[Jesse Nooijen]

I agree. With AIC and BIC, lower numbers are better. In addition to comparing IC, you can run a likelihood ratio test (lrtest in Stata) that compares the two log likelihoods given that the models are nested. By nested I mean that one can get from the random slope model to the random intercept model by constraining the random slope variance and random slope-intercept covariance to 0. This gives you two additional degrees of freedom. The likelihood ratio test follows a chi-square distribution except that the null hypothesis is on the boundary of the parameter space since variances from these models are > 0. Accordingly, you divide the p-value reported by lrtest by 2.

mixed ln_Revenue CharismaticValuebased c.CharismaticValuebased#i.crisis i.crisis HHI GDPG ///
Inflation Political ln_Assets ln_GDP Charismat icControls || Countryid: crisis , cov(un) ///
residuals(ar 1, t(time_variable)) vce(robust)
eststo ri
mixed ln_Revenue CharismaticValuebased c.CharismaticValuebased#i.crisis i.crisis HHI GDPG ///
Inflation Political ln_Assets ln_GDP Charismat icControls || Countryid: crisis , cov(un) ///
residuals(ar 1, t(time_variable)) vce(robust)
eststo rc
lrtest rc ri, stats

Thanks for your reply. I was quite confused initially regarding the AIC and BIC values in a negative sense, but I found an older thread on here that explained more negative is better as well. Given that, I was wondering whether the more negative BIC value for the model without random slopes (-23 vs -17) then would indicate to use that model if you ony look at AIC and BIC, given that those for BIC are more negative for the model without and that I read that BIC is preferred over AIC as it places a more stringent penalty on model complexity if I am not mistaken.. Given the feedback and paper provided in this thread, I was planning to continue with the random slopes model, but the last test made me curious again.

I ran the model you presented, and I modified the second model code to exclude the random slopes and I removed the vce(robust) option as it prevented the lrtest from computing.

 
 mixed ln_Revenue CharismaticValuebased c.CharismaticValuebased#i.crisis i.crisis HHI GDPG ///                 Inflation Political ln_Assets ln_GDP CharismaticControls || Countryid: crisis , cov(un) ///                 residuals(ar 1, t(year))  eststo ri mixed ln_Revenue CharismaticValuebased c.CharismaticValuebased#i.crisis i.crisis HHI GDPG ///                 Inflation Political ln_Assets ln_GDP CharismaticControls || Countryid: , ///                 residuals(ar 1, t(year))  eststo rc lrtest rc ri, stats

which resulted in output:

. do "C:\Users\jesse\AppData\Local\Temp\STD58c0_000000.tmp"

. mixed ln_Revenue CharismaticValuebased c.CharismaticValuebased#i.crisis i.crisis HHI GDPG ///
>                 Inflation Political ln_Assets ln_GDP CharismaticControls || Countryid: crisis , cov(un) ///
>                 residuals(ar 1, t(year)) 

Obtaining starting values by EM ...

Performing gradient-based optimization: 
Iteration 0:  Log likelihood =  46.857674  
Iteration 1:  Log likelihood =  49.990457  
Iteration 2:  Log likelihood =  50.852714  
Iteration 3:  Log likelihood =  50.896779  
Iteration 4:  Log likelihood =  50.897016  
Iteration 5:  Log likelihood =  50.897016  

Computing standard errors ...

Mixed-effects ML regression                         Number of obs    =     199
Group variable: Countryid                           Number of groups =      25
                                                    Obs per group:
                                                                 min =       7
                                                                 avg =     8.0
                                                                 max =       8
                                                    Wald chi2(10)    = 1728.95
Log likelihood =  50.897016                         Prob > chi2      =  0.0000

------------------------------------------------------------------------------------------------
                    ln_Revenue | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------------------------+----------------------------------------------------------------
         CharismaticValuebased |   1.183889   .3365705     3.52   0.000     .5242225    1.843555
                               |
crisis#c.CharismaticValuebased |
                            1  |   .0066603   .1103049     0.06   0.952    -.2095334     .222854
                               |
                      1.crisis |  -.4462518   .6383311    -0.70   0.484    -1.697358    .8048542
                           HHI |   .0000466   .0000497     0.94   0.349    -.0000509     .000144
                          GDPG |   .0114903    .002893     3.97   0.000     .0058202    .0171604
                     Inflation |   .0398219   .0048346     8.24   0.000     .0303463    .0492974
                     Political |   .0491699   .0750792     0.65   0.513    -.0979826    .1963224
                     ln_Assets |   .7857556   .0258404    30.41   0.000     .7351093    .8364018
                        ln_GDP |   .2019008   .0564514     3.58   0.000      .091258    .3125435
           CharismaticControls |  -.4559882   .1321677    -3.45   0.001    -.7150322   -.1969442
                         _cons |   -9.99609   2.402726    -4.16   0.000    -14.70535   -5.286834
------------------------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects parameters  |   Estimate   Std. err.     [95% conf. interval]
-----------------------------+------------------------------------------------
Countryid: Unstructured      |
                 var(crisis) |   .0132031   .0094471      .0032481    .0536692
                  var(_cons) |    .120477    .037722      .0652211    .2225461
           cov(crisis,_cons) |   .0103949   .0132997      -.015672    .0364617
-----------------------------+------------------------------------------------
Residual: AR(1)              |
                         rho |   .4009766   .1355596      .1077968    .6299944
                      var(e) |   .0263192   .0056278      .0173086    .0400205
------------------------------------------------------------------------------
LR test vs. linear model: chi2(4) = 280.25                Prob > chi2 = 0.0000

