An easy solution is to apply a logit or probit transformation to y (i.e. logit y is a real number).

This is Quite technical

When your outcome is a true proportion in (0, 1): go with beta regression.

Because beta regression is statistically principled, interpretable, and prevents the logical issues of unbounded predictions

Beta regression is appropriate when your dependent variable takes on values between 0 and 1. If your DV can only be either 0 or 1, then you should use logistic regression.

When the target variable y is continuous and bounded between 0 and 1, applying linear regression (i.e., y = X\beta + \varepsilon) poses limitations:

• Linear regression predictions are unbounded and can fall outside (0,1), making them inappropriate for strictly bounded outcomes.
• Even if linear regression assumptions are met, its structure doesn’t inherently constrain predictions to (0,1).
• This can lead to invalid or misleading results, especially near the boundaries.

A better approach is beta regression, which:

• Models y using the beta distribution, suitable for values strictly between 0 and 1.
• Uses a link function (e.g., logit) to ensure predicted means stay within (0,1).
• Provides more accurate modelling of both the mean and variance structure of the data.

In conclusion, for a continuous outcome in (0,1), beta regression is more appropriate and statistically robust than linear regression.