Hello! I apologize if some of the questions in this post seem simple, but this has to do with my thesis and I appreciate any assistance!

I am dealing with panel data involving some banks over 63 time periods(N=4736 T=63) balanced. I used the Hausman test to determine I needed to use a FE model. But, this is where the trouble begins and my questions begin! So I run a basic xtreg fe model and get this result
Code:
xtreg zscore lnasset lnassetsq diverse leverage eeffqr DGS10 CPIAUCSL_PCH GDPC1_PC1, fe

Fixed-effects (within) regression               Number of obs     =    298,355
Group variable: cert                            Number of groups  =      4,736

R-sq:                                           Obs per group:
     within  = 0.0179                                         min =         62
     between = 0.0242                                         avg =       63.0
     overall = 0.0227                                         max =         63

                                                F(8,293611)       =     670.17
corr(u_i, Xb)  = -0.0196                        Prob > F          =     0.0000

------------------------------------------------------------------------------
      zscore |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     lnasset |   .4193168   .0420687     9.97   0.000     .3368634    .5017702
   lnassetsq |  -.0071421   .0016294    -4.38   0.000    -.0103357   -.0039485
     diverse |   .0000122    .000058     0.21   0.834    -.0001015    .0001259
    leverage |   -.013206   .0004068   -32.46   0.000    -.0140033   -.0124087
      eeffqr |  -.0001518   8.33e-06   -18.23   0.000    -.0001681   -.0001355
       DGS10 |   .0478444   .0021882    21.86   0.000     .0435557    .0521332
CPIAUCSL_PCH |   .0676272   .0032808    20.61   0.000     .0611969    .0740575
   GDPC1_PC1 |   .0310941   .0008967    34.68   0.000     .0293366    .0328516
       _cons |  -1.795671   .2740217    -6.55   0.000    -2.332746   -1.258596
-------------+----------------------------------------------------------------
     sigma_u |  1.7745564
     sigma_e |  .99459682
         rho |  .76095759   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(4735, 293611) = 196.07              Prob > F = 0.0000
so now I go about running some test to check for serial correlation and heteroscedasticity.
Code:
xttest3

Modified Wald test for groupwise heteroskedasticity
in fixed effect regression model

H0: sigma(i)^2 = sigma^2 for all i

chi2 (4736)  =   379.43
Prob>chi2 =      1.0000
Which if I interpret this correctly means my model doesn't suffer from hetero.
Now running xtserial I get
Code:
xtserial zscore lnasset lnassetsq diverse leverage eeffqr DGS10 CPIAUCSL_PCH GDPC1_PC1

Wooldridge test for autocorrelation in panel data
H0: no first-order autocorrelation
    F(  1,    4735) =    226.237
           Prob > F =      0.0000
Meaning my model does suffer from autocorrelation. So what now with this information do I run a vce(robust) fe or is there some other tests I should run or other model or option I should use. I am outside my statistical chops currently but I am trying to learn.

Secondly if I were to use a VCE(Robust) model why when I run it using areg as such do I get such a different significance on some of my variables than using xtreg. It was my impression they were so similar that they should not differ by much?
Results from areg note that cert is just a unique identifier for each individual bank:
Code:
areg zscore lnasset lnassetsq diverse leverage eeffqr DGS10 CPIAUCSL_PCH GDPC1_PC1, a(cert) vce(robust)

Linear regression, absorbing indicators         Number of obs     =    298,355
Absorbed variable: cert                         No. of categories =      4,736
                                                F(   8, 293611)   =     343.16
                                                Prob > F          =     0.0000
                                                R-squared         =     0.7691
                                                Adj R-squared     =     0.7654
                                                Root MSE          =     0.9946

------------------------------------------------------------------------------
             |               Robust
      zscore |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     lnasset |   .4193168   .0686047     6.11   0.000     .2848535    .5537801
   lnassetsq |  -.0071421   .0025061    -2.85   0.004    -.0120541   -.0022301
     diverse |   .0000122   .0000432     0.28   0.778    -.0000725    .0000969
    leverage |   -.013206   .0102436    -1.29   0.197    -.0332831    .0068711
      eeffqr |  -.0001518   .0001045    -1.45   0.146    -.0003566     .000053
       DGS10 |   .0478444   .0046113    10.38   0.000     .0388064    .0568824
CPIAUCSL_PCH |   .0676272   .0041492    16.30   0.000     .0594948    .0757596
   GDPC1_PC1 |   .0310941   .0011823    26.30   0.000     .0287769    .0334113
       _cons |  -1.795671   .4056216    -4.43   0.000    -2.590678   -1.000664
------------------------------------------------------------------------------
Now using a fe model:
Code:
xtreg zscore lnasset lnassetsq diverse leverage eeffqr DGS10 CPIAUCSL_PCH GDPC1_PC1, fe vce(robust)

Fixed-effects (within) regression               Number of obs     =    298,355
Group variable: cert                            Number of groups  =      4,736

R-sq:                                           Obs per group:
     within  = 0.0179                                         min =         62
     between = 0.0242                                         avg =       63.0
     overall = 0.0227                                         max =         63

                                                F(8,4735)         =     184.31
corr(u_i, Xb)  = -0.0196                        Prob > F          =     0.0000

                               (Std. Err. adjusted for 4,736 clusters in cert)
------------------------------------------------------------------------------
             |               Robust
      zscore |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     lnasset |   .4193168    .134761     3.11   0.002     .1551224    .6835111
   lnassetsq |  -.0071421   .0051472    -1.39   0.165    -.0172331    .0029488
     diverse |   .0000122   .0000445     0.27   0.784     -.000075    .0000994
    leverage |   -.013206   .0102906    -1.28   0.199    -.0333804    .0069684
      eeffqr |  -.0001518   .0001059    -1.43   0.152    -.0003594    .0000558
       DGS10 |   .0478444   .0067429     7.10   0.000     .0346253    .0610636
CPIAUCSL_PCH |   .0676272   .0037723    17.93   0.000     .0602317    .0750227
   GDPC1_PC1 |   .0310941   .0014933    20.82   0.000     .0281667    .0340216
       _cons |  -1.795671   .8540467    -2.10   0.036        -3.47   -.1213422
-------------+----------------------------------------------------------------
     sigma_u |  1.7745564
     sigma_e |  .99459682
         rho |  .76095759   (fraction of variance due to u_i)
------------------------------------------------------------------------------
I do not understand exactly why in particular lnassetsq became so insignificant in the FE model with robust errors but not in the areg? I am sorry if I am missing something elementary here.