I am running a panel data analysis of geographic diversification (quadratic) on company performance (winsorized ROA.) I noticed that the the effect of the quadratic geographic diversification term (cL.GSD#cL.GSD ) is significant (P>|t| of 0.014) if I run it without clustering SEs. However, if I cluster the SEs the effect of cL.GSD#cL.GSD is not significant (P>|t| of 0.063) I presume this is happening on account of heteroskedasticity and autocorrelation. My questions are as follows:
1. For checking for heteroskedasticity, I presume I run xttest3 after running the non-clustered xtreg fe and if Prob>chi2 is < 0.05, I should conclude that the data is heteroskedastic.
2. For autocorrelation, what would be the most appropriate test to use?

(I am neither a statistician nor an expert on STATA, so please pardon me if my question is too basic.)

For your reference, my output is pasted below:

Code:
. xtreg ROA_win05 Ln_Revenue Ln_LTDTA CoAge TPSD Ind_GSD wGDPpc wCPI wDCF wExpgr wGDPgr wCon
> s c.l1.GSD##c.l1.GSD if Mfg_Ratio>80 &FOREIGNSALESTOTALSALES >10 & Year_ < YearInactive, f
> e

Fixed-effects (within) regression               Number of obs     =      1,099
Group variable: n_CUSIP                         Number of groups  =        122

R-sq:                                           Obs per group:
     within  = 0.0303                                         min =          1
     between = 0.2820                                         avg =        9.0
     overall = 0.1260                                         max =         18

                                                F(13,964)         =       2.31
corr(u_i, Xb)  = 0.0634                         Prob > F          =     0.0050

---------------------------------------------------------------------------------
      ROA_win05 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
----------------+----------------------------------------------------------------
     Ln_Revenue |   2.422417   .7250379     3.34   0.001     .9995821    3.845251
       Ln_LTDTA |  -.5803542    .207356    -2.80   0.005    -.9872754    -.173433
          CoAge |  -.0860849   .1004581    -0.86   0.392    -.2832267     .111057
           TPSD |  -.4280329   1.105539    -0.39   0.699    -2.597573    1.741507
Ind_GSD_by_Year |   2.389643    2.10156     1.14   0.256    -1.734516    6.513803
         wGDPpc |  -.0000434   .0001677    -0.26   0.796    -.0003726    .0002857
           wCPI |  -.3764426   .2832035    -1.33   0.184     -.932209    .1793239
           wDCF |   7.03e-13   1.28e-12     0.55   0.585    -1.82e-12    3.22e-12
         wExpgr |   .1760092   .1066144     1.65   0.099    -.0332138    .3852322
         wGDPgr |  -.4340272   .2912374    -1.49   0.136     -1.00556    .1375052
          wCons |  -3.26e-13   4.59e-13    -0.71   0.478    -1.23e-12    5.75e-13
                |
            GSD |
            L1. |   8.412533   3.788138     2.22   0.027     .9785863    15.84648
                |
  cL.GSD#cL.GSD |  -7.214446   2.919635    -2.47   0.014    -12.94402   -1.484873
                |
          _cons |  -41.19891   14.33343    -2.87   0.004    -69.32723    -13.0706
----------------+----------------------------------------------------------------
        sigma_u |   17.97445
        sigma_e |  7.1515211
            rho |  .86333291   (fraction of variance due to u_i)
---------------------------------------------------------------------------------
F test that all u_i=0: F(121, 964) = 18.73                   Prob > F = 0.0000
Code:
. xtreg ROA_win05 Ln_Revenue Ln_LTDTA CoAge TPSD Ind_GSD wGDPpc wCPI wDCF wExpgr wGDPgr wCon
> s c.l1.GSD##c.l1.GSD if Mfg_Ratio>80 &FOREIGNSALESTOTALSALES >10 & Year_ < YearInactive, f
> e cluster(n_CUSIP)

Fixed-effects (within) regression               Number of obs     =      1,099
Group variable: n_CUSIP                         Number of groups  =        122

R-sq:                                           Obs per group:
     within  = 0.0303                                         min =          1
     between = 0.2820                                         avg =        9.0
     overall = 0.1260                                         max =         18

                                                F(11,121)         =          .
corr(u_i, Xb)  = 0.0634                         Prob > F          =          .

                                 (Std. Err. adjusted for 122 clusters in n_CUSIP)
---------------------------------------------------------------------------------
                |               Robust
      ROA_win05 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
----------------+----------------------------------------------------------------
     Ln_Revenue |   2.422417   1.262198     1.92   0.057    -.0764376    4.921271
       Ln_LTDTA |  -.5803542   .2804897    -2.07   0.041    -1.135658   -.0250509
          CoAge |  -.0860849   .1138875    -0.76   0.451    -.3115552    .1393855
           TPSD |  -.4280329   1.173237    -0.36   0.716    -2.750766      1.8947
Ind_GSD_by_Year |   2.389643   2.990261     0.80   0.426    -3.530367    8.309653
         wGDPpc |  -.0000434   .0002331    -0.19   0.852    -.0005049    .0004181
           wCPI |  -.3764426   .3020473    -1.25   0.215    -.9744249    .2215398
           wDCF |   7.03e-13   1.29e-12     0.55   0.586    -1.84e-12    3.25e-12
         wExpgr |   .1760092   .1363525     1.29   0.199    -.0939365    .4459549
         wGDPgr |  -.4340272   .3859219    -1.12   0.263    -1.198061     .330007
          wCons |  -3.26e-13   4.73e-13    -0.69   0.492    -1.26e-12    6.10e-13
                |
            GSD |
            L1. |   8.412533   6.196508     1.36   0.177    -3.855088    20.68015
                |
  cL.GSD#cL.GSD |  -7.214446   3.837988    -1.88   0.063    -14.81275    .3838632
                |
          _cons |  -41.19891   23.68022    -1.74   0.084    -88.08015    5.682326
----------------+----------------------------------------------------------------
        sigma_u |   17.97445
        sigma_e |  7.1515211
            rho |  .86333291   (fraction of variance due to u_i)
---------------------------------------------------------------------------------