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<a name="Deterministic-simulation"></a>
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Next: <a href="Stochastic-solution-and-simulation.html#Stochastic-solution-and-simulation" accesskey="n" rel="next">Stochastic solution and simulation</a>, Previous: <a href="Getting-information-about-the-model.html#Getting-information-about-the-model" accesskey="p" rel="prev">Getting information about the model</a>, Up: <a href="The-Model-file.html#The-Model-file" accesskey="u" rel="up">The Model file</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Command-and-Function-Index.html#Command-and-Function-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Deterministic-simulation-1"></a>
<h3 class="section">4.12 Deterministic simulation</h3>

<p>When the framework is deterministic, Dynare can be used for models
with the assumption of perfect foresight. Typically, the system is
supposed to be in a state of equilibrium before a period &lsquo;<samp>1</samp>&rsquo; when
the news of a contemporaneous or of a future shock is learned by the
agents in the model. The purpose of the simulation is to describe the
reaction in anticipation of, then in reaction to the shock, until the
system returns to the old or to a new state of equilibrium. In most
models, this return to equilibrium is only an asymptotic phenomenon,
which one must approximate by an horizon of simulation far enough in
the future.  Another exercise for which Dynare is well suited is to
study the transition path to a new equilibrium following a permanent
shock.  For deterministic simulations, the numerical problem consists of solving
 a nonlinar system of simultaneous equations in <code>n</code> endogenous
 variables in <code>T</code> periods. Dynare offers several algorithms for
 solving this problem, which can be chosen via the
 <code>stack_solve_algo</code>-option. By default (<code>stack_solve_algo=0</code>),
Dynare uses a Newton-type method to solve the simultaneous equation
system. Because the resulting Jacobian is in the order of <code>n</code> by
<code>T</code> and hence will be very large for long simulations with many
variables, Dynare makes use of the sparse matrix capacities of
MATLAB/Octave. A slower but potentially less memory consuming alternative
(<code>stack_solve_algo=6</code>) is based on a Newton-type algorithm first
proposed by <cite>Laffargue (1990)</cite> and <cite>Boucekkine (1995)</cite>, which
uses relaxation techniques. Thereby, the algorithm avoids ever storing
the full Jacobian. The details of the algorithm can be found in
<cite>Juillard (1996)</cite>. The third type of algorithms makes use of block
decomposition techniques (divide-and-conquer methods) that exploit the
structure of the model. The principle is to identify recursive and
simultaneous blocks in the model structure and use this information to
aid the solution process.  These solution algorithms can provide a
significant speed-up on large models.
</p>

<dl>
<dt><a name="index-perfect_005fforesight_005fsetup"></a>Command: <strong>perfect_foresight_setup</strong> <em>;</em></dt>
<dt><a name="index-perfect_005fforesight_005fsetup-1"></a>Command: <strong>perfect_foresight_setup</strong> <em>(<var>OPTIONS</var>&hellip;);</em></dt>
<dd>
<p><em>Description</em>
</p>
<p>Prepares a perfect foresight simulation, by extracting the information in the
<code>initval</code>, <code>endval</code> and <code>shocks</code> blocks and converting them into
simulation paths for exogenous and endogenous variables.
</p>
<p>This command must always be called before running the simulation with
<code>perfect_foresight_solver</code>.
</p>
<p><em>Options</em>
</p>
<dl compact="compact">
<dt><code>periods = <var>INTEGER</var></code></dt>
<dd><p>Number of periods of the simulation
</p>
</dd>
<dt><code>datafile = <var>FILENAME</var></code></dt>
<dd><p>If the variables of the model are not constant over time, their
initial values, stored in a text file, could be loaded, using that
option, as initial values before a deterministic simulation.
</p></dd>
</dl>

<p><em>Output</em>
</p>
<p>The paths for the exogenous variables are stored into <code>oo_.exo_simul</code>.
</p>
<p>The initial and terminal conditions for the endogenous variables and the
initial guess for the path of endogenous variables are stored into
<code>oo_.endo_simul</code>.
</p>
</dd></dl>


