# Simple trend analysis with unbalanced & short panel data

I have the following (unbalanced) panel data: yearly sustainability ratings (ESG) of ca. 2000 individual firms over a 11-year period. The average observations per firm only covers 5.3 periods. These firms can be grouped into 3 categories (i.e. 1=financial services, 2=industrial products, 3=oil and gas).

I now want to investigate the differences in ESG ratings across these categories. In particular, I want to:

1. determine if a trend exists (regarding these ratings) over my observation period for each of the three individual categories (i.e. no trend/positive/negative)
2. compare the trends determined in 1) across the 3 categories (i.e. statistically show if categories show similar trends and if not, how they differ)

As for now, I fitted a fixed-effects linear regression model for the panel data (depvar: ESG rating, indepvar: year), without yet accounting for the different categories:

. xtreg esgscore year, fe

Fixed-effects (within) regression               Number of obs     =     10,467
Group variable: firmid                          Number of groups  =      1,964

R-sq:                                           Obs per group:
within  = 0.2498                                         min =          1
between = 0.0002                                         avg =        5.3
overall = 0.0370                                         max =         11

F(1,8502)         =    2831.28
corr(u_i, Xb)  = -0.1414                        Prob > F          =     0.0000

------------------------------------------------------------------------------
esgfundy |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
year |   1.045261   .0196442    53.21   0.000     1.006754    1.083768
_cons |  -2032.695   39.44217   -51.54   0.000    -2110.011   -1955.379
-------------+----------------------------------------------------------------
sigma_u |  9.3688475
sigma_e |  4.6491274
rho |   .8024093   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(1963, 8502) = 16.15                 Prob > F = 0.0000


For 1), I am unsure whether to proceed with a panel model approach (for each of the three categories) or if potentially a multilevel model with repeated measures is better suitable for my trend analysis, especially because I only have few observations per firm. I assume that a panel unit root test isn't really suitable for the same reason. What do you think? Am I missing other options?

For 2), I suggest checking for differences in the slopes of the regression lines from 1). Is this sufficient to determine statistically significant differences across the respective trends?

Please bear in mind that determining the change over time and the comparison across these 3 categories are my only objectives.

Thank you!