Note: LR test is conservative and provided only for reference.

. eststo ri

. mixed ln_Revenue CharismaticValuebased c.CharismaticValuebased#i.crisis i.crisis HHI GDPG ///
>                 Inflation Political ln_Assets ln_GDP CharismaticControls || Countryid: , ///
>                 residuals(ar 1, t(year)) 

Obtaining starting values by EM ...

Performing gradient-based optimization: 
Iteration 0:  Log likelihood =  39.646494  
Iteration 1:  Log likelihood =  39.646494  (not concave)
Iteration 2:  Log likelihood =  47.679019  
Iteration 3:  Log likelihood =  48.591585  
Iteration 4:  Log likelihood =  48.619727  
Iteration 5:  Log likelihood =   48.61981  
Iteration 6:  Log likelihood =   48.61981  

Computing standard errors ...

Mixed-effects ML regression                         Number of obs    =     199
Group variable: Countryid                           Number of groups =      25
                                                    Obs per group:
                                                                 min =       7
                                                                 avg =     8.0
                                                                 max =       8
                                                    Wald chi2(10)    = 1743.46
Log likelihood =   48.61981                         Prob > chi2      =  0.0000

------------------------------------------------------------------------------------------------
                    ln_Revenue | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------------------------+----------------------------------------------------------------
         CharismaticValuebased |    1.26116   .3489299     3.61   0.000     .5772698     1.94505
                               |
crisis#c.CharismaticValuebased |
                            1  |   .0049884   .0912919     0.05   0.956    -.1739404    .1839172
                               |
                      1.crisis |  -.4319053   .5281149    -0.82   0.413    -1.466991    .6031808
                           HHI |    .000043   .0000515     0.84   0.403    -.0000578    .0001439
                          GDPG |   .0120583   .0029826     4.04   0.000     .0062126    .0179041
                     Inflation |    .039591   .0049799     7.95   0.000     .0298306    .0493515
                     Political |   .0349166   .0781253     0.45   0.655    -.1182062    .1880393
                     ln_Assets |   .7891284   .0261221    30.21   0.000       .73793    .8403267
                        ln_GDP |    .202211   .0589909     3.43   0.001     .0865909    .3178311
           CharismaticControls |  -.4968404   .1365651    -3.64   0.000    -.7645031   -.2291778
                         _cons |  -10.48755   2.491459    -4.21   0.000    -15.37072   -5.604378
------------------------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects parameters  |   Estimate   Std. err.     [95% conf. interval]
-----------------------------+------------------------------------------------
Countryid: Identity          |
                  var(_cons) |    .129686   .0401278       .070715    .2378343
-----------------------------+------------------------------------------------
Residual: AR(1)              |
                         rho |   .4436219   .1062349      .2141314    .6267011
                      var(e) |   .0305616   .0055765      .0213727    .0437013
------------------------------------------------------------------------------
LR test vs. linear model: chi2(2) = 275.70                Prob > chi2 = 0.0000

Note: LR test is conservative and provided only for reference.

. eststo rc

. lrtest rc ri, stats

Likelihood-ratio test
Assumption: rc nested within ri

 LR chi2(2) =   4.55
Prob > chi2 = 0.1026

Note: The reported degrees of freedom assumes the null hypothesis is not on the boundary of the parameter space. If this is not true, then
      the reported test is conservative.

Akaike's information criterion and Bayesian information criterion

-----------------------------------------------------------------------------
       Model |          N   ll(null)  ll(model)      df        AIC        BIC
-------------+---------------------------------------------------------------
          rc |        199          .   48.61981      14  -69.23962  -23.13335
          ri |        199          .   50.89702      16  -69.79403  -17.10116
-----------------------------------------------------------------------------
Note: BIC uses N = number of observations. See [R] IC note.

. 
end of do-file

Given this, if I understand correctly, it would indicate according to the test I should be using the simpler model without the specified covariance structure given the non-signifcant result of prob>chi2 = 0.1026. Now the thing is, I am not sure if I can rely on this result given the fact that I had to remove my robustness options which I am certain of I need to include given my present heteroskedasticity. Given this, I am tempted to disregard the test and still continue with the cov(un) options included, how would you interpret these results given my constraints?

