Exogenous variable as observable variable in estimation

Generally, you want to use all relevant information that you have. If you can perfectly observe a relevant variable, you may want to use it. That should improve your model estimates, not deteriorate them. If you were to not observe that variable, the model would need to infer its putative dynamics from the other observed data, which should only happen with errors.
The only case where that may not be advisable is if there is a gap between the actual object in the data and the concept measured. For example, there is not just one oil price, but there are prices for a variety of sorts (e.g. Brent vs. WTI). But even in that case, you may simply add measurement error.

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