4.12 Deterministic simulation

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 ‘1’ 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 n endogenous variables in T periods. Dynare offers several algorithms for solving this problem, which can be chosen via the stack_solve_algo-option. By default (stack_solve_algo=0), Dynare uses a Newton-type method to solve the simultaneous equation system. Because the resulting Jacobian is in the order of n by T 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 (stack_solve_algo=6) is based on a Newton-type algorithm first proposed by Laffargue (1990) and Boucekkine (1995), which uses relaxation techniques. Thereby, the algorithm avoids ever storing the full Jacobian. The details of the algorithm can be found in Juillard (1996). 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.

Command: perfect_foresight_setup ;

Command: perfect_foresight_setup (OPTIONS…);

Description

Prepares a perfect foresight simulation, by extracting the information in the initval, endval and shocks blocks and converting them into simulation paths for exogenous and endogenous variables.

This command must always be called before running the simulation with perfect_foresight_solver.

Options

periods = INTEGER

Number of periods of the simulation

datafile = FILENAME

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.

Output

The paths for the exogenous variables are stored into oo_.exo_simul.

The initial and terminal conditions for the endogenous variables and the initial guess for the path of endogenous variables are stored into oo_.endo_simul.

Command: perfect_foresight_solver ;

Command: perfect_foresight_solver (OPTIONS…);

Description

Computes the perfect foresight (or deterministic) simulation of the model.

Note that perfect_foresight_setup must be called before this command, in order to setup the environment for the simulation.

Options

maxit = INTEGER

Determines the maximum number of iterations used in the non-linear solver. The default value of maxit is 50.

tolf = DOUBLE

Convergence criterion for termination based on the function value. Iteration will cease when it proves impossible to improve the function value by more than tolf. Default: 1e-5

tolx = DOUBLE

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 tolx. Default: 1e-5

stack_solve_algo = INTEGER

Algorithm used for computing the solution. Possible values are:

0

Newton method to solve simultaneously all the equations for every period, using sparse matrices (Default).

1

Use a Newton algorithm with a sparse LU solver at each iteration (requires bytecode and/or block option, see Model declaration).

2

Use a Newton algorithm with a Generalized Minimal Residual (GMRES) solver at each iteration (requires bytecode and/or block option, see Model declaration; not available under Octave)

3

Use a Newton algorithm with a Stabilized Bi-Conjugate Gradient (BICGSTAB) solver at each iteration (requires bytecode and/or block option, see Model declaration).

4

Use a Newton algorithm with a optimal path length at each iteration (requires bytecode and/or block option, see Model declaration).

5

Use a Newton algorithm with a sparse Gaussian elimination (SPE) solver at each iteration (requires bytecode option, see Model declaration).

6

Use the historical algorithm proposed in Juillard (1996): it is slower than stack_solve_algo=0, but may be less memory consuming on big models (not available with bytecode and/or block options).

7

Allows the user to solve the perfect foresight model with the solvers available through option solve_algo (See solve_algo for a list of possible values, note that values 5, 6, 7 and 8, which require bytecode and/or block options, are not allowed). For instance, the following commands:

perfect_foresight_setup(periods=400);
perfect_foresight_solver(stack_solve_algo=7, solve_algo=9)

trigger the computation of the solution with a trust region algorithm.

robust_lin_solve

Triggers the use of a robust linear solver for the default stack_solve_algo=0.

solve_algo

See solve_algo. Allows selecting the solver used with stack_solve_algo=7.

no_homotopy

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.

markowitz = DOUBLE

Value of the Markowitz criterion, used to select the pivot. Only used when stack_solve_algo = 5. Default: 0.5.

minimal_solving_periods = INTEGER

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 stack_solve_algo = 5. Default: 1.

lmmcp

Solves the perfect foresight model with a Levenberg-Marquardt mixed complementarity problem (LMMCP) solver (Kanzow and Petra 2004), 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 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 Model declaration) 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:

model;
   ...
   [mcp = 'r > -1.94478']
   r = rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e;
   ...
end;

where 1.94478 is the steady state level of the nominal interest rate and r 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 r<=-1.94478, in which case the r is fixed at -1.94478 (thereby being equivalent to a complementary 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 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.

Note that in the current implementation, the content of the mcp 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).

endogenous_terminal_period

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 stack_solve_algo==0.

linear_approximation

Solves the linearized version of the perfect foresight model. The model must be stationary. Only available with option stack_solve_algo==0.

Output

The simulated endogenous variables are available in global matrix oo_.endo_simul.

Command: simul ;

Command: simul (OPTIONS…);

Description

Short-form command for triggering the computation of a deterministic simulation of the model. It is strictly equivalent to a call to perfect_foresight_setup followed by a call to perfect_foresight_solver.

Options

Accepts all the options of perfect_foresight_setup and perfect_foresight_solver.

MATLAB/Octave variable: oo_.endo_simul

This variable stores the result of a deterministic simulation (computed by perfect_foresight_solver or simul) or of a stochastic simulation (computed by stoch_simul with the periods option or by extended_path).

The variables are arranged row by row, in order of declaration (as in M_.endo_names). Note that this variable also contains initial and terminal conditions, so it has more columns than the value of periods option.

MATLAB/Octave variable: oo_.exo_simul

This variable stores the path of exogenous variables during a simulation (computed by perfect_foresight_solver, simul, stoch_simul or extended_path).

The variables are arranged in columns, in order of declaration (as in M_.exo_names). Periods are in rows. Note that this convention regarding columns and rows is the opposite of the convention for oo_.endo_simul!