Hi everyone

Smets and Wouters (2007) use nominal variables and adjust all variables by the GDP deflator since the economy has only one final good. In this case P_{t} represents the price of the final good (GDP deflator), and at the same time \Pi_{t} = \frac{P_{t}}{P_{t-1}}, so they use the log-difference of the GDP deflator as the inflation rate.

In my case, I want to define the measurement equations for the price indexes of the different goods, and I would like to know if I should adjust. For example, for tradable goods in real terms we have p_{t}^T = \frac{P^T_{t}}{P_{t}}. So, if I take the first log-difference of the export price index (FRED: IQ) as observable, which of the following measurement equations is correct?

Let X_{t} = 100 \times [ln(\text{FRED: IQ}_{t}) - ln(\text{FRED: IQ}_{t-1}) ]

- X_{t} = ln(\frac{p_{t}^T}{p_{t- 1}^T})
- X_{t} = ln(\frac{p_{t}^T}{p_{t- 1}^T}) - ln(\Pi_{t})

Thanks in advance.

Another question: is it correct to transform the nominal to real observable variables using the GDP deflator, but to use the CPI as the observable variable of inflation?