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Merge branch 'doc_mcp' of git.dynare.org:JohannesPfeifer/dynare 

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bgp-dev
Sébastien Villemot 2022-09-14 17:32:40 +02:00
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doc/manual/source/the-model-file.rst
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@ -2993,7 +2993,9 @@ Finding the steady state with Dynare nonlinear solver
``10``
Levenberg-Marquardt mixed complementarity problem
(LMMCP) solver (*Kanzow and Petra (2004)*).
(LMMCP) solver (*Kanzow and Petra (2004)*). The complementarity
conditions are specified with an ``mcp`` equation tag, see
:opt:`lmmcp`.
``11``
@ -3764,16 +3766,27 @@ speed-up on large models.
the endogenous variables (such as a ZLB on the nominal interest
rate or a model with irreversible investment). This option is
equivalent to ``stack_solve_algo=7`` **and**
``solve_algo=10``. Using the LMMCP solver requires a particular
model setup as the goal is to get rid of any min/max operators
and complementary slackness conditions that might introduce a
singularity into the Jacobian. This is done by attaching an
equation tag (see :ref:`model-decl`) with the ``mcp`` keyword
to affected equations. This tag states that the equation to
which the tag is attached has to hold unless the expression
within the tag is binding. For instance, a ZLB on the nominal
interest rate would be specified as follows in the model
block::
``solve_algo=10``. Using the LMMCP solver avoids the need for min/max
operators and explicit complementary slackness conditions in the model
as they will typically introduce a singularity into the Jacobian. This is
done by setting the problem up as a mixed complementarity problem (MCP) of the form:
.. math::
LB = X &\Rightarrow F(X)>0\\
LB\leq X \leq UB &\Rightarrow F(X)=0\\
X =UB &\Rightarrow F(X)<0.
where :math:`X` denotes the vector of endogenous variables, :math:`F(X)` the equations
of the model, :math:`LB` denotes a lower bound, and :math:`UB` an upper bound. Such a setup
is implemented by attaching an equation tag (see :ref:`model-decl`)
with the ``mcp`` keyword to the affected equations. This tag states that
the equation to which the tag is attached has to hold unless the inequality
constraint within the tag is binding.
For instance, a ZLB on the nominal interest rate would be specified
as follows in the model block::
model;
...
@ -3791,20 +3804,27 @@ speed-up on large models.
slackness condition). By restricting the value of ``r`` coming
out of this equation, the ``mcp`` tag also avoids using
``max(r,-1.94478)`` for other occurrences of ``r`` in the rest
of the model. It is important to keep in mind that, because the
of the model. Two things are important to keep in mind. First, because the
``mcp`` tag effectively replaces a complementary slackness
condition, it cannot be simply attached to any
equation. Rather, it must be attached to the correct affected
equation as otherwise the solver will solve a different problem
than originally intended. Also, since the problem to be solved
is nonlinear, the sign of the residuals of the dynamic equation
matters. In the previous example, for the nominal interest rate
rule, if the LHS and RHS are reversed the sign of the residuals
(the difference between the LHS and the RHS) will change and it
may happen that solver fails to identify the solution path. More
generally, convergence of the nonlinear solver is not guaranteed
when using mathematically equivalent representations of the same
equation.
than originally intended. Second, the sign of the residual of the dynamic
equation must conform to the MCP setup outlined above. In case of the ZLB,
we are dealing with a lower bound. Consequently, the dynamic equation
needs to return a positive residual. Dynare by default computes the residual
of an equation ``LHS=RHS`` as ``residual=LHS-RHS``, while an implicit equation
``LHS`` is interpreted as ``LHS=0``. For the above equation this implies
``residual= r - (rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e);``
which is correct, since it will be positive if the implied interest rate
``rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e`` is
below ``r=-1.94478``. In contrast, specifying the equation as
``rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e = r;```
would be wrong.
Note that in the current implementation, the content of the
``mcp`` equation tag is not parsed by the preprocessor. The