I am working with pseudo-panel data (UK Cost of Living and Food Survey), which means I am observing expenditure data for different households every month over ~10 years with data on demographic controls. I want to find out what the price elasticity of demand $\eta$ is for a certain item.
I have the following model, where expenditure on an item $E$ over time is determined by some constant $\gamma$ and its price $p$. There are two types of consumers, where group $P$ is facing a price that is higher/lower than group $C$ with $p_t^\text{P}=\lambda_t p_t^C$. The "price gap" $\lambda_t>0$ is exogenous and varies over time.
$$\ln(E^{\text{C}}_{i,t})=(1+\eta)\ln(p^{\text{C}}_t)-\eta\ln(\gamma_i)$$
$$ \ln(E^{\text{P}}_{i,t})=(1+\eta)\ln(\lambda_t p^{\text{C}}_t)-\eta\ln(\gamma_i) $$
We assume that the constant $\gamma$ varies between individuals $i$ but and can be somewhat predicted by demographic controls (household size, income etc.). The constant does not depend on the group ($P,C$).
This specification allows me to take differences and (in theory) estimate the following model, where the difference is no longer a function of the actual price:
$$ \ln(E_{i,t}^{\text{P}})-\ln(E_{j,t}^{\text{C}})=(1+\eta)\ln(\lambda_t )-\eta\ln\left(\frac{\gamma_i}{\gamma_j}\right) $$
This way I should be able to avoid simultaneity problems (since price might also be determined by demand, we cannot just regress $E$ on $p$).
Is there a way to estimate $\eta$ based on pseudo-panel data? I was thinking of something along the lines of estimating a counterfactual $E_{j,t}^C$ for every $E_{i,t}^P$ based on demographic controls and then running a regression with the results, but I am not sure if that would work. Has anybody ever done anything similar to this?
