Data and measurement equations

Estimation
1

Hello,

I am estimating a medium-scale NK model without trend. I’ve read posts in this forum about data transformation and measurement equations, as well as Johannes’ A Guide to Specifying Observation Equations for the Estimation of DSGE Models, but I still have some doubts.

I am trasforming my data, first dividing the raw data by a price deflator and a population index, second taking logs and third multiplying by 100: X_obs =100* ln( (X/def)/popindex) ). For the interest rate (FFR): r_obs= FFR/4. For inflation: pi_obs=ln(def_t/def_t-1)*100.
Finally, I detrend all the data using the one-sided HP filter. My questions are:

  1. Is it correct to multiply by 100 before detrending? Or, should I multiply after detrending?

  2. Is it correct to also detrend inflation and the interest rate? I thought they are already stationary, but then I read in post that one could demean (or detrend?) them.

    With my detrended observables, my measurement equations will look like this: X_obs=100*x_t, for the interest rate and inflation as well.

  3. Is this correct? I believe the interest rate should not be multiplied by 100, but in this case I can only estimate a couple of shocks and the obtained mode check plots are flat. When it is multiplied by 100 instead, estimation results look better.

  4. Would it be correct not to multiply by 100 neither in the data, nor the model variables? In that case, the interpretation of shocks, for example, would be not percentage points, but percent, is that correct?

  5. Also, could I demean instead of detrending? Would 100*( ln(X) -mean(ln(X)) ) be equivalent to my transformation above for output, consumption, etc.? For the interest rate: r_obs= FFR/4, and for inflation: pi_obs=100*( ln(def_t/def_t-1) - mean(ln(def_t/def_t-1) ) ?

Thanks to anyone who can help me.

2
  1. The (one-sided) HP filter is a linear filter. It does not matter whether you scale before or after detrending.
  2. Whether you detrend them mostly depends on whether you want to work with mean 0 data or not. That in turn often depends on whether you want to identify parameters also via steady state restrictions. Regarding X_obs=100*x_t: that makes no sense. You usually scale by 100 in order to scale the standard deviations by 100 as well. If you put the 100 in the observation equation, it would act as if you did not multiply by 100 in the first place.
  3. I cannot answer that question without seeing the model. I would recommend simulation the model and comparing the simulated observables with the actual ones. That should show you problems in scaling as well as issues with net vs. gross concepts.
  4. Yes, if you are uncertain, do not multiply by 100. But be careful. Interest rates are often measured with 4 meaning 4%. Here, you need to scale to assure consistency with the model.
  5. That depends. For trending variables like output, the mean is not well-defined. It will not be equivalent. For stationary variables, you could demean instead of filtering. That will generally lead to different estimates because you focus on different frequency components. You need to decide what you want.