Dear all,

I am having trouble with understanding the measurement equations in Justiniano and Preston model.

I have found a mod file used in Rubaszek and Kolasa (2018) who worked with J&P model variants in their paper.

The J&P model is a log-linearised model with variables expressed as deviations from their steady states. This concept is pretty clear to me. As for the data, I would normally take the log of each variable and detrend it (e.g. one-sided HP filter). Thus, SS of all variables is 0 and I can directly match the data with the variables in the model as following (I will only refer to output for simplicity): **y_obs = y**

The problem is that Rubsazek and Kolasa used the following measurement equation for output (and similarly for other variables): **100*log(1+y_obs/100) = y-y(-1) + mu_y**.

The output data enter the model as (log(Yt) - log(Yt-1))*100.

**1)** I don’t understand why there is this type of transformation on the left side. The transformed data is nearly the same as the original data. Why did they use it?

**2)**

**a)** They didn’t detrend or demean the data prior to estimation so I assume the **mu_y** should reflect the trend/steady state of the variable. The transformation of their output data was (log(Yt) - log(Yt-1))*100, thus growth rates. So they used the term **mu_y** to get rid of the trend, is that right?

**b)** However, there are two things I don’t understand. Why did they inicialize the **mu_y = 0** and why they assumed **mu_y, uniform_pdf, 0, 0.5, 0, 10** in the estimated_params block. For example, Smets and Wouters (2007) did similar thing (I think) regarding the measurement equations but they certainly did not pressume the mean of 0 and a uniform pdf. It seems a bit odd to me.

**c)** Also, if I demean the data prior to estimation, delete the **mu_y** term from the model and estimate it as *model(linear)*, do I get the same results?

**3)** In general, I think I don’t understand the usage of the gap data and growth rates data in this model. I would assume that since the model is a log-linear model built in a way that variables reflect the deviations from their SS, thus their cyclical parts, why are the approx. growth rates used? Log difference transformation and detrended data (e.g. via HP filter) are two different concepts giving diffent results. Please, could you explain it to me?

In the original paper Justiniano and Preston (2004), they used a log deviation from a linear trend for output so why Rubaszek and Kolasa (2018) used different approach - growth rates for nearly the same model (only extended a little bit)?

*I attach the exact .mod file that Rubaszek and Kolasa used in their paper.*

Thank you very much for any help.

Martin

jpd.mod (6.5 KB)