Optimal Policy

Hello everyone!
Could you please explain me how to estimate optimal policy rule and how Dynare estimates it? For example, I have a linear system of equations. Suppose that it’s of the form A0E{x(t+1)}+A1x(t)+A2x(t-1)+Be(t) = 0. Suppose that the last equattion is Taylor Rule. I can estimate all parameters (Sims algorithm + Maximum likelihood(or bayesian inference))/ But I want to estimate optimal policy using some quadratic loss function (e.g. with inflation and output). As far as I understand I should transfrom this system into Linear-Quadratic framework. But how to do this, cause I have expectation terms in the system of equations? And what should I do if I want to restrict admissible policy functions somehow (for example let it be functions of only some subset of variables, cause LQG yields control variable as function of all state variables)?

Any references are appreciated.

Excuse me for my Engish.
Thank you in advance!

you need to be more precise what type of optimal policy you want? Commitment (Ramsey) vs. discretion vs Simple Rule. Depending on that choice, the algorithms differ. For Ramsey, you don’t need to do LQ yourself.

Thank you for your answer!
I am interested at all these types. Can you explain me please why we don’t need for Ramsey LQ? Am I correctly understand that for other two type we should use LQ framework? Could you please give me relevant references with clear explanations? And what are main functions of Dynare that do main work for corresponding problems? I’d like to trce code files.
Thank you in advance

  1. For optimal simple rules, the relevant files is osr_1.m. The reason for going LQ in OSR is that the objective function is specified as a linear combination of (co)-variances, subject to a linear law of motion. Therefore, it is by construction linear quadratic. More details are in the Dynare manual.
  2. For discretionary_policy, the relevant file is discretionary_policy_1.m. The reference is Dennis (2007). The algorithm described in that paper is only applicable for LQ problems, therefore the restriction.

In both these case, you don’t have to linearize the model by hand, but you cannot go higher than

in the approximation of the constraints.

  1. For Ramsey policy, you enter the first order conditions of the private economy and the planner objective in nonlinear terms (the constraints can be linear, if desired), and Dynare will construct the Lagrangian of the planner’s problem, derive the FOC’s and then approximate those. For that reason, you don’t need to do LQ yourself. However, what you will end up with is essentially an LQ problem. The relevant function is ramsey_policy.m, although the derivation of the Lagrangian is done in the preprocessor.