Multiple regression with model $Y = (1 + c_1X_1)(1 + c_2X_2)\ldots(1 + c_nX_n)$

Question

I'm currently working with data contained in $Y, X_1, X_2, \ldots, X_n$ and wish to fit it to the model:

$Y = (1 + c_1X_1)(1 + c_2X_2)\ldots(1 + c_nX_n)$

where the $c_i$ are coefficients to be determined through regression. How can I do this? I've tried playing with logarithms (i.e. looking at $Y$ vs. $\log (1 + c_iX_i)$), but due to the pesky $1$, I can't figure out how to follow through.

Thanks in advance!

Answer 1

If you multiple out you will find it has a standard linear regression form except that it includes many high order interaction terms with related coefficients. To see this note that

Y=(1+c$_1$X$_1$)(1+c$_2$X$_2$)(1+c$_3$X$_3$)=

1+c$_1$X$_1$+c$_2$X$_2$+c$_3$X$_3$ +c$_1$c$_2$X$_1$X$_2$+c$_1$c$_3$X$_1$X$_3$+c$_2$c$_3$X$_2$X$_3$+c$_1$c$_2$c$_3$X$_1$X$_2$X$_3$.