[Erik Ruzek]

If testing a cross-level interaction is an important part of your analysis and you want to draw conclusions from this interaction, then I would keep the random slope. The evidence is pretty strong that abandoning it leads to problematic tests of the interaction coefficient.

[Jesse Nooijen]

If testing a cross-level interaction is an important part of your analysis and you want to draw conclusions from this interaction, then I would keep the random slope. The evidence is pretty strong that abandoning it leads to problematic tests of the interaction coefficient.

The cross-level interaciton is needed to answer my hypotheses, so I will keep them. Thanks so much for clarifying and helping me. My deadline is next week and I am confident I can hand in my work with the proper methodologies. I actually learned quite a bit regarding stata and statistics throughout this process, which is very much appreciated and mostly due to the responses here.

[Jesse Nooijen]

Hey I am actually back with a few last questions. I am trying to understand the interpretation of the mixed effects model, but after my variables it specifies several constant variables. If I understand correctly, the Constant variable at the end of my output (see below) presents the expected value of the dependent variable when all independent variables are set to zero. So it's the point where the regression line crosses the y-axis in a simple linear regression. Given this, I am rather confused as to what the other constant variables are indicative of. I tried looking at other papers that used mixed effects model, but I couldn't find them reporting it similarly and clearly. The code I ran is;

 mixed ln_Revenue CharismaticValuebased c.CharismaticValuebased#i.crisis i.crisis HHI GDPG ///
Inflation Political ln_Assets ln_GDP CharismaticControls || Countryid: crisis , cov(un) ///
residuals(ar 1, t(year)) vce(robust)
est store CharRev
estat ic
//charismatic current
mixed ln_Current CharismaticValuebased c.CharismaticValuebased#i.crisis i.crisis HHI GDPG ///
Inflation Political ln_Assets ln_GDP CharismaticControls || Countryid: crisis , cov(un) ///
residuals(ar 1, t(year)) vce(robust)
est store CharCur
estat ic

//TeamOriented\\
//TeamOriented Revenue
mixed ln_Revenue TeamOriented c.TeamOriented#i.crisis i.crisis HHI GDPG ///
Inflation Political ln_Assets ln_GDP TeamControls || Countryid: crisis , cov(un) ///
residuals(ar 1, t(year)) vce(robust)
est store TeamRev
estat ic
//Team Current
mixed ln_Current TeamOriented c.TeamOriented#i.crisis i.crisis HHI GDPG ///
Inflation Political ln_Assets ln_GDP TeamControls || Countryid: crisis , cov(un) ///
residuals(ar 1, t(year)) vce(robust)
est store TeamCur
estat ic

//Participative\\
//Participative Revenue
mixed ln_Revenue Participative c.Participative#i.crisis i.crisis HHI GDPG ///
Inflation Political ln_Assets ln_GDP ParticipativeControls || Countryid: crisis , cov(un) ///
residuals(ar 1, t(year)) vce(robust)
est store PartRev
estat ic
//Participative Current
mixed ln_Current Participative c.Participative#i.crisis i.crisis HHI GDPG ///
Inflation Political ln_Assets ln_GDP ParticipativeControls || Countryid: crisis , cov(un) ///
residuals(ar 1, t(year)) vce(robust)
est store PartCur
estat ic

esttab CharRev CharCur TeamRev TeamCur PartRev PartCur using "C:\Users\jesse\Documents\Synology Drive Documents\Tilburg University\Master International Management\Thesis\Draft Output\FinalRegResults.rtf", compress cells(b(star fmt(3)) se(par fmt(3))) label noobs nogap nonotes title("Regression Results") mtitle("Model 1" "Model 2" "Model 3" "Model 4" "Model 5" "Model 6") legend replace

Which gives the following regression table (I modified it for my thesis to fit on a page):

I was wondering if anyone would be willing to explain the other constant variables and what their interpretation is, or if anyone knows a relevant paper I can read into regarding this. Thanks!

[Erik Ruzek]

Those are the random effect parameters, including the rho (ar1) estimate. Check out this thread, Jesse. Short story, when esttab grabs the results from the mixed models it does so in the form of how the estimates are stored in e(b). See also the estout/esttab documentation for examples. If you are not sure which estimates are which, you can add the estmetric option at the end of your mixed output and then go back and forth to figure it out.

[Jesse Nooijen]

Those are the random effect parameters, including the rho (ar1) estimate. Check out this thread, Jesse. Short story, when esttab grabs the results from the mixed models it does so in the form of how the estimates are stored in e(b). See also the estout/esttab documentation for examples. If you are not sure which estimates are which, you can add the estmetric option at the end of your mixed output and then go back and forth to figure it out.

Thanks for clarifying, I quickly looked through the provided sources and it looks very insightful. Will be the first thing I study tomorrow, thanks!