I have a set of variables $\displaystyle x_1, x_2, \ldots , x_n$ which I calculate from 100 samples. For each sample, I get a score, $\displaystyle y_1, y_2, \ldots , y_n$, where $\displaystyle y_i = w_1 x_{1i} + w_2 x_{2i} + \ldots w_n x_{ni}$, where the w's are subjective relative weights I have determined for each x in the i-th sample.

Now, given the subjective weights (fixed for all 100 samples) and the fixed x-values which I get from some calculations, I have a matrix of 100 rows, where each row comprises y (the output) as well as the $\displaystyle n$ dependent variables. Given this matrix, I do multiple regression and test for collinearity to see which x's cause multicollinearity and, thus, are to be removed from the final equation.

I have only one question: is there anything wrong with my approach? That is, conventionally, you have the y-values and x-values and you do multiple regression to get the equation. In my case, however, I do NOT have the y-values, but just the x-values. Then, I assign some weights, get the y-values, and do a multiple regression and collinearity test just to get other weights in the final equation for the significant variables. Is this right?