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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!

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    That product expands to 1+all possible combinations using $1,2,\dots,n$ variables. Are you fitting it with some software like R? Considering logarithms I guess you could use $ \log(1+x) = x - \frac{x^2}2 + \frac{x^3}3 + {O}(x^4)\! $2012-08-07
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    I'm trying to fit with MATLAB - I think the Taylor series method is promising, although $\log(1 + cx) = cx - c^2x^2/2 + c^3x^3/3 + O(cx^4)$, and the powers of $c$ get in the way.2012-08-07
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    It is not possible so select comment as the answer, but in these situations you might express your wish and commentators usually repost their comments as answers. Is it my comment you are talking about? Edit: I will post full answer soon.2012-08-07
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    If you do a polynomial approximation to the log where do you truncate and since the model is nonlinear in the coefficients there are issues of existence and uniqueness of solutions with this approach as well?2012-08-07

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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$_3$.

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    How can you estimate $c_1$,$c_2$ and $c_3$ then without estimating $d=c_1c_2c_3$ as well?2012-08-07
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    But since the coefficients on the interaction terms are related, is there an algorithmic way to determine these coefficients?2012-08-07
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    @K.Hu You can certainly form the likelihood equations and attempt to maximize it with respect to the parameters. But it becomes complicated in terms of solutions as it is not simply a system of k linear equations in k unknowns. So there would be issues of existence and uniqueness of solutions.2012-08-07