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I face the following simple regression model: $$ \widehat{rdexp}=\hat{\beta}_{0}+\hat{\beta}_{1}log(sales) $$ where $$ \widehat{rdexp}=\frac{RD}{sales} $$ and $RD$ is some positive number.

I am having a hard time coming up with a null hypothesis for $\hat{\beta}_{1}$. I know that the $H_{0}$ value shall be negative, as the increase in $log(sales)$, will impact the $\widehat{rdexp}$ negatively via the denominator. I suspect that the $\hat{\beta}_{1}$ under $H_{0}$ shall be equal to -1, because this is the result I obtain when I take the natural logarithm of $\widehat{rdexp}$ and differentiate it wrt. $log(sales)$. Is this approach (and by extension the result) correct?

borninthenorth
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  • I suspect you would have better luck with [a Poisson model with a logarithmic offset](https://stats.stackexchange.com/questions/52352/poisson-regression-dealing-with-widely-different-population-sizes/52364#52364). You can also test the proportionality assumption. Having a variable both not eh right and the left seems problematic to me. – dimitriy Apr 07 '21 at 21:45
  • Unfortunately that's the model I was given. – borninthenorth Apr 08 '21 at 05:31

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