The choice between fractional logit regression or Tobit regression

[Fimi Karimi]

Hello everyone,
I faced some questions regarding my research, and I would really appreciate if you could assist me with. It's my first research experience of research and I'm new to econometrics. I have panel data with the percentage of women on corporate boards across different countries as dependent variable. This ratio ranges from 0.7% to 49.8% globally. My data doesn't include 0 or 100% but I used Tobit regression and censored data between 0 to 100. I followed the literature in BGD which were firm-level studies that used Tobit. For firm level studies, it makes sense to use Tobit since some firms have no women on their boards, but at the country level, it's rare (Although some countries like Qatar, Saudi Arabia, and Kuwait only report their women on boards for certain recent years, I can't set the ratio to zero for years when no data is reported). One of the reviewers on my research mentioned that Tobit might not be suitable for my data and suggested I use fractional logit instead. He asked me to justify my choice of method or use fractional logit for robustness testing and wanted to know why I didn't use country fixed effects. I’d really appreciate your help with following questions:

1. I tried the tests with OLS and got almost the same results, but I’m wondering: is it completely wrong to use Tobit for this data? I only used year fixed effects with Tobit because when I tried country fixed effects, some of my control variables showed unexpected signs or high values.
2. When I re-ran the tests with fractional logit, the results confirmed what I got with Tobit and OLS. However, when I applied fixed effects, I got unexpected signs or insignificant results for my main explanatory variables. Is it correct to use country or time fixed effects with fractional logit?
3. For my robustness test, I constructed PCA from four of my independent variables. I’m not sure if I can use the residuals from PCA into bit or fractional logit. I think OLS works better for that, but I’m also unsure if I can use fixed effects with PCA.
4. Regarding the issue of endogeneity, I used IV regression. I think I can't apply IV to fractional logit, right?

Best, Fimi

[Jeff Wooldridge]

There are a few somewhat subtle issues here. First, if y never hits zero or one, then the coefficient estimates from the two-limit Tobit are identical to the OLS estimates. The subtle part is that the conditional mean function derived from the Tobit with corners at zero and one is not the same thing as x*b. The proper formula can be obtained but using the predict command after the Tobit estimation.

Fractional response uses a different functional form for the mean, and it is a bit easier to manipulate. You can use logit or probit. A benefit is that it doesn't require a full distributional assumption (unlike Tobit). Fractional logit or probit model E(y|x) for 0 <= y <= 1.

Neither approach works well when you include unit (country) fixed effects due to the incidental parameters problem. You can use the correlated random effects approach for fractional regression or Tobit. In my 2010 MIT Press book I describe these approach. Or, Papke and Wooldridge (2008, Journal of Econometrics) shows the fractional probit case. You just have to include the time averages of the time-varying explanatory variables, run a pooled frac logit or probit, and use vce(cluster country).

[Fimi Karimi]

Thank you so much for your detailed explanations, Professor. Your comments clarified a lot for me.