Beta regressione: parameters interpretation

[Leonardo Alaimo]

Hi all.
I contact you because I have a problem with the interpretation of a beta regression.

i read different work about the regression with a dependent variable bounded between 0 and 1.
I use three different models to my data:

1. Linear Model FGLS
2. Logit Model (in this case, I trasform my dependent variable in a dummy variable)
3. Beta regression model.

All the models used are a good fitting to data, but I think that the best one is the beta regression model.
My problem is that I don't understand how I have to interpret the coefficient of the output of betareg Stata command and how to use post estimation commands.
I'm a new Stata user and I became to study non-linear models recently.

[Leonardo Alaimo]

Thanks all

leonardo

[Maarten Buis]

You probably misunderstood advise for solution 2. I suspect what model is supposed to be is a fractional logit. In that case you just keep the fractional dependent variable unchanged and use glm to estimate a quasi-likelihood model with a logit link function and the vce(robust) option.

Lets consider the example below:

. clear all

. use http://fmwww.bc.edu/repec/bocode/k/k401.dta
(source: Papke and Wooldridge 1996)

. gen totemp100 = totemp/100

. glm prate mrate totemp100 age sole, ///
>     link(logit) family(binomial) vce(robust) eform nolog
note: prate has noninteger values

Generalized linear models                         No. of obs      =      4,734
Optimization     : ML                             Residual df     =      4,729
                                                  Scale parameter =          1
Deviance         =  1023.737134                   (1/df) Deviance =   .2164807
Pearson          =  1377.971352                   (1/df) Pearson  =   .2913875

Variance function: V(u) = u*(1-u/1)               [Binomial]
Link function    : g(u) = ln(u/(1-u))             [Logit]

                                                  AIC             =   .5794217
Log pseudolikelihood = -1366.491144               BIC             =  -38995.55

------------------------------------------------------------------------------
             |               Robust
       prate | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       mrate |   1.774365   .1418167     7.17   0.000     1.517087    2.075274
   totemp100 |   .9994222   .0001146    -5.04   0.000     .9991976    .9996468
         age |   1.031377   .0028755    11.08   0.000     1.025756    1.037028
        sole |   1.438494   .0684727     7.64   0.000      1.31036    1.579157
       _cons |   2.927246   .1431645    21.96   0.000     2.659678    3.221733
------------------------------------------------------------------------------

I used the eform option to help in the interpretation. The variable prate (participation rate) is the number of employees that participate in a pension plan divided by the total number of employees in a firm. The constant is the baseline odds or relative proportion. It says that when all explanatory variables are 0 we expect to find 2.9 employees that participate in a pension plan for every employee that does not participate. The variable mrate (match rate) is the rate at which the employer matches the contribution of the employee. If that match rate increases by 1, then we expect the odds or relative proportion to increase by a factor 1.77 or (1.77-1)*100%=77%. If the firm gets a 100 employees larger then we expect the relative proportion to decrease by a factor 0.9994 or (0.9994 - 1)*100% = -0.06%.

Alternatively you can look at the average marginal effects:

. margins, dydx(*)

Average marginal effects                        Number of obs     =      4,734
Model VCE    : Robust

Expression   : Predicted mean prate, predict()
dy/dx w.r.t. : mrate totemp100 age sole

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       mrate |   .0632799   .0086066     7.35   0.000     .0464113    .0801485
   totemp100 |  -.0000638   .0000127    -5.01   0.000    -.0000887   -.0000388
         age |   .0034092   .0003149    10.83   0.000      .002792    .0040265
        sole |   .0401232   .0052106     7.70   0.000     .0299106    .0503357
------------------------------------------------------------------------------

You can see that the proportion of employees that participate increases by 0.06 when the match rate increases by 1, etc.

[Leonardo Alaimo]

Dear Maarten,
Thanks for your answer.
My question is if I use the command fracreg of Stata 14 I obtain the same results of glm.

Thanks
Leonardo

[Maarten Buis]

That is as it should be; they are the exact same model.

I have been working with fractional logit models long before fracreg existed but not so much recently, so the glm way of estimating is still most salient to me. That is the only reason why I used glm for the estimation in the example, and you should feel perfectly free to use fracreg instead.