Theoretical Questions Related to Bayesian Estimation

I am not sure I understand your last question. As long as you correctly specify the observation equation/measurement equation, scaling a variable with GDP is not a problem.

I have a very general question/inquiry about the motivations behind the estimation of a model. It is an issue that I think is important for people to keep in mind and so I figured I would post it in this thread I had previously created.

When one builds a model and estimates it, there is the assumption that this model portrays the structure of the economy or to some level is a good representation of the behavior of the economy. You then feed data to figure out the fit. There is an implicit assumption that throughout the sample data period, the economy has behaved according to the imposed model. Right?

If however, one believes that there is some structural change after a period (say the fin. crisis of 08-09) does it make sense to estimate a model using data before that period. Essentially, doesn’t that imply a contradiction between the motivation of the estimation and the data used with the structure of the model? The model is based on a recent development, but the data used in estimation does not capture this recent development.

Regardless of results or overall model fit, wouldn’t that kill the whole premise of the estimation of the model?

You typically assume that the model is (close to) the data generating process. Estimation relies on the data being covariance stationary and ergodic. Only in this case the past is a good guidance for the future. If you have a structural break, the data will not satisfy this and there will be a problem. In that case, you can only try to model this break, e.g. by allowing for time-varying parameters.
Regarding the financial crisis, Stock/Watson (2012) have argued that we did not have a structural break, but rather just big shocks within the old structure. In that case, estimating the model on a longer data set should be fine.

If you are still looking for an answer, I would recommend you to add either a government spending or tax shock. In the end of the day, you are trying to have something that exogenously change the government debt. The exogenous decision of government to choose how to finance its deficit can do the job.
These are some references: Chen, Curdia and Ferrero (2012) and Alpanda and Kabaca (2019).