Observation equation of tax income

Hello, Dynare.
In my DSGE model government Tax Labor Revenue is T=W * L * tau_l;
I collected the data on labor income (demeaned, detrended) and wanted to match with the model equation.
So, I put it as:

  1. Tobs=W * L * tau_l-steady_state(W)*steady_state(L)*steady_state(tau_l);

  2. But since my model is log-linearized model with exp(x), I reconsidered it to be:


I estimated the model with both observation equations, first one gives adequate response for positive productivity shock e (it increases output). The second one does not (it decreases output to positive productivity shock e).

Could you please give some suggestions on observation equation for labor tax revenue :pray:

Without more information on the model specification and the data treatment, it is impossible to tell.

Dear @jpfeifer ,

Sorry for the late response, I could not get into Dynare forum for sometime, it was showing some error.
As for the labor tax income I found in data “labor tax revenue per capita”, I took a log and detrended and also demeaned (my model is stationary, I ignore pop. growth).
I did similar for consumption tax revenue .
My mod file is attached (observation equations #50-51)
DSGE_dynare.mod (8.1 KB)

You should have
\begin{align} TA{X_t} &= {e^{{W_t}}}{e^{{L_t}}}{e^{{\tau _t}}} \hfill \\ \ln \left( {TA{X_t}} \right) &= \ln \left( {{e^{{W_t}}}{e^{{L_t}}}{e^{{\tau _t}}}} \right) = {W_t} + {L_t} + {\tau _t} \hfill \\ \ln \left( {TA{X_t}} \right) - \ln \left( {TAX} \right) &= {W_t} + {L_t} + {\tau _t} - \left( {\bar W + \bar L + \bar \tau } \right) \end{align}

Thank you very much for the confirmation, I was not sure before.