS would cause omitted variables bias). The fact that POLS and RE are equivalent in this case is not a recommendation of POLS. Low rho just indicates there is little variation induced by between-group differences so the group-level terms will be very very close to one another, and in POLS they are required to be exactly equal, so there isn't much difference.

[Fabian Wagner]

Carlo:
That makes sense! - I would therefore now close this thread

I appreciate you taking the time and thank you very much for your help!

\Fabian

[Fabian Wagner]

Jackson: Thank you for the background information!

\Fabian