<dl>
<dt><a name="index-perfect_005fforesight_005fsolver"></a>Command: <strong>perfect_foresight_solver</strong> <em>;</em></dt>
<dt><a name="index-perfect_005fforesight_005fsolver-1"></a>Command: <strong>perfect_foresight_solver</strong> <em>(<var>OPTIONS</var>&hellip;);</em></dt>
<dd>
<p><em>Description</em>
</p>
<p>Computes the perfect foresight (or deterministic) simulation of the model.
</p>
<p>Note that <code>perfect_foresight_setup</code> must be called before this command, in
order to setup the environment for the simulation.
</p>
<p><em>Options</em>
</p>
<dl compact="compact">
<dt><code>maxit = <var>INTEGER</var></code></dt>
<dd><p>Determines the maximum number of iterations used in the non-linear solver. The
default value of <code>maxit</code> is 50. 
</p>
</dd>
<dt><code>tolf = <var>DOUBLE</var></code></dt>
<dd><p>Convergence criterion for termination based on the function value. Iteration will cease when it proves impossible to
improve the function value by more than <code>tolf</code>. Default: <code>1e-5</code>
</p>
</dd>
<dt><code>tolx = <var>DOUBLE</var></code></dt>
<dd><p>Convergence criterion for termination based on the change in the function argument. Iteration will cease when the solver 
attempts to take a step that is smaller than <code>tolx</code>. Default: <code>1e-5</code>
</p>
</dd>
<dt><code>stack_solve_algo = <var>INTEGER</var></code></dt>
<dd><p>Algorithm used for computing the solution. Possible values are:
</p>
<dl compact="compact">
<dt><code>0</code></dt>
<dd><p>Newton method to solve simultaneously all the equations for every
period, using sparse matrices (Default).
</p>
</dd>
<dt><code>1</code></dt>
<dd><p>Use a Newton algorithm with a sparse LU solver at each iteration
(requires <code>bytecode</code> and/or <code>block</code> option, see <a href="Model-declaration.html#Model-declaration">Model declaration</a>).
</p>
</dd>
<dt><code>2</code></dt>
<dd><p>Use a Newton algorithm with a Generalized Minimal Residual (GMRES)
solver at each iteration (requires <code>bytecode</code> and/or <code>block</code>
option, see <a href="Model-declaration.html#Model-declaration">Model declaration</a>; not available under Octave)
</p>
</dd>
<dt><code>3</code></dt>
<dd><p>Use a Newton algorithm with a Stabilized Bi-Conjugate Gradient
(BICGSTAB) solver at each iteration (requires <code>bytecode</code> and/or
<code>block</code> option, see <a href="Model-declaration.html#Model-declaration">Model declaration</a>).
</p>
</dd>
<dt><code>4</code></dt>
<dd><p>Use a Newton algorithm with a optimal path length at each iteration
(requires <code>bytecode</code> and/or <code>block</code> option, see <a href="Model-declaration.html#Model-declaration">Model declaration</a>).
</p>
</dd>
<dt><code>5</code></dt>
<dd><p>Use a Newton algorithm with a sparse Gaussian elimination (SPE) solver
at each iteration (requires <code>bytecode</code> option, see <a href="Model-declaration.html#Model-declaration">Model declaration</a>).
</p>
</dd>
<dt><code>6</code></dt>
<dd><p>Use the historical algorithm proposed in <cite>Juillard (1996)</cite>: it is
slower than <code>stack_solve_algo=0</code>, but may be less memory consuming
on big models (not available with <code>bytecode</code> and/or <code>block</code>
options).
</p>
</dd>
<dt><code>7</code></dt>
<dd><p>Allows the user to solve the perfect foresight model with the solvers available
through option <code>solve_algo</code> (See <a href="Steady-state.html#solve_005falgo">solve_algo</a> for a list of possible
values, note that values 5, 6, 7 and 8, which require <code>bytecode</code> and/or
<code>block</code> options, are not allowed). For instance, the following commands:
</p><div class="example">
<pre class="example">perfect_foresight_setup(periods=400);
perfect_foresight_solver(stack_solve_algo=7, solve_algo=9)
</pre></div>
<p>trigger the computation of the solution with a trust region algorithm.
</p>
</dd>
</dl>

</dd>
<dt><code>robust_lin_solve</code></dt>
<dd><p>Triggers the use of a robust linear solver for the default <code>stack_solve_algo=0</code>. 
</p>
</dd>
<dt><code>solve_algo</code></dt>
<dd><p>See <a href="Steady-state.html#solve_005falgo">solve_algo</a>. Allows selecting the solver used with <code>stack_solve_algo=7</code>.
</p>
</dd>
<dt><code>no_homotopy</code></dt>
<dd><p>By default, the perfect foresight solver uses a homotopy technique if it cannot
solve the problem. Concretely, it divides the problem into smaller steps by
diminishing the size of shocks and increasing them progressively until the
problem converges. This option tells Dynare to disable that behavior. Note that
the homotopy is not implemented for purely forward or backward models.
</p>
</dd>
<dt><code>markowitz = <var>DOUBLE</var></code></dt>
<dd><p>Value of the Markowitz criterion, used to select the pivot. Only used
when <code>stack_solve_algo = 5</code>. Default: <code>0.5</code>.
</p>
</dd>
<dt><code>minimal_solving_periods = <var>INTEGER</var></code></dt>
<dd><p>Specify the minimal number of periods where the model has to be
solved, before using a constant set of operations for the remaining
periods. Only used when <code>stack_solve_algo = 5</code>. Default: <code>1</code>.
</p>
</dd>
<dt><code>lmmcp</code></dt>
<dd><a name="lmmcp"></a><p>Solves the perfect foresight model with a Levenberg-Marquardt mixed complementarity problem (LMMCP) solver
(<cite>Kanzow and Petra 2004</cite>), which allows to consider inequality constraints on the endogenous variables
(such as a ZLB on the nominal interest rate or a model with irreversible
investment). This option is equivalent to <code>stack_solve_algo=7</code> <strong>and</strong>
<code>solve_algo=10</code>. Using the LMMCP solver requires a particular model setup as the goal is to get rid of 
any <code>min/max</code> operators and complementary slackness conditions that might introduce 
a singularity into the Jacobian. This is done by attaching an equation tag (see <a href="Model-declaration.html#Model-declaration">Model declaration</a>)
with the <code>mcp</code> 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:
</p><div class="example">
<pre class="example">model;
   ...
   [mcp = 'r &gt; -1.94478']
   r = rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e;
   ...
end;
</pre></div>
<p>where 1.94478 is the steady state level of the nominal interest rate and
<code>r</code> is the nominal interest rate in deviation from the steady state. This construct implies that
the Taylor rule is operative, unless the implied interest rate <code>r&lt;=-1.94478</code>, in which case the 
<code>r</code> is fixed at <code>-1.94478</code> (thereby being equivalent to a complementary slackness
condition). By restricting the value of <code>r</code> coming out of this equation, the
<code>mcp</code>-tag also avoids using <code>max(r,-1.94478)</code> for other occurrences of <code>r</code> in the 
rest of the model. It is important to keep in mind that, because the <code>mcp</code>-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.
</p>
<p>Note that in the current implementation, the content of the <code>mcp</code> equation tag is not parsed by the
preprocessor. The inequalities must therefore be as simple as possible: an endogenous
variable, followed by a relational operator, followed by a number (not a
variable, parameter or expression). 
</p>
</dd>
<dt><code>endogenous_terminal_period</code></dt>
<dd><p>The  number of  periods is  not constant  across Newton  iterations when
solving the perfect foresight model. The size of the nonlinear system of
equations  is  reduced  by  removing  the  portion  of  the  paths  (and
associated equations) for which the solution has already been identified
(up to the  tolerance parameter). This strategy can be  interpreted as a
mix  of the  shooting and  relaxation  approaches. Note  that round  off
errors are  more important with  this mixed strategy (user  should check
the reported value  of the maximum absolute error).  Only available with
option <code>stack_solve_algo==0</code>.
</p>
</dd>
<dt><code>linear_approximation</code></dt>
<dd><p>Solves the linearized version of the perfect foresight model. The model must be
stationary. Only available with option <code>stack_solve_algo==0</code>.
</p>
</dd>
</dl>

<p><em>Output</em>
</p>
<p>The simulated endogenous variables are available in global matrix
<code>oo_.endo_simul</code>.
</p>
</dd></dl>

<dl>
<dt><a name="index-simul"></a>Command: <strong>simul</strong> <em>;</em></dt>
<dt><a name="index-simul-1"></a>Command: <strong>simul</strong> <em>(<var>OPTIONS</var>&hellip;);</em></dt>
<dd>
<p><em>Description</em>
</p>
<p>Short-form command for triggering the computation of a deterministic simulation
of the model. It is strictly equivalent to a call to
<code>perfect_foresight_setup</code> followed by a call to
<code>perfect_foresight_solver</code>.
</p>
<p><em>Options</em>
</p>
<p>Accepts all the options of <code>perfect_foresight_setup</code> and
<code>perfect_foresight_solver</code>.
</p>
</dd></dl>
<a name="oo_005f_002eendo_005fsimul"></a><dl>
<dt><a name="index-oo_005f_002eendo_005fsimul"></a>MATLAB/Octave variable: <strong>oo_.endo_simul</strong></dt>
<dd><p>This variable stores the result of a deterministic simulation (computed by
<code>perfect_foresight_solver</code> or <code>simul</code>) or of a stochastic simulation
(computed by <code>stoch_simul</code> with the <code>periods</code> option or by
<code>extended_path</code>).
</p>
<p>The variables are arranged row by row, in order of declaration (as in
<code>M_.endo_names</code>). Note that this variable also contains initial
and terminal conditions, so it has more columns than the value of
<code>periods</code> option.
</p></dd></dl>

<a name="oo_005f_002eexo_005fsimul"></a><dl>
<dt><a name="index-oo_005f_002eexo_005fsimul"></a>MATLAB/Octave variable: <strong>oo_.exo_simul</strong></dt>
<dd><p>This variable stores the path of exogenous variables during a simulation
(computed by <code>perfect_foresight_solver</code>, <code>simul</code>, <code>stoch_simul</code>
or <code>extended_path</code>).
</p>
<p>The variables are arranged in columns, in order of declaration (as in
<code>M_.exo_names</code>). Periods are in rows. Note that this convention
regarding columns and rows is the opposite of the convention for
<code>oo_.endo_simul</code>!
</p>
</dd></dl>

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