Suppose the true relationship is

[y=f(x_1,…,x_k)]

with (x_1,…,x_k) factors explaining the (y). Then the first order Taylor approximation of (f) around zero is:

[f(x_1,…,x_k)=f(0,…,0)+sum_{i=1}^{{k}frac{partial} f(0)}{partial x_k}x_k+varepsilon,]

where (varepsilon) is the approximation error. Now denote (alpha_0=f(0,…,0)) and (alpha_k=frac{partial{f}(0)}{partial x_k}) and you have a regression:

[y=alpha_0+alpha_1 x_1+…+alpha_k x_k + varepsilon]