Dynare++: Decision Rule Form

[gmartins]

Hi,

I am still not sure about what is the mathematical problem solved by Dynare++ to compute the “stochastic fixed point” or to express the decision rule “centralized around the stochastic fixed point” or when it takes “Steps in Volatility Dimension”. (all terms here are taken from the “DSGE Models with Dynare++. A Tutorial” ). Therefore, I have three questions about it.

1) What are the best references to understand the mathematical problem when computing these two rules and the stochastic fixed point? The “DSGE Models with Dynare++. A Tutorial” is not very clear about it.

2) How can Dynare++ replicate the asset pricing example in Collard and Julliard (2001, JEDC, Table 2, O2 rows) and replicated in Schmitt-Groh and Uribe (2004, JEDC, Table 1, Fixed-point algorithm rows)? I tried both methods and none of them give the same solution as reported in those tables.

3) Is it correct to assume that with the “--non-centralize” option the decision rule is always the same as the solution of the perturbation method in Schmitt-Groh and Uribe (2004, JEDC)?

Tks

Guilherme

[SébastienVillemot]

Hi,

• the --no-centralize does exactly what Dynare does: an approximation of the decision rule around the deterministic steady state
• by default, if the --no-centralize option is not given, then Dynare++ computes the fix point of the approximated decision rule mentionned above. This fix point is called the stochastic fix point: it is the point where agents will want to stay if current shocks are zero, but knowing that future shocks have a non-zero variance. Of course the true stochastic fix point is unknown, since the true decision rule is also unknown. Then Dynare++ rewrites the expression of the approximated decision rule, in order to express it in terms of deviation from the stochastic fix point, instead of deviations from the deterministic steady state. Note that it is the same rule than the one obtained with --no-centralize: it is just expressed with a different organization of the terms, but it is still an approximation around the deterministic steady state
• the use of the --steps option has a different effect: the idea is to iterately compute an approximation of the stochastic fix point, and to compute an approximated decision rule around this point (instead of around the deterministic steady state as above). I don’t have the details of the algorithm (but they are probably given in the file kord.pdf which is distributed with Dynare++). My understanding is that the algorithm rescales the variance matrix, starting from a zero scale factor (deterministic setup) then going to a one scale factor by small steps; at each step it recomputes an approximation of the decision rule, and then the fix point of that rule, and then uses that fix point as the point used for the Taylor expansion of next step; convergence is obtained when the fix point of the rule is the one around which it is computed.
• I can’t help you concerning the papers: it is always hard to reproduce the results obtained by authors, since methods may differ or be unexplicited