Model/link function to deal with dependent variable in range [-1,1]? My dependent variable, $Y$, contains values anywhere from -1 to 1 (i.e. it is bounded continuously on the range $[-1,1]$). I know that a regular OLS regression on such a variable would sometimes create predictions that would go under -1, or above 1, and I immediately thought a link function could be of use here (as it is in the case of logistic regression with binary variables), but it seems no link functions specifically constrain the predicted output to be in the aforementioned range.
Neither Python's statsmodels', not even wikipedia, mention a link function that could help out in this situation.
What would be the way to approach such a modelling problem? If a link function does not exist for this specific case, are there other ways to handle such a dependent variable?
Thank you ahead of time for any advice.
 A: I am going to assume you want a link function on the open interval $(-1,1)$, since you want this to map continuously onto the real numbers.  There are infinitely many monotonic link functions $g: (-1,1) \rightarrow \mathbb{R}$ that map from your desired domain to the set of all real numbers.  Taking the function to be strictly increasing, every link function of this form can be characterised in terms of a strictly increasing CDF $F$ for a random variable with support given by the set of real numbers.  Specifically, you have:
$$g(\mu) = 2 ( F^{-1}(\mu) - \tfrac{1}{2} ).$$
Inverting this relationship gives the mean function:
$$\mu = F \bigg( \frac{1}{2} \big( 1+g(\mu) \big) \bigg) = \mathbf{X} \boldsymbol{\beta}.$$
Since there are an infinite number of strictly increasing CDFs on the reals there are an infinite number of link functions of this form.  Some simple examples that are sometimes used in GLMs are the following:
$$\begin{equation} \begin{aligned}
g(\mu) &= \frac{2}{\pi} \cdot \tan (\mu), \\[10pt]
g(\mu) &= \text{artanh} (\mu), \\[10pt]
g(\mu) &= 2 ( \Phi (\mu) - \tfrac{1}{2} ). \\[10pt]
\end{aligned} \end{equation}$$
