Dear Johannes,
First thank you for your previous guidance, I am grateful.
In my PhD thesis, I have modelled an economy with state-owned firm production and privately firm production, however, in real data, i only have aggregate production data, I am wondering that in measurement equation, can I model observed GDP (observable variable Y) as a nonlinear combination of state-owned production (state variable Y_s) and privately production (state variable Y_p), specifically,
Y=(\alpha Y_s^{1/\beta}+(1-\alpha) Y_p^{1/\beta})^{\beta}
where \alpha is the share for state-owned production, (1-\alpha) is the share of privately-owned production. \beta is elasticity of substitution between state-owned production and privately-owned production.
Alternatively, should i connect between observable variable Y and state variable Y_s and state variable Y_p using a simple linear combination:
Y=Y_s+Y_p
Thank you very much and look forward to hearing from you.
Best regards,
Jesse