This is how I treat my variables and would like to ask you if you think is correct:

-First of all it is a log-linearized model.

-Output, investment, wage, are all in log differences, measurement equations account for cointegration.

-Real return on rented capital and Real interest rate r_obs= log(1+r_data/400) - mean(log(1+r_data/400));

-I need hours but I have employment so I used log differences on the employment time series. Not sure about;

-I have public budget surplus and deficit +/- percentage numbers on GDP. I am not sure how to treat this and I just take log(1+surplus) to deal with negative percentages.

Sorry for the late reply. The first part looks sensible, but

regarding hours: using first difference is correct to get rid of the unknown levels. However, substituting employment for hours mean that you are only capturing the extensive margin (how many people work), but not the intensive margin (how much an employed person works). Depending on your model, this can be problematic

regarding surplus: I would not log-linearize this variable, but enter it as a percentage of GDP, i.e. you observe surplus/GDP. That way you get rid of the scaling problem and deal with negative numbers.

Regarding surplus, I will try to use percentage/gdp as you suggest. My concern is about stationarity. My model is log-linearized and if I try to simulate variables with my benchmark parameters I get stationary time series. The true surplus/deficit time series is not stationary (having observables for europe in the last 10 years is almost always negative).

Is this an issue? Should I demean data to obtain a time series that deviates from zero? Alternatively I guess I can use a measurement equation of the type

s_obs=s+Sf where Sf is the steady state surplus in the model, that however is still close to zero.

I know I already wrote a similar question but I am still dealing with this issue.

Again, this is hard to tell. If you think your data is non-stationary using a stationary model is wrong. If you think that the deficits in the last 10 years are just the result of bad shocks, but the intertemporal budget constraint of the government is still satisfied (i.e. there exists an equilibrium with finite debt), then having deficits most of the time is not an issue.

Rescaling the deficit as you suggest is a possibility, but usually hard to justify. The better way often is to just have lump sum taxes in the model which deal with the residual budget deficit. That way, even if the observed deficit is really high, the model will be stable.