Shocks in stochastic simulations

Dear Prof Pfeifer,

I have a couple of questions relating to stochastic simulations.

  1. I have a model with a positive government spending (gc) shock financed by lump sum taxes. Given this, following the gc shock, the labour income tax rate (tauW) should be constant at 0. Its percentage deviation from steady state goes from 0 to -3e^-17, which is also approximately zero. However, this issue does not happen with the capital income tax shock (tauK), which stays at 0 following the shock. I just want to confirm that there is no issue, i.e., that the value of tauW is actually staying at its steady state value even though its percentage deviation from steady state is going to -3e^-17?

Please run the run.m file, which in turn runs cycle_irf.mod to show the results for the gc shock.

  1. Secondly, if I model a tax process as follows:

tauW = (1 - rhoW)tauW_SS + rhoWtauW(-1) + (1 - rhoW)gammaN(D/GDP - D_SS) + epsW;

If I want to write this equation in exp(logs), do I express the steady state values of the tax rate (tauW_SS) and debt (D_SS) as the exponent of their steady state values exp(tauW_SS) and exp(D_SS), respectively, or just simply as their steady state values? It is not clear to me how to treat steady state values in this context.

Many thanks in advance for your reply. (1.4 MB)

Dear Prof. Pfeifer,

Wondering if you have had a chance to take a look at this issue?

Many thanks.

  1. Yes, that is only numerical noise that you can safely ignore.
  2. I don’t really understand the question. Please use the \LaTeX capabilities (with Dollar signs) to make the equation readable and define all objects.

Many thanks, Prof. Pfeifer.

Here is the equation again in the \LaTeX format.

\tau^k_t = (1 - \rho_k) \tau^k + \rho_k \tau^k_{t-1} + (1 - \rho_k) \gamma_k (\frac{D_t}{gdp_t} - \frac{D}{gdp}) + \varepsilon_{k,t}

\tau^k_t is the tax process for capital income tax in the model. \tau^k is its steady state value. \rho_k and \gamma_k are parameters and \varepsilon_{k,t} is the exogenous shock. D_t and gdp_t are debt and GDP at time t, respectively. D and gdp are their corresponding steady state values.

If I want to write the model (and therefore this equation) in exp(log) in Dynare so that the output is in percentage deviations from steady state, do I leave the steady state values as they are in the .mod file - for example, will \tau^k be written as \tau^k, not \exp(\tau^k) and same for D and gdp? If so, why is this the case?

Lastly, if the steady state value \tau^k shows up in another equation, would it appear the same way as it does in the \tau^k_t equation?

Thanks in advance.

You need to be systematic. Whether you need to have logs here depends on the type of feedback you. Is it the percentage point tax rate that reacts to absolute deviations of debt? Or is it the percent change in tax rate relative to its steady state that reacts the the percentage deviation of debt from its steady state? That will determine where to put exp or log. Note that I would not recommend general exp() substitutions as there are better options. See Question about understanding irfs in dynare