# Problem with variable matching

Dear Professor Pfeifer,

When I was learning DSGE model with housing sector , I feel a bit confused about the two problems as bellow. Would you please give any advice ?

1. Like the model in Iacoviello(2010), they generally use the data of residental investment to match new housing {{IH}_{t}} when running the bayesian estimation. While I thought the residental investment should be {{q}_{t}}{{IH}_{t}} , why don’t they use the data of residental investment to match {{q}_{t}}I{{H}_{t}} ?
2. In a DSGE model, we often use output {{Y}_{t}} to define GDP. After adding housing sector, the definition of GDP becomes {{Y}_{t}}+\bar{q}I{{H}_{t}} . Is there any tips on how to define GDP in the model ? What kind of variables should be included in GDP ?

Thank you for your time.

1. q_t is the relative price of residential investment relative to output. That is, real units of investment goods are transformed into unit of the final good. But if your observable is real investment measured in units of the investment good, then obviously, that is only IH_t. In contrast, q_t maps into the price of investment.
2. I don’t understand this point. Residential investment is already part of GDP, so there is no need to add it.

Dear Professor Pfeifer,

1. I thought all the empirical data we get is measured in units of the final good. It seems that they are using the data of residental investment measured in units of the final good to match new housing {{IH}_{t}} in Iacoviello(2010). That’s the confusing part. Did I misunderstand this ?

2. It seems that GDP is defined as {{Y}_{t}}+\bar{q}I{{H}_{t}} in Iacoviello(2010).

Thanks again for your time.

Which paper exactly are you referring to? I was looking at Iacoviello/Neri (2010)

Dear Professor Pfeifer,

Yes, it’s Housing Market Spillovers: Evidence from an Estimated DSGE Model by Iacoviello/Neri (2010).

1. No, the data in real US national accounts are chain-weighted. So they are measured in different units. See e.g. Whelan (2003).
2. Footnote 15 in the paper shows their mapping. They do not use GDP in the model as an observable, because the mapping is not straightforward as it includes imputed rents in practice.