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 $t_t$.