# Toolbox API¶

## MATLAB Interface¶

iter_modname([options])

Run policy iterations.

Parameters

options – a Matlab struct that contains options and parameters to be overwritten. Notice only options that do not require recompiling can be overwritten. See Options

Returns

a Matlab struct that contains the structure of the problem and converged policy and state transition functions

simulate_modname(IterRslt[, options])

Simulate models using policy/state transition functions contained in Matlab struct IterRslt.

Parameters
• IterRslt – results returned by the policy iteration procedure

• options – a Matlab struct that contains options and parameters to be overwritten.

Returns

a Matlab struct that contains simulated panels of variables defined in var_simu

## Variable declaration¶

parameters var1 var2 ...

Declare parameters. Parameters are variables that do not change across states or over time. A parameter can be a vector. A vector parameter can be accessed in the model block using round brackets. For example,

parameters var1, var2;
var1 = 1.0;         % Scalar parameter
var2 = [2.0,3.0];   % Vector parameter with two elements
...
model;
a = var1;       % This assigns a the scalar parameter var1 (1.0)
b = var2(1);    % This assigns b the first element of parameter var2 (2.0)
...
end;

var_shock var1 var2 ...

Declare exogenous state variables. The number of elements of the cartesian set of var_shock is specified by shock_num. The full transition matrix is specified by shock_trans.

var_state var1 var2 ...

Declare endogenous state variables. A fixed grid for each state variable should be defined. The grid will be used for fixed-grid function approximations such as splines and linear interpolations. The range of the grid will be used for adaptive grid methods.

var_tensor var1 var2 ...

Declare tensor variables that are simple functions of var_shock and var_state.

Some variables are simple functions of var_shock and var_state, and can be evaluated out of the model block. For example, in the simple RBC model, output needs to enter the resource constraint and is a simple function of productivity and capital. We can declare output Y as a var_tensor and use it as following:

parameters alpha;
var_shock z;
var_state K;
var_tensor Y;

... % Definition of z and K here
Y = z.*K.^(alpha); % Assign Y

var_policy c K_next;
inbound c 0 Y;  % var_tensor can be used in inbound
inbound K_next 0 Y;

model;
resource_residual = Y + (1-delta)*K - c - K_next;
...
end;


As illustrated above, the advantage of using *var_tensor*s and constructing them before the model block is that they can be used to specify the bounds of policy variables, and they are used to avoid redundant evaluations, since these variables are functions of exogenous and endogenous state variables and do not depend on policy variables.

var_interp var1 var2 ...

Declare policy functions from the last iteration to be evaluated in the current iteration. An initial value should be specified for each var_interp. An update procedure should be specified for each var_interp after a policy iteration. A var_interp can be used as a function which take values of var_state as arguments in the model block, to evaluate future policy variables.

Convergence for policy iterations is reached when the maximum absolute difference of var_interp between two iterations is smaller than TolEq.

var_policy var1 var2[len2] ...

Declare policy variables that directly enter the system of equations.

A var_policy can be declared as a vector, for example, var2 of length len2 is declared as var2[len2] in the example. To access elements of a vector var_policy in the model block, use round bracket to index or use the prime (‘) operator to refer to the whole vector.

For each var_policy, its lower and upper bounds entering the equation solver should be defined as

inbound var1 var1_lb var1_ub;


If the lower and upper bounds of a var_policy cannot be determined ex-ante, specify tight bounds and use the adaptive bound option such as

inbound var1 var1_lb var1_ub adaptive(2.0);


This will adjust bounds by expanding the lower and upper bounds by a factor of 2.0 after each policy iteration, if UseAdaptiveBound is set to one. If option UseAdaptiveBoundInSol is set to one, after a failed attempt in finding a solution within the bounds and the equation solver returns a failed solution that hits the lower or upper bound, the bound will be expanded by a factor of 2.0.

var_aux var1 var2[len2] ...

Some policy variables of interests are simple functions of var_state and var_policy, and thus do not need to enter the system of equations as unknowns. These variables can be declared as var_aux.

They need to be evaluated in the model block so as to be returned.

var_output var1 var2 ...

A subset of var_policy or var_aux of which the function approximation (such as splines) parameters should be constructed. These function approximation parameters will be used in simulations if SIMU_INTERP is set to one.

var_others var1 var2 ...

Any variables in the MATLAB workspace that needs to be returned.

model; ... end;

Declare the model block.

The model;…end; block defines the system of equations for each collocation point of endogenous states and exogenous states. The equations should be eventually specified in the equation;…end; block, in which each line corresponds to one equation in the system. Any calculations in order to evaluate these equations are included in the model block preceding the equations block.

model_init; ... end;

Declare the model_init block.

This is similar to the model;…end; block, but is called only once at the start of the policy iteration. This is often used to define a last period problem as a starting point of the iteration, which potentially solves a different system of equations.

One can set option:SkipModelInit=1 to skip this block, so the policy iteration starts with a WarmUp specified in the option.

simulate; ... end;

Declare the simulation block.

The simulation block should define the initial exogenous state index and endogenous states (e.g. var1 and var2) as follows:

initial shock 1;
initial var1 some_value1;
initial var2 some_value2;


The simulation block should declare the transition of each endogenous variable as follows:

var1' = some_variable1;
var2' = some_variable2;


If the transition of an endogenous variable is given by indexing a vector var_policy or var_aux with the future exogenous state index, specify the transition as follows:

var1' = some_var_policy';
var2' = some_aux_policy';


The simulate block should declare variables to be recorded following keyword var_simu. A var_simu must be contained in var_policy or var_aux if SIMU_RESOLVE=1, or must be contained in var_output if SIMU_INTERP=1.

The simulate block can overwrite options num_samples (default 1) and num_periods (default 1000).

## Built-in functions¶

GDSGE_INTERP_VEC[vec_index](shock, var_state1, var_state2, ...)

Return each var_interp evaluated at (var_state1, var_state2, …) for the exogenous shock index referred by argument shock, according to the order defined in var_interp. For example,

...
var_shock z;
var_state x1 x2;
var_interp y1 y2 y3;
...
model;
x1_future = z;
x2_future = z^2;
[y1_future,y2_future,y3_future] = GDSGE_INTERP_VEC(shock,x1_future,x2_future);
end;


The optional vec_index specifies a subset of var_interp to be evaluated. For example, the following skips the evaluation of y2.

...
[y1_future,y3_future] = GDSGE_INTERP_VEC[1,3](shock,x1_future,x2_future);

Parameters
• shock – the index of exogenous state at which the evaluation is done. Keyword shock refers to the index at the current collocation point.

• var_state – values of endogenous states

• vec_index – matlab integer vector that specifies the index of var_interp to be evaluated and returned. Return all var_interp if omitted.

GDSGE_INTERP_VEC'[vec_index](var_state1, var_state2, ...)

Return each var_interp evaluated at (var_state1, var_state2, …) for each realization of exogenous states, returned according to the order defined in var_interp. The returned results should be assigned to a vector of variables (i.e., variables followed by a prime (‘)). For example,

...
shock_num = 2;
var_shock z;
var_state x1 x2;
var_interp y1 y2 y3;
...
model;
x1_future' = z';
x2_future' = z'^2;
[y1_future',y2_future',y3_future'] = GDSGE_INTERP_VEC'(x1_future',x2_future');
end;


The line with GDSGE_INTERP_VEC’ is equivalent to calling

[y1_future(1),y2_future(1),y3_future(1)] = GDSGE_INTERP_VEC(1,x1_future(1),x2_future(1));
[y1_future(2),y2_future(2),y3_future(2)] = GDSGE_INTERP_VEC(2,x1_future(2),x2_future(2));

Parameters
• var_state – values of endogenous states

• vec_index – matlab integer vector that specifies the index of var_interp to be evaluated and returned. Return all var_interp if omitted.

GDSGE_EXPECT{expression | trans_matrix=shock_trans}

Calculate the conditional expectation of an expression.

Parameters
• expression – mathematical expression to be calculated

• trans_matrix – the Markov transition matrix used to form conditional probability. The default value is shock_trans.

GDSGE_MAX{expression}

Calculate the maximum of expression across all realizations of exogenous states.

Parameters

expression – mathematical expression to be calculated

GDSGE_MIN{expression}

Calculate the minimum of expression across all realizations of exogenous states.

Parameters

expression – mathematical expression to be calculated

## Macros¶

The toolbox supports a set of macros to facilitate developing. The macro is going to be first preprocessed before the file is passed into the parser.

#define

Define a literal that will be replaced in the preprocessing. For example

#define N 8
shock_num = 8;
var_policy w1n[N];


The block is equivalent to

shock_num = 8;
var_policy w1n[8];


Currently, #define can only appear at the beginning of a gmod file, and cannot be “undefined”.

#for ... #end

Define a for block that will be expanded in the preprocessing. For example

#define N 3
#for i=1:N
var_state K_#i;
#end


The block expands to

var_state K_1;
var_state K_2;
var_state K_3;


Notice the iterator (i in the example) appears in the block preceded by a #.

This is convenient to write multi-agent models with agents sharing similar problems, or problems with equilibrium conditions that depend on the current realization of exogenous states. See example Heaton and Lucas with Transition Function Iterations.

Currently the toolbox does not support nested #for loops.

## Options¶

An option named in all caps (e.g., USE_SPLINE) requires recompiling (via a local or remote compiler) when its value is changed. Other options can be simply specified through a structure to overwrite existing values without recompiling. For example,

>> options.SaveFreq = 100;      % Change saving frequency in policy iterations
>> IterRslt = iter_modname(options);
...
>> options.num_samples = 100;   % Change number of sample paths in simulations
>> SimuRslt = simulate_modname(IterRslt,options);


### Warm ups¶

WarmUp

Pass a converged policy iteration solution (returned by the iter file) as the starting point of the policy iteration. For example,

>> IterRslt = iter_modname;     % Solve the initial problem
>> options.WarmUp = IterRslt;   % Specify the starting point
>> options.alpha = 0.5;         % Change the parameter value
>> IterRslt = iter_modname(options)     % Starting from the previous converged solution (contained in *IterRslt*) and solving under the new parameter


The WarmUp passed in does not need to be solved over the same grids as the current problem, and thus can be used as starting point to refine solutions over finer grids. See example Mendoza (2010).

Default: Empty

### Policy iterations¶

SkipModelInit

Skip the model_init block. Start iterations with var_interp in the WarmUp.

TolEq

Convergence criterion for the policy iteration. The policy iteration stops when the maximum absolute distance of var_interp across all collocation points between two iterations is smaller than TolEq. Default: 1e-6

MaxIter

Maximum iterations for policy iterations. Default: inf

MaxMinorIter

Maximum minor iterations for randomizing initial guesses when solutions are not found. Default: inf

UseAdaptiveBound

Activate the adaptive bound procedure after each policy iteration. For example, when UseAdaptiveBound is set to one, the inbound derivative specified as following

inbound x 0.0 1.0 adaptive(2);


expands the lower and upper bounds by a factor of 2 from the converged solutions of each collocation point, after each policy iteration.

Takes value 0 or 1 (default).

UseAdaptiveBoundInSol

Use adaptive bound in randomization after a failed search of solutions within the current bounds, and the search fails at an immature step that hits the current bounds. