Hello everyone,

I apologize in advance for the length of this post, I hope it will receive an answer anyway. I am working on panel data and I am looking for suggestions on which diagnostic tests I should (and should not) perform to correctly specify my model. More particularly I would like to ask:

1. Do you think I left out some relevant diagnostic test?
2. Do you think I should mention these tests and their results in writing the paper?

My panel counts 104 cross-sectional units (countries) and 15 time periods (months), the dependent variable is new monthly deaths due to COVID-19 per million habitants.
I have carried out the following tests:

1. Test for multicollinearity (variance inflation factors (VIFs) for the independent variables): the highest VIF I got is 9.54 which is lower than the 10.00 threshold suggested from the literature (Hair et al., 1995 "Multivariate Data Analysis (3rd ed)"), hence, I do exclude control variables from my model.

2. Wooldridge (2002) test for serial correlation (autocorrelation): xtserial reports a Prob > F = 0.0000 and xtistest reject the H0 till the second lag.

3. Test for Heteroskedasticity: a plot of the residuals suggest the presence of heteroskedasticity, which is also confirmed by the tests White (1980) test (estat imtest, white), by the test proposed in the Stata FAQ.

4. Test for cross-sectional dependence: the Pesaran (2015) test for weak cross-sectional dependence (xtcd2 residual,) reported a p-value = 0.000, suggesting the presence of cross-sectional dependence, which is confirmed also by the p-value = 0.000 of the test (xtcdf dependent_variable).

5. Test whether to use Pooled OLS or panel (RE): this test was performed by the post-estimation command xttest0 which reported chibar2(01) = 201.18 and Prob > chibar2 = 0.0000, suggesting a panel-wise effect.

6. Test whether to employ a RE model or FE model: I performed both the Mundlak test and the Hansen-Sargan test (xtoverid), which suggested a FE model

7. Test the inclusion of Time-Fixed Effects (with testparm): should include them in the model

8. Test the inclusion of squared terms: I included squared terms in the model and tested u-relationship with the command utest

9. Ramsey's RESET test: yielded Prob > F = 0.4581, suggesting that the model does not suffer from omitted variable bias and misspecification.

Also this is the regression with my final model:

Code:
. xtscc new_deaths_per_million dt2-dt15 month2-month15 new_tests_per_thousand people_va
> ccinated_ph people_vaccinatedsquared population population_density median_age cardiov
> asc_death_rate diabetes_prevalence hospital_beds_per_thousand life_expectancy gdp_per
> _capita health_exp_percap urbanization_share internet_users air_passengers smokers_sh
> are, fe lag(2)

Regression with Driscoll-Kraay standard errors   Number of obs     =      1362
Method: Fixed-effects regression                 Number of groups  =       103
Group variable (i): n_country                    F( 44,   102)     =  8.67e+08
maximum lag: 2                                   Prob > F          =    0.0000
                                                 within R-squared  =    0.2814

--------------------------------------------------------------------------------------
                     |             Drisc/Kraay
new_deaths_per_mil~n |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
---------------------+----------------------------------------------------------------
                 dt2 |   .3138264    .001512   207.55   0.000     .3108273    .3168255
                 dt3 |   1.584465    .003592   441.11   0.000     1.577341     1.59159
                 dt4 |   .5226887   .0661408     7.90   0.000     .3914987    .6538786
                 dt5 |   .2736537   .0660774     4.14   0.000     .1425894     .404718
                 dt6 |     .32476   .0026076   124.55   0.000     .3195879    .3299321
                 dt7 |   .4208554   .0035925   117.15   0.000     .4137297    .4279811
                 dt8 |   .3413074   .0080331    42.49   0.000     .3253738    .3572411
                 dt9 |    .527191   .0622785     8.47   0.000     .4036619    .6507201
                dt10 |   .2857736   .0656563     4.35   0.000     .1555447    .4160025
                dt11 |  -.0484925    .075562    -0.64   0.522    -.1983693    .1013844
                dt12 |  -.1315471   .0793195    -1.66   0.100     -.288877    .0257827
                dt13 |  -.5313282   .3175867    -1.67   0.097     -1.16126    .0986035
                dt14 |  -1.191948   .0840271   -14.19   0.000    -1.358615    -1.02528
                dt15 |  -1.323201   .3152522    -4.20   0.000    -1.948503   -.6979001
              month2 |    .259636   .0031567    82.25   0.000     .2533746    .2658974
              month3 |   1.127022   .0129004    87.36   0.000     1.101435     1.15261
              month4 |   .5801031   .0172902    33.55   0.000     .5458081    .6143981
              month5 |   .5247511   .0210133    24.97   0.000     .4830712    .5664309
              month6 |   .7252116   .0296244    24.48   0.000     .6664517    .7839714
              month7 |   .7215047   .0352729    20.45   0.000      .651541    .7914684
              month8 |   .7877615   .0463446    17.00   0.000     .6958371    .8796858
              month9 |    1.13178   .0533886    21.20   0.000     1.025884    1.237676
             month10 |   3.049781   .0671831    45.40   0.000     2.916524    3.183038
             month11 |    3.69505   .0825455    44.76   0.000     3.531322    3.858779
             month12 |   3.458879   .1158383    29.86   0.000     3.229115    3.688644
             month13 |   2.350512   .1901172    12.36   0.000     1.973416    2.727609
             month14 |   2.002837   .3156656     6.34   0.000     1.376716    2.628959
             month15 |   2.498445   .3994487     6.25   0.000      1.70614    3.290749
new_tests_per_thou~d |   .1133193   .0338773     3.34   0.001     .0461237    .1805148
people_vaccinated_ph |   .1089296   .0475717     2.29   0.024     .0145712    .2032879
people_vaccinateds~d |  -.0022201   .0008529    -2.60   0.011    -.0039119   -.0005283
          population |          0  (omitted)
  population_density |          0  (omitted)
          median_age |          0  (omitted)
cardiovasc_death_r~e |          0  (omitted)
 diabetes_prevalence |          0  (omitted)
hospital_beds_per_~d |          0  (omitted)
     life_expectancy |          0  (omitted)
      gdp_per_capita |  -2.50e-07   4.50e-07    -0.56   0.580    -1.14e-06    6.42e-07
   health_exp_percap |          0  (omitted)
  urbanization_share |          0  (omitted)
      internet_users |          0  (omitted)
      air_passengers |          0  (omitted)
       smokers_share |          0  (omitted)
               _cons |          0  (omitted)
--------------------------------------------------------------------------------------
in which new_deaths_per_million is the dependent variable and dt2-dt15 are interaction terms of time units and a dummy variable to distinguish between federal and non-federal countries.
Thanks in advance to whoever is willing to help
I wish you a nice weekend