Observation equations when model contains both linearized and log-linearized variablies

Hi all,

I’m working through the paper “Inflation as a fiscal limit” by Francesco Bianchi and Leonardo Melosi (available at: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4205158). In footnote 2 on page 9, they note that \hat{x} indicates a variable which is log-linearized, while \tilde{x} indicates a variable which is in linear deviations. Linear deviations are used for variables which are normalized with respect to GDP (debt, expenditure, and taxes). It’s not clear to me how exactly to specify the observation equations for these normalized variables - is the data in log-levels or levels?


In levels, of course. You match e.g. the debt to GDP ratio in the data to the one in the model (potentially demeaned).