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    $\begingroup$ Your question is slightly ambiguous. Are you only asking how a multiplicative model can be estimated, or are you asking how the causal effects can be isolated from these observational data? (Explicitly, it seems the former, but the story implies you want to provide causal information to the Duke to convince him.) Furthermore, in the true data generating process, where is the error? Is it multiplicative as well, or not (see: How to tell the difference between linear and non-linear regression models?)? $\endgroup$ Commented Sep 16, 2019 at 16:21
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    $\begingroup$ This is not a multiplicative model at all: upon taking logarithms, it's a standard additive model with Normal errors. $\endgroup$ Commented Sep 16, 2019 at 17:03
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    $\begingroup$ You shouldn't have any problems with the deity. As @whuber correctly notes, you take logs and you have a bog-standard additive linear model. Given that that is true, & that the deity knows this well (by virtue of being omniscient), the deity will not give you any grief for providing the correct answer. $\endgroup$ Commented Sep 16, 2019 at 18:20
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    $\begingroup$ The analysis you perform at "worthless but misleading" appears to have nothing to do with your stated question, making the post confusing and occult. Could you restate what you mean by "crafted according to the art"? $\endgroup$ Commented Sep 18, 2019 at 17:50
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    $\begingroup$ I still have to read through your question but in any case +1 for Dune. $\endgroup$ Commented Sep 19, 2019 at 10:08