# Smoothed values for TS not included in the estimation

Dear prof. Pfeifer,

**First (regarding observation equation for public debt):
**

See file attached to see how time series of public debt looks like. If I understand you correctly I need to drop out the steady state quarterly growth rate, i.e. ln(mu_z) which is calculated as a sample mean of quarterly GDP growth, so that my observation equation reads as:

delta_log_public_debt_observed = pd_hat - pd_hat(-1) + mu_z_hat

where
delta_log public_debt_observed = delta_log_public debt - mean(delta_log_public_debt) + mean(delta_log_public_GDP) - mean(delta_log_public_GDP)?

**Second (regarding debt brake rule):
**You wrote

[quote=“jpfeifer”]
Expenditures go up if tax revenues go up and go down when output is below trend.[/quote]

Hope that the debt brake rule I specified correctly incorporates the following feature:
in times of recession deficit is allowed whereas in expansion phase surplus is required.

What is interesting is that in this type of model (when using unit-root technology level, i.e. z_t as a potential output) expenditure ceiling is indeed equal to tax-to-GDP ratio:
gex_hat = t_hat - y_hat.
Nominal_debt.pdf (35 KB)

I am saying that you should use demeaned growth rates for debt to get rid of the potentially different mean growth rates of debt and GDP.

Note that your debt brake implies symmetry. Thus, it does not matter whether you are in expansion or recession. The fiscal rule is the same. You rather seem to have a penalty function approach in mind that punishes deviations in only one direction, but this is not what you implement.

The debt brake (Swiss) rule states:
The expenditure ceiling (GEX) is linked to the amount of receipts (T), which are adjusted using a factor that takes the economic environment into account (cyclical factor, i.e. Y_potential/Y_actual). When the economy is booming, the expenditure ceiling is lower than receipts and the government generates a surplus. Conversely, the formula tolerates a deficit in times of recession. Balanced finances are achieved over the entire economic cycle. So, the general formula is given by:

GEX = T*(Y_potentital/Y_actual).

Do you have any suggestions how to implement this in my log-linearized model that features a stochastic unit-root technology shock?

The way the text describes that particular rule, your formula looks correct. There is no asymmetry here.

So, in log-linearized form the debt brake (Swiss) rule described above would be:

gex_hat = t_hat - y_hat?

This follows from:

GEX/Pz = (T/Pz)*(z/Y)

Y_potential = z

z= unit-root technology level

Look correct if yhat is the output gap. Just make sure that the output gap has the correct sign. In the New Keynesian literature a positive output gap often means that output is above its trend, i.e. in a boom.

I think so. In the code I have the following expression:

y_hat = lambda_d*(epsilon + alpha*(k_hat-mu_zhat) + (1-alpha)*H_hat);

This expression comes from the production function. To obtain this expression, first, all real variables in the production function are scaled by z_t and then the expression is log-linearized. Thus y_hat = ln(Y_t/z_t) - ln(Y_ss/z_ss)] is my output gap which uses unit-root technology level, i.e. z_t as a potential output. I think that this is ok.

Regardless of this, I doubt that

gex_hat = t_hat - y_hat

is correct representation of the debt brake (Swiss) rule described below,

because this expression (i.e. t_hat - y_hat) is just an ordinary tax-to-GDP ratio:

t_hat = ln(T_t/P_tz_t) - ln(T_ss/P_ssz_ss)

y_hat = ln(Y_t/z_t) - ln(Y_ss/z_ss)

=====> t_hat - y_hat = [ln(T_real_t) - ln(T_real_ss)] - [ln(Y_t) - ln(Y_ss)]

Again, all real variables in the model are expressed as deviations around the common unit-root technology level i.e. z_t.

Do you have any suggestions how to specify debt brake (Swiss) rule i.e. GEX = T*(Y_potentital/Y_actual), taking into account these specific features of my model?