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Say I have some data which are sets of variables: (y, x_1, x_2, x_3, x_4), and y = ax_1 + bx_3 + error, and x_2 = cx_1 + error, x_4 = dx_1 + e*x_3 + error.

Then I do a linear regression and find the best values for y = w1*x_1 + w2*x_2 + w3*x_3 + w4*x_4 and the correlations between them.

Can I recover a,b,c,d,e from the w_i (EDIT: all of them, not just a and b)? Or do I need to do PCA or some other technique?

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"Say I have some data which are sets of variables: (y, x_1, x_2, x_3, x_4), and y = ax_1 + bx_3 + error, and x_2 = cx_1 + error, x_4 = dx_1 + e*x_3 + error."

If the above implies you are generating your dataset as stated in the above equations, then YES using linear regression you should be able to find the values of a,b,c ... by using regression.

For example, when you perform y = w1*x_1 + w2*x_2 + w3*x_3 + w4*x_4 , you should find that w1=a,w3=b, w2=0,w4=0.

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  • $\begingroup$ Yes, but can I recover a,b,c,d,e, not just a and b? $\endgroup$ – David Gregory Oct 20 '16 at 10:21
  • $\begingroup$ You would have to regress with x_2 as your target to recover c, and x_4 to recover d and e. In practice however you might run into problems with multicollinearity since your independent variables are actually not independent but derived from each other. $\endgroup$ – Arun Jose Oct 20 '16 at 10:37

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