Mesurement system

Hallo everyone
I have a question related to mesurement equation
For example: Based on quarterly data, I have

  • y_obs: observable real output growth
  • i_obs: observable interest rate
  • y: output level is a model variable
  • i: interest rate is a model variable
    Why the obs. variable should relate to model variables such as

100log(1+y_obs/100) = y-y(-1) + mu_y; % obs. domestic output
log(1+i_obs/400) = 4*i + mu_i; % obs. domestic interest rate

What is the meaning when taking log like the above equations

Thank you so much indeed

The log is there to compute percentage deviations as \log(1+x)\approx x. But without more context, it is hard to judge

Thank you so much Prof. Pfeifer