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Say I have a multivariate model which I expect to have linear fit but with a relative (not absolute) error term:

$ y = \beta X + \epsilon y$

and for which the distributions of the $X$s and $y$ are all approximately exponential.

Is it appropriate to model this data with a Poisson regression? Or if not, what are my alternatives? Use a linear regression but weight each data point by $1/y$?

UPDATE

The dependent variable is total traffic flow at various points on a network. The independents are predicted traffic flow from different submodels (e.g. home-business, business-business, short range traffic, long range traffic etc). So I literally expect the independent to be a weighted sum of the dependent variables, but the distributions are exponential, and error terms higher for higher flows.

Sideshow Bob
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  • A logarithmic distribution is a particular discrete distribution (e.g. https://en.wikipedia.org/wiki/Logarithmic_distribution). Do you mean lognormal? – Nick Cox Dec 17 '15 at 17:59
  • Oops my bad. I meant exponential. Corrected – Sideshow Bob Dec 17 '15 at 18:05
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    I guess you can, but do you want to? See my answer here: http://stats.stackexchange.com/questions/142338/goodness-of-fit-and-which-model-to-choose-linear-regression-or-poisson/142353#142353 – kjetil b halvorsen Dec 17 '15 at 18:38
  • I'm not managing to see the relevance of that answer to this problem. Have updated my question above with details on the data set. – Sideshow Bob Dec 18 '15 at 10:09
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    Question now seems contradictory. Title says Poisson, text says exponential. Do you mean exponential strict sense, or exponential family wide sense? – Nick Cox Dec 18 '15 at 11:43
  • In a broad sense. I have not formally tested the distributions. Most roads carry little traffic but a few roads carry a lot. Viewing a histogram, the distribution looks approximately exponential (if the mode is anywhere other than the minimum value it's hard to see it) but logically it's probably more Poisson like (the mode will not be at zero). – Sideshow Bob Dec 18 '15 at 14:55

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