IRF matching with multiple goods

I have a question about real prices and outputs. I want to replicate a model like Pieschacon (2012), and I am confused with the varibles in the model and in the data. I was wondering if you could help. Since it is estimated using IRF matching, it is very important to be sure what in the model is what in data.

My question:

If we have three types of goods in the economy (e.g., Tradable, Nontradable and oil) in the economy and model. We choose the tradable sector’s product to be the nummeraire. The GDP in the model will be GDP_t= Y^T_t+P^{NT}_t*Y^{NT}_t+P^O_t*Y^O_t; My question is that whether this model GDP is the one we see in the data?
My dout roots in the nummeraire selection of the tradable goods.

Also, I do not know with what I should math the P_NT and Y_NT. Are they real or nominal? Real with what deflator?

Note: Pieschacon (2012):The value of fiscal discipline for oil-exporting countries

The crucial assumption in that paper is that the numeraire has a dollar price of 1. Put differently, the equation for GDP can be interpreted as nominal GDP.

Thank you so much.
How about P^{NT}_t and Y^{NT}_t? How should be they calculated or made real to have the correct answer? I mean, Is P^{NT}_t just the ratio of prices? Should Y^{NT}_t be realized with gdp deflator or with its own deflator?

What do you want to do? P^{NT}Y^{NT} is simply nominal spending on non-tradable goods.

We want to have the seperate variables P and Y to match each ones IRF with data.

Then usually, you use the real output measure for Y and the deflator for the price P. Most national accounts have this data

What about the price of oil P^O_t? in the paper, while it should be the relative price to tradable price, it is the oil it self. wouldn’t that be a problem?

That should be the price of oil relative to the GDP deflator.

Why? The tradable is the nummeraire. Shouldn’t all the prices be relative to that?
you mentioned that it has assumed to be one. Why this has been assumed? Is it a good assumption?

You are right. Nominal GDP is measured in Dollars, which is the price of the tradable good. But the GDP deflator is a composite of tradable and non-tradable prices. My hunch is that the whole Dollar equation is divided by the 1993 base US GDP deflator (below equation (19)). You then end up with the oil price in terms of real dollars.

Thank you.
I have problem with the assumption that the price of tradable is assumed to be one. Why? I understand the choice of nummeraire, but the prices in the time of estimation should not be calculated in the form of related to the price of nummerarie?
It is measured in the data as is P^T_t and all real world prices should be made real using that. It has assumed that for all t, P^T_t=1.

This means that prices measured in dollars are equivalent to relative prices expressed in units of the tradable good at the wholesale level.

I was thinking that may be the price of the oil, which is an endowment here and not the outcome of general equilibrium should not be made relative to the nummeraire. Am I right?

Here, you need to ask the author of the paper.