Interpretation of Rho

[Georgina Milliot]

Hi Statalisters, I have a question regarding the interpretation of "Rho". I understand that Stata defines it as the variance of the sigmas. However, what does that mean for my model? For some background, I tested my FE model for autocorrelation and heteroskedasticity, which both results turned out positive. In efforts to address these, I used the cluster-robust standard errors and Driscoll-Kraay. The cluster-robust standard errors generated a high Rho of 95 %. I noticed the standard errors of Driscoll Kraay are also much lower than the cluster-robust. Can anyone please explain what the Rho means for my model? Or should I focus more on the standard errors between the two estimations?

Thank you!!

Fixed-effects (within) regression               Number of obs      =       259
Group variable: unit_id                         Number of groups   =        27
 
R-sq:  within  = 0.2539                         Obs per group: min =         1
       between = 0.3563                                        avg =       9.6
       overall = 0.4015                                        max =        14
 
                                                F(11,26)           =         .
corr(u_i, Xb)  = 0.3156                         Prob > F           =         .
 
                                (Std. Err. adjusted for 27 clusters in unit_id)
-------------------------------------------------------------------------------
              |               Robust
     econ_gov |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
      lag_aid |  -.0000726   .0000995    -0.73   0.473    -.0002772     .000132
   lag_aid_sq |   2.10e-08   2.29e-08     0.92   0.369    -2.61e-08    6.80e-08
  lag_aid_gdp |    .714177   .5446976     1.31   0.201     -.405465    1.833819
lag_aid_gdp~q |  -.5419295   .4934509    -1.10   0.282    -1.556232    .4723733
    gdp_grwpc |   .0014575    .002655     0.55   0.588    -.0039999     .006915
     conflict |   .0005238   .0032868     0.16   0.875    -.0062324    .0072799
        trade |  -.0012793   .0008535    -1.50   0.146    -.0030336     .000475
   state_frag |  -.0090594   .0105918    -0.86   0.400    -.0308311    .0127123
      tax_rev |  -.0025985   .0044017    -0.59   0.560    -.0116464    .0064494
      log_pop |  -.0772481   .2916359    -0.26   0.793    -.6767144    .5222181
      pop_urb |   -.004556   .0095538    -0.48   0.637    -.0241941     .015082
    inst_qual |   .0231438   .0044136     5.24   0.000     .0140715    .0322161
        _cons |  -.1813826   4.684948    -0.04   0.969     -9.81143    9.448665
--------------+----------------------------------------------------------------
      sigma_u |  .46398543
      sigma_e |    .101959
          rho |  .95393599   (fraction of variance due to u_i)
-------------------------------------------------------------------------------
 
Regression with Driscoll-Kraay standard errors   Number of obs     =       259
Method: Fixed-effects regression                 Number of groups  =        27
Group variable (i): unit_id                      F( 12,    13)     =    919.12
maximum lag: 2                                   Prob > F          =    0.0000
                                                 within R-squared  =    0.2539
 
-------------------------------------------------------------------------------
              |             Drisc/Kraay
     econ_gov |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
      lag_aid |  -.0000726   .0000847    -0.86   0.407    -.0002554    .0001103
   lag_aid_sq |   2.10e-08   1.46e-08     1.44   0.174    -1.05e-08    5.24e-08
  lag_aid_gdp |    .714177   .4087988     1.75   0.104    -.1689792    1.597333
lag_aid_gdp~q |  -.5419295   .3156461    -1.72   0.110    -1.223841    .1399824
    gdp_grwpc |   .0014575   .0019099     0.76   0.459    -.0026686    .0055837
     conflict |   .0005238   .0044359     0.12   0.908    -.0090595    .0101071
        trade |  -.0012793   .0003259    -3.93   0.002    -.0019834   -.0005752
   state_frag |  -.0090594   .0059308    -1.53   0.151    -.0218722    .0037535
      tax_rev |  -.0025985   .0032915    -0.79   0.444    -.0097093    .0045123
      log_pop |  -.0772481   .1913638    -0.40   0.693    -.4906646    .3361683
      pop_urb |   -.004556   .0054977    -0.83   0.422    -.0164331     .007321
    inst_qual |   .0231438   .0033416     6.93   0.000     .0159247     .030363
        _cons |  -.1813826   2.824954    -0.06   0.950    -6.284325     5.92156
-------------------------------------------------------------------------------

[Carlo Lazzaro]

Georgina:
as far as the interpretation of -rho- is concerned, it is better read as the following ratio:

e(sigma_u)^2/(e(sigma_u)^2+e(sigma_e)^2)

That said, -rho- (or intraclass correlation) gives you the proportion of the variation in the dependent variable that can be explained by differences between u_i.
As an aside, I would not be particularly concerned about which regression model to perform, as they give back similar results.

[Georgina Milliot]

Dear Carlo,

Thank you for your response and the information on Rho. Regarding the two regression models, does it make a difference that Driscoll-Kraay generates lower standard errors? I was under the assumption that due to this fact, the DK model was preferred over the cluster-robust SE.

Best regards,
Georgina

[Carlo Lazzaro]

Georgina:
it is true that having lagged predictors can justify DK (lower) SEs. However, I do not see any relevant difference in the precision of the coefficients estimates across the two models.

[Georgina Milliot]

Dear Carlo,

Thank you again for further clarification and your response!

Best regards,
Georgina

[Carlo Lazzaro]

Georgina:
you're welcome!

[Alfonso Sánchez-Peñalver]

On a totally different matter, just some advice on using factor variables. I see that you enter two quadratic relationships with lag_aid and lag_aid_gp. The parameters in those variables seem insignificant, but if you were to want to estimate the marginal effects on each of the variables later it would be more convenient if you entered c.lag_aid##c.lag_aid and c.lag_aid_gp##c.lag_aid_gp, instead of entering both the level and squared terms for each variable. If you were to use margins, dydx(lag_aid) after, for example, you would get the right marginal effect on the variable.