Consider a standard "log-log" linear regression model like this:

$\log(y_i) = \log(a_i + b_i)\delta + \epsilon_i$,

where $y$ is the dependent variable, $a$ and $b$ are two independent variables, and $\epsilon$ is an i.i.d. error term. Assume that $a$ and $b$ are uncorrelated with each other, and all variables are indexed by observations at the level of $i$. $\delta$ is the parameter of interest -- a regression coefficient to be estimated.

Here, the parameter $\delta$ is an approximate elasticity; the percent change in $y$ given a percent change in $(a+b)$. Formally, $\delta \equiv \frac{\partial\log(y)}{\partial\log(a+b)} \approx \frac{\partial y}{y} \frac{(a+b)}{\partial \,(a+b)}$

But, what if we already knew the elasticity for one of the independent variables (i.e., the elasticity of $y$ w.r.t $b$), and wanted to use that to solve for $\delta$.

That is to say, if given the following definition of $\gamma$, the value of which is known:

$\gamma \equiv \frac{\partial\log(y)}{\partial\log(b)} \approx \frac{\partial y}{y} \frac{b}{\partial b}$

Can we define the parameter of interest, $\delta$, as a function of $\gamma$ (and, perhaps, $y, a,$ or $b$)?

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    $\begingroup$ I don't recognize this as Poisson regression: could you explain what $a_i,$ $b_i,$ and $\delta$ refer to? In particular, you appear to use "$b_i$" both as a variable and a parameter. $\endgroup$ – whuber Dec 30 '19 at 23:08
  • $\begingroup$ I am following the notation of a Poisson regression such as shown here: link. I changed the names of the variables and parameters slightly to clarify what is what. English letters are variables (a,b,y) , greek letters are parameters (delta, gamma). $\endgroup$ – km5041 Dec 31 '19 at 0:07
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    $\begingroup$ You have not followed/adapted the model at the link correctly. $\endgroup$ – Glen_b Dec 31 '19 at 12:02
  • $\begingroup$ Because it seemed to be causing some confusion, I removed the Poisson reference since it is irrelevant to what I am interested in -- defining the $\delta$ parameter as a function of the $\gamma$ parameter. $\endgroup$ – km5041 Dec 31 '19 at 13:22
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    $\begingroup$ Maybe you should remove the poisson-regression tag, and write a more informative title? $\endgroup$ – kjetil b halvorsen Dec 31 '19 at 14:22

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