# Smoothed values for TS not included in the estimation

Dear Dynare users

I estimate my model using data on 15 macroeconomic variables. There are also other time series that could be used in the estimation (e.g. fiscal variables – public debt, etc.), but they are not at this moment. My question is: how should the smoothed values of, let me say, public debt look like (when public debt is not included in the estimation as observable variable)? Should the smoothed values reflect the actual (historical) data (of course, adjusted/prepared in such a way as if they would be actually used in the estimation)?

Many thanks.

They should ideally look simular. This is kind of a tests for overidentifying restrictions. Does your model imply the correct movement for a variable not used to estimate the model?

Dear

Check your model. Debt will fluctuate around its steady state in the model. Apparently you set up your model so that the steady state for debt is negative. Depending on your definition it might be correct (a positive debt level in the households budget constraint is their savings, but the governments debt, but you can also define it the other way round).

Ok. But I’m working with log-linearized model (SS of all variables with a hat is 0, except for some observation equations which contain constant terms). Steady state values/ratios that enter my log-linearized equations are specified within the parameters block of the model. How does Dynare compute smoothed values in this case?

If your model is linearized and you get one variable to be permanently below steady state, some other shock must be permanently above/below steady state. Most of the time this is due to using a non mean 0 variable for estimating a linear model

If I understand you correctly, constants in my observation equations (again, public debt is not included as observable variable) affect the smoothed values of public debt, so that they do not fluctuate around the steady state of \hat{public_debt} which is 0 by definition?

Put differently, if I would demean my observable variables, the smoothed values of public debt would fluctuate around 0?

Regards.

No, what I am saying is: for one variable to be permanently below steady state, you need to have a sequence of shocks that all go in one direction that drive this. Usually, shocks are both positive and negative so that in a longer sample this (almost) never happens. The most common reason you get something like this is that you forgot to correctly account for the mean of one observed variable in the observation equation, thereby forcing Dynare to account for this mean by assuming a sequence of one-sided shocks. There may be other reasons, but this is the most common source of problems.

I have looked at the smoothed shock innovations to see whether they are roughly mean zero - all of them fluctuate around zero (see attachment).

Should I check my observation equations, i.e. if my data are correctly related to the model’s concepts?
Smoothed_shocks.pdf (171 KB)

The smoothed shocks look good. Are the other smoothed observed variables mean 0?

Yes and no. When estimating the model I use both demeaned and non-demeaned data. For example, data on employment has been demeand prior to estimation and the observation equation is the following:

e_obs = \hat{E}_{t},

while some other variables (output, consumption, etc.) are not demeaned (i.e. they contain a constant term). In this case the observation equations look like:

dy_obs (GDP growth) = ln(mu_z) + \hat{y}{t} - \hat{y}{t-1} + \hat{mu_z}_{t}.

I have also checked whether all model’s variables are indeed mean 0. So they are, except for some observed variables which have a mean equal to constant term.

Was debt a mean 0 observed variable? If not, this could explain the negative values as the mean is added.

Debt is not included in the estimation as observed variable at this moment. It is just one of the endogenous variables in my log-linearized model, i.e. \hat{debt}_{t} with SS equal to 0 by definition.

Therefore also the smoothed values of debt should fluctuate around 0. Am I right?

Yes. It should fluctuate around 0.

In the model the expression for real debt dynamics (scaled by unit-root technology level) is as follows (in log-linearized form):

pd_hat = pd_hat(-1)/(mu_z_sspi_ss) - (1/(mu_z_sspi_ss))*(pi_hat + mu_zhat) + def_hat/pd_ss;

When solving the model, I found large coefficient of autocorrelation for debt, i.e. 0.999. Could this cause problems with Kalman smoother and consequently explain the negative smoothed values?

Yes, that could be a reason.

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

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

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?

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

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.