Estimation problem of ratio values

Estimation ML/Bayesian estimation
1

Dear Professor,

I have a question regarding the estimation of a DSGE model. Specifically, I am using C_{obs} and Y_{obs} as observables to estimate C_t and Y_t. I would like to ask if Scheme A for the measurement equations is generally preferable to Scheme B:

Scheme A:

Yobs_{HP} = y + e_y

\log(C_{obs}/Y_{obs}) = c - y + \log(c_{ss}/y_{ss}) + e_{cy}

Scheme B:

Yobs_{HP} = y + e_y

Cobs_{HP} = c + e_c

In these equations, y and c represent the log-linearized variables of Y_t and C_t, while Yobs\_{HP} and Cobs\_{HP} denote the cyclical components obtained via the HP filter.

Does the use of the consumption-to-output ratio in Scheme A offer better identification or estimation properties compared to filtering both series independently as in Scheme B?

Thank you for your time and guidance.

2

Scheme A has the advantage of still providing information about cointegration relationships. However, it involves mixing frequency components. The y in the first equation will only contain frequency components from the HP filter passband. In the second equation, it will contain all frequency components. For that reason (consistency), I would opt for scheme B.