I have a simple model heterogeneous agents model, where the distribution of wealth is not a relevant state variable, thus the model is tractable, and I guess can be solved in dynare. My question is the following.
Consider that the aggregate shock, is the standard AR(1) process for the aggregate productivity, and then I parametricaly introduce two idiosyncratic shocks as follows:
[ul]1. A_{t} = rhoA_{t-1} + e_{t} where e_{t} white noise —> (Aggregate Shock )
2. theta_1 is an iid process with distribution (1, sigma^2_{t}), i.e mean 1 and some time varying variance sigma^2_{t} which follows this rule:
2.1 sigma^2_t = sigma(1-eta*A_t) — > (that is the idiosyncratic shock is a simple heteroskedastic iid process which depends only on the value of the aggregate shock)[/ul]
similar assumptions as in (2) are imposed for the second idiosyncratic shock. Of course, the correlation between the idiosyncratic shocks are zero.
The model has a recursive structure, where essentially the uncertainty here will be modeled by a simple joint markov process for the aforementioned processes.
The problem then is not with the idiosyncratic shock, but with the particular shock structure you want to preserve. This pretty much looks a stochastic volatility process, suggesting you need to go to a third order approximation to get IRFs that capture shocks to the volatility. For more on this, see e.g. Section B.5 “Model Solution” in the Appendix to Born/Pfeifer (2014): Policy risk and the business cycle at sites.google.com/site/pfeiferecon/Born_Pfeifer_Policyrisk_Appendix.pdf?attredirects=0
Yes, it looks like stochastic volatility, but i only want to shock the TFP productivity process, there is no explicit noise for the time-varying variance, my only condition is that the variance depends on the value of the TFP. So, i guess maybe it would be slightly easier in practice that the “pure” stochastic volatility models. I assume from your answer, that dynare is able solve stochastic volatility models (if i take third order approximation) and therefore I will try and see whether there any other issues.
In general, I do not see why you should not be able to solve the model using Dynare. It could be that your model is easier. In that case, second order might be sufficient. You could check this by inspecting the IRFs. You can find replication files for Born Pfeifer (2014): “Risk Matters: A comment” on my homepage. There, a model with stochastic volatility is solved using third order.
Further to my previous posts, my model it essentially boils down to a recursive equation of the following form:
where F(.) is a function that embraces another function G(.), where inside G (.) there is a maximization problem to be solved before each iteration for b(t) (is the variable to be found by the recursion and depends only on the aggregate state (That is the total factor productivity ). Think that before each iteration there is a portfolio choice problem to be solved (where R) are the gross returns that depend on the total factor productivity and the idiosyncratic shock as calibrated before., E_t] is the conditional expectations operator that depend only on the aggregate state and theta the vector of the portfolio shares. So, before each iteration for b_{t} there is an objective to be maximized. Are there chances for dynare to handle this ?
Second, the model features endogenous growth, which implies that when the aggregate shock hits the economy does not necessary converge to its old steady state, but to a new one, as the growth rate of the economy permanently changed. For the impulse response functions will dynare calculate the convergence to the new steady state or the ‘‘old one’’?
As long as you can define a recursive law of motion for your objects (in particular your expectations), Dynare is able to handle this. The trick is similar to transforming the infinite sequence problem arising in utility maximization into a recursive value function problem. I don’t know if this can be done in your case.
Regarding non-stationarity, if there is a unit root the IRFs will converge to the new steady state (as long as this conditional steady state is uniquely determined). The Blanchard-Kahn conditions do not exclude unit roots.