# Grouped Logit vs. Non-Linear Least Squares vs. Negative Binomial w/ Exposure?

I'm currently working on a project looking at the effect of competition on the type of appeals candidates make in their campaign advertisements.

My dataset consists of a list of candidates, the number of ads with some characteristic (ex: ads_prtymen, # of ads that mention party labels), the total number of ads aired (total_aired), and a proportion (prp_prtymen). The proportions does include 0 and 1 as candidates can air no ads with party labels or all their ads include party labels. The dataset does not include candidates who aired 0 ads, so zero-inflated models to model the 0s isn't necessary.

What I am stuck on at the moment is the best approach to estimating this. My advisor has been suggesting a grouped logit model, but digging into advice on my own I've gotten a few other suggestions.

For grouped logit, each candidate is a "group", with the total number of ads aired as the # of trials and the # of ads with party mentions the "successes". From what I could understand of implementing this STATA 14, that could would look something like this:

glm total_prty_men ip_margin , link(logit) family(binomial num_air) vce(cluster statdist_cen)


This deals with the boundedness of the model, but one other thing I was looking at was inclusion of fixed effects for congressional district. In my dataset, however, average times a district is included is about 2-3, which apparently does not produce good results in a logit model (I tested it with the dummy variable approach: many of the fixed effects could not get standard errors, for example).

However, when I asked about this problem elsewhere, a few other suggestions came up that I'm not as sure about how to implement or if appropriate:

1) My advisor also suggested non-liner least square regression. However, I've not done this before, so I don't know how to code it. STATA offers some common functions, and I'm presuming one of their logit ones would do, but I'm not sure which would be appropriate. It also has the benefit of allowing fixed effects for my dataset (if I upgrade to STATA 16).

2) Another person had suggested using a negative binomial regression, presumably using the num_air (number of ads aired) as an offset/exposure. Tried that in STATA, but it only runs if I set it as an exposure. I'm presuming that, like grouped logit, it would have issues if I ran a fixed effects model.

3) OLS, which is simple, can allow for fixed effects, but doesn't address the bounded issue. That might be something I just bite the bullet on (it'd take really extreme values of in-party vote for predicted effects to break the bounds), however, and perhaps leave the other models to an appendix for those really that are bothered by the other issues.

Edit: Earlier I had asked about fractional regression, but someone elsewhere informed me it is generally used if you don't know the denominator of your DV as it makes certain assumptions about the denominator. Since I have the denominator, a grouped logit is apparently preferable to fractional.