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I have seen many applications of beta regression when dependent variables are bounded between 0 and 1 (proportions, probabilities, etc.).

However, would beta regression be appropriate when my dependent variable is bounded between 0 and 1, but is neither a proportion nor a probability? For example, say that my dependent variable is an index of democracy (this is the strategy that V-Dem uses) that ranges from 0-1 (the values themselves are rarely ever 0 or 1 exactly). This is basically a continuous measure that has a hard lower and upper-bound. I could just as easily transpose scores to range between 0 and 100, for example, but the problem with hard cut-offs still applies (it would not be theoretically possible for a value to ever be <0 or >100).

With this situation, should one use OLS or beta regression? Or is there another sort of specialized tool for outcome variables such as this?

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Beta regression still makes perfect sense. It is in no way limited to, or even mainly motivated by, data that are probabilities or proportions. (If your data are "far" away from the theoretical limits and you do not plan on extrapolating, OLS might be a possibility - it is easier to explain and understand. But beta regression would absolutely be the first thing that comes to mind.)

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