Hello everyone!

I am trying to analyse a panel data of 30 countries and each for 2007-2016. I have gone through the fixed effect models and specifications but still got some questions and will be very thankful if anyone could give some hints.
I am trying several options to come up with correct model. In all cases I have significant coefficient, this is not an issue, but there are still some issues I cannot handle.
First - can you please say what xtreg does when we do not specify re, fe and just leave like that. Is it doing just pooled regression?
Code:
. xtreg growth_rate ln_prod_lagged i.year

Random-effects GLS regression                   Number of obs     =        260
Group variable: id                              Number of groups  =         26

R-sq:                                           Obs per group:
     within  = 0.3517                                         min =         10
     between = 0.0569                                         avg =       10.0
     overall = 0.3303                                         max =         10

                                                Wald chi2(10)     =     123.74
corr(u_i, X)   = 0 (assumed)                    Prob > chi2       =     0.0000

--------------------------------------------------------------------------------
   growth_rate |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------+----------------------------------------------------------------
ln_prod_lagged |  -.0156277   .0076401    -2.05   0.041     -.030602   -.0006535
               |
          year |
         2008  |  -.0651795   .0183597    -3.55   0.000    -.1011638   -.0291951
         2009  |  -.1140758   .0183588    -6.21   0.000    -.1500583   -.0780932
         2010  |    .068735   .0183539     3.74   0.000     .0327621    .1047079
         2011  |  -.0280067    .018377    -1.52   0.128    -.0640249    .0080116
         2012  |  -.0567071   .0183908    -3.08   0.002    -.0927524   -.0206619
         2013  |  -.0330568   .0183925    -1.80   0.072    -.0691054    .0029918
         2014  |  -.0457379   .0184061    -2.48   0.013    -.0818132   -.0096626
         2015  |  -.0348612   .0184136    -1.89   0.058    -.0709512    .0012288
         2016  |  -.0325653   .0184291    -1.77   0.077    -.0686857    .0035551
               |
         _cons |   .2311222    .082884     2.79   0.005     .0686725    .3935719
---------------+----------------------------------------------------------------
       sigma_u |  .00594337
       sigma_e |  .06168062
           rho |   .0091993   (fraction of variance due to u_i)
--------------------------------------------------------------------------------
In this model I get -0.016 which is the meaningful speed I would expect to have.

Second-if I have only 30 countries in my database is it valid to use fe. I get good results with it but the coeficient is too big and not intuitive, though significant.

Code:
. xtreg growth_rate ln_prod_lagged, fe

Fixed-effects (within) regression               Number of obs     =        260
Group variable: id                              Number of groups  =         26

R-sq:                                           Obs per group:
     within  = 0.1395                                         min =         10
     between = 0.0569                                         avg =       10.0
     overall = 0.0172                                         max =         10

                                                F(1,233)          =      37.78
corr(u_i, Xb)  = -0.9701                        Prob > F          =     0.0000

--------------------------------------------------------------------------------
   growth_rate |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
---------------+----------------------------------------------------------------
ln_prod_lagged |  -.2549382   .0414766    -6.15   0.000    -.3366553    -.173221
         _cons |   2.788079   .4491071     6.21   0.000     1.903249    3.672908
---------------+----------------------------------------------------------------
       sigma_u |  .14074613
       sigma_e |  .07530423
           rho |  .77744573   (fraction of variance due to u_i)
--------------------------------------------------------------------------------
F test that all u_i=0: F(25, 233) = 2.06                     Prob > F = 0.0030

I also checked testparm i.year and use dummies to control time effects.
Code:
. xtreg growth_rate ln_prod_lagged i.year, fe

Fixed-effects (within) regression               Number of obs     =        260
Group variable: id                              Number of groups  =         26

R-sq:                                           Obs per group:
     within  = 0.4450                                         min =         10
     between = 0.0569                                         avg =       10.0
     overall = 0.0622                                         max =         10

                                                F(10,224)         =      17.96
corr(u_i, Xb)  = -0.9479                        Prob > F          =     0.0000

--------------------------------------------------------------------------------
   growth_rate |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
---------------+----------------------------------------------------------------
ln_prod_lagged |  -.2972772   .0458798    -6.48   0.000    -.3876885    -.206866
               |
          year |
         2008  |  -.0481278   .0173312    -2.78   0.006    -.0822808   -.0139748
         2009  |  -.0984127   .0172964    -5.69   0.000    -.1324971   -.0643283
         2010  |   .0687048   .0171071     4.02   0.000     .0349933    .1024163
         2011  |   .0059763   .0179805     0.33   0.740    -.0294562    .0414088
         2012  |  -.0137751   .0184814    -0.75   0.457    -.0501948    .0226445
         2013  |   .0108456   .0185418     0.58   0.559    -.0256931    .0473843
         2014  |   .0053554   .0190243     0.28   0.779    -.0321342    .0428449
         2015  |   .0197774    .019284     1.03   0.306    -.0182238    .0577787
         2016  |   .0287665   .0198108     1.45   0.148     -.010273    .0678059
               |
         _cons |   3.248588   .4916846     6.61   0.000     2.279669    4.217507
---------------+----------------------------------------------------------------
       sigma_u |  .16449166
       sigma_e |  .06168062
           rho |  .87672547   (fraction of variance due to u_i)
--------------------------------------------------------------------------------
F test that all u_i=0: F(25, 224) = 2.60                     Prob > F = 0.0001
Here I get even higher coeficient, which is really weird to have.


Might it be because of the small database. One of my assumptions is that I have only 10 years periods for 30 countires and it comes 300 datapoints, that is why i cannot properly capture the fixed effects. Please share your thought and maybe there are some suggestions how to deal with small panel datasets?
Thank you very much for the support beforehand!
Sara