[John Mullahy]

For Q2 I would simply note that these are two different exercises. In essence, in one case (log or logit transformed d.v.) you are estimating a model for E[t(y)|x], in the other (fracreg) you are estimating a model for E[y|x]. These are not the same models, and the degree to which their predictions, estimated marginal effects, and estimates' significance diverge depends on the nature of the transformation and the specifics of your data. To me, the issue comes down to considering the phenomenon whose properties you are trying to understand, and recognizing that t(y) and y are different phenomena. In my experience this point is so fundamental that it is often un- or under-appreciated.

[Prateek Bedi]

Ok John Sir. In light of your argument, if I am interested in E[y|x], should I avoid a logit transformation? Further, even if I go for a logit transformation, how can I comment on the relationship (sign and significance of the coefficients) of my independent variables with the original (not transformed) dependent variable?

[Neeraj Jain]

I think you are more interested in E(y|x), sign and significance of estimated coefficients, than predicted values. If this is the case then you should strictly avoid log-odd transformation (which I think you are referring as logit transformation) of your fractional response variable. As pointed by Wooldridge (2010):

“even if y is strictly in the unit interval, b (beta) is difficult to interpret: without further assumptions, it is not possible to estimate E(y|x) from a model for E(log[y/(1-y)]|x)”

An alternative is you can directly specify E(y|x) that take predicted value in [0,1] (Fractional regression approach). You can use logit function, probit function or any other mathematical function that ensure predicted values are in [0,1]. It should be noted that log-odd transformation and logit function for E(y|x) are different method (you might find them similar). Now, your model is non-linear, therefore you can not interpret your beta coefficients directly (as you do in linear model), but you have to consider average partial effects (APE). Since, this method does not transform response variable, therefore you do not need to worry about change in sign and significance of coefficients (concern raised in point 1 of your post).

If you use either logit or probit function for E(y|x), then both estimation and interpretation become easier (Wooldridge, 2010; p.n. 750). For estimation, you can use QMLE fractional logit/probit regression (see Papke and Wooldridge, 1996). Further, as pointed by Wooldridge (2010, p.n. 752);

“Further, if we are given data on proportions but do not know ni, it makes sense to use a fractional logit or probit analysis. If we observe the ni, we might use a binomial regression model instead.”

I have no idea about how to implement panel fractional regression in STATA. For more detail, you can read Chapter 15 and Chapter 18 from Wooldridge (2010). Also read Papke and Wooldridge (2008) for implementation of fractional regression for panel data.

[Prateek Bedi]

Thanks a lot Neeraj Jain for your valuable inputs.