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I am trying to estimate $\lambda$ from this equation:

$(1+ t_t) = \left(\dfrac{C_t}{Y_t}\right)^{\frac{1-\lambda}{\lambda}}$

After taking logs and approximations, I get:

$t_t \approx \left( \frac{1-\lambda}{\lambda} \right)(\ln C_t - \ln Y_t) $

I can get data on the consumption (C) and GDP (Y), but I wasn't sure about what to do with the taxes (t) and asked my professor about it, and this is what I heard back:

" ... won't need to estimate the tax level. The tax is basically a proxy variable for capital market frictions. If you assume that it is a function of the difference between domestic consumption and domestic income (in logs) then you need only data on those variables, not on any tax rates. ..."

I was wondering whether you would happen to know of any techniques that I can use here for estimating $\lambda$? GMM or MLE or whatever helps, without needing to get data for $t_t$.

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    $\begingroup$ The way you describe it, it appears that your professor spelled to you nothing more than the equation you already have in your hands, telling you that it is ok if one equation has two unknowns, you still can estimate the one unknown. So if $z = aw$ you can estimate $a$ even if you only know $w$. Is it possible that this is the whole story? $\endgroup$ – Alecos Papadopoulos Jun 5 '15 at 1:59
  • $\begingroup$ I don't think econometrics will help you here. I think you should look for a theoretical argument like the one you professor gave you. $\endgroup$ – Rud Faden Jun 5 '15 at 22:00
  • $\begingroup$ You should provide a little more context. It seems, that it has something to do with the Cobb-Douglas function. $\endgroup$ – callculus Jun 7 '15 at 1:32

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