Smoothed values for TS not included in the estimation

[jpfeifer]

Yes, that could be a reason.

[econ86]

Is there any way to solve this problem? What do you suggest?

[jpfeifer]

You need to understand the economic intuition behind debt being estimated to be so persistent by the model

[econ86]

Dear all,

I have two questions.

1. I would like to use public debt as observed variable when estimating my DSGE model. The model I'm working on is log-linearized and contains a stochastic unit-root technology shock. To match the data with model variables I would proceed as follows. First I would take the time series of nominal public debt (not SA because SA series is not available for my country) and divide it by GDP deflator (not SA) to produce a real time series. Then I would adjust my data for seasonal patterns. Next I would use the following two approaches:

First approach
I would use the first differences in logs and specify the following observation equation:

delta_log_public_debt_observed = ln(mu_z) + pd_hat - pd_hat(-1) + mu_z_hat

where
mu_z = Steady state quarterly gross growth rate, i.e. 1.006
mu_z_hat = unit-root technology shock, i.e. mu_z_hat = rho*mu_z_hat(-1) + epsilon_muz

and

delta_log public_debt_observed = delta_log_public debt - mean(delta_log_public_debt) + mean(delta_log_GDP).

Is this correct?

Second approach
As a second approach I would specify the observation equation of the following form:

HP filtered log of real public debt = pd_hat

Is this correct? Should I do any other transformation to the data in this case?

2. Additionally I would like to specify the debt brake rule. See file attached. Is this correct?

[jpfeifer]

First:

1. Do not use the HP-filter.
2. In principle, you should use the market value of debt if that is available. See Leeper/Plante/Traum (2010)
3. Your model is going to imply a balanced growth path where everything grows at the same rate in the long run, including debt. If you see a clear trend in debt to GDP (usually upwards), you might want to work with demeaned growth rates as the model will have a hard time to deal with tis.


Second:
that looks ok. Expenditures go up if tax revenues go up and go down when output is below trend.

[econ86]

Dear prof. Pfeifer,

thank you for your reply.

First (regarding observation equation for public debt):

Do not use the HP-filter.

How the data (on the left side of equation below) should be prepared if one want to specify the following observation equation:
data = pd_hat, where pd = PD/P*z (i.e. real debt scaled by unit-root technology level z)

Your model is going to imply a balanced growth path where everything grows at the same rate in the long run, including debt. If you see a clear trend in debt to GDP (usually upwards), you might want to work with demeaned growth rates as the model will have a hard time to deal with tis.

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

Expenditures go up if tax revenues go up and go down when output is below trend.

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.

[jpfeifer]

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.

[econ86]

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?

[jpfeifer]

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

[econ86]

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

gex_hat = t_hat - y_hat?

This follows from:

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

Y_potential = z

z= unit-root technology level

[jpfeifer]

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.

[econ86]

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,

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).

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

t_hat = ln(T_t/P_t*z_t) - ln(T_ss/P_ss*z_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?

Thank you for your help!

[jpfeifer]

Write you wrote about the definition of yhat is exactly what I said. If yhat is positive, there is a boom.

I don't get your second point. If
GEX = T*(Y_potentital/Y_actual) and everything is date comtemporaneously, then everything can be treated as being stationary. Now, if yhat=log(Y_potentital/Y_actual), you will have

Code: Select all

gex_hat = t_hat - y_hat

where gex_hat and t_hat are percentage deviations from steady state.