# Using empirical estimates (from %-deviation) for simulation

I’d like to start out (my first post) by thanking you all for the wonderful resource you have provided us. It is appreciated.

I have quarterly data which has been transformed to percent-deviation from steady state (using an H.P filter w/standard smoothing parameter 1600). I have a fairly complex non-linear search model I have implemented into dynare; I have written the non-linear equilibrium equations into the .mod file. It has a stable steady state and I am thus having dynare run the linearization and simulation. The simulation currently runs splendidly. I am using the model to explain certain aspects of the data and thus would like to compare the output of the simulation directly with the data.
There are variables in the model (output: y and the labor match separation rate: sigma) which are exogenous to the model; I have written the equations for y and sigma in the .mod file as:

y = y_barexp(yhat)
sigma = sigma_bar
exp(sighat)

where y_bar and sigma_bar are the empirical steady state values for y and sigma (they are parameters in the .mod file), yhat and sighat are the disturbances which I model as mean-zero AR(1) processes:

yhat = phi_yyhat(-1) + epsilon_y
sighat = phi_sig
sighat(-1) + epsilon_sig

where epsilon_y and epsilon_sig are varexos in the .mod file. I think I am doing something stupid, although I may be overly paranoid. For calibrations for the parameters phi_y and phi_sig, I am using the estimated AR(1) coefficients from the empirical data for those variables (the % deviation data I mentioned earlier), and the strandard deviations for the varexos directly from the respective frequency distributions of the error terms from the AR(1) estimations (by the way, I am performing the time series estimations outside of dynare), however I think the theoretical moments are not consistent as I read somewhere that the output from dynare (theoretical moments & impulse response functions) is not in percent-deviation from a stationary steady-state. I would also like to stay consistent with the variables and have levels and rates defined accordingly i.e., I would like the percent deviation for y (a level variable) to be log(y) - log(y*) and the percent deviation for sigma (a rate variable) should be sigma – sigma*, where starred variables are H.P. filtered versions of the variable.

Apologies for the long-winded post, but how do I ensure that what I am inputting in my file for the AR(1) coefficients and for the S.D. of the varexos results in theoretical movements in y and sigma that match their empirical counterparts perfectly (so that I can make a correct inference using the simulation output of the endogenous variables in my model)?

I tried defining a logged version of y to see what would happen, but noticed that the IRF for the logged version looked no different than the IRF for the non-logged version, in terms of both shape and scale (but again…I’ve been working for a while and perhaps am not thinking clearly here). I am using dynare 4.3.2. Any help is appreciated

Chris

I am not completely following what you are doing here. When you define

then yhat measures percentage deviations from y_bar. See [Comparing model with data after log-linearization) on how to compare model and data moments

What you need to get straight is the filtering issue. You are not supposed to estimate AR1-processes on HP-filtered data. What you seem to be aiming for is a moment matching.

Regarding

The IRF to the logged version is the IRF to the unlogged version divided by the steady state (Jacobian transformation). Thus, if your steady state is close to 1 they will be (almost) the same.