Continuous independent variables that sum up to 100% As I am studying the capital structure of firms, the total of proportions owned by different ownership types (free float, institutional ownership ... etc.) will be 100% of the company shares. In my study each of those ownership types is an independent variable. Hence, when running regressions, one of these variables will be dropped due to collinearity.
Is there any transformations that can be done to these variables in order to have all of them in the model? If not, which variable (highest/ lowest correlated with the dependent variable) is to be dropped?
 A: What follows is a possible approach.   Suppose you have independent variables $x_i,y_i,z_i$ with $x_i+y_i+z_i=1$ and you are trying to predict the dependent variable $w_i$ by linear regression.
Drop one of the independent variables (perhaps the $x_i$s, and I would choose the one with the greatest variance) and get a result of the form $\hat w_i=\beta_0+\beta_y y_i+\beta_z z_i$
You could use this directly, or you could reintroduce $x_i$ so the coefficients of the independent variables add up to something, and I would choose $0$, so of the form  $\hat w_i=\gamma_0+\gamma_x x_i+\gamma_y y_i+\gamma_z z_i$  where

*

*$\gamma_x = \frac13(1-\beta_y-\beta_z)$

*$\gamma_y =\beta_y+\gamma_x$

*$\gamma_z =\beta_z+\gamma_x$

*$\gamma_0 =\beta_0-\gamma_x$
You would than have to be careful interpreting $\hat w_i=\gamma_0+\gamma_x x_i+\gamma_y y_i+\gamma_z z_i$, but the way to read it is to say that, for example, an increase in $y$ and equal reduction in $z$ has an effect on $\hat w$ of $(\gamma_y-\gamma_z)$ times the size of the change
A: A "traditional" transformation for $n$ variables that must add up to a constant (usually we use 1 corresponding to 100%), is to work with $n-1$ parameters $\alpha_i \in (-\infty, \infty)$, to define the $n$th one as $\alpha_n = 1 - \sum_{i=1}^n \alpha_i$ ("sum to zero constraint") and to use the softmax function to define
$$w_i = \frac{e^{\alpha_i}}{\sum_{i=1}^n e^{\alpha_i}}.$$
We will then have $\sum_{i=1}^n w_i =1$ and all $w_i$ will be in $(0,1)$.
The $\alpha_i$ for $i=1,\ldots,n-1$ could then each have a set of linear regression terms behind them (or some other kind of model, e.g. if you like a neural network or whatever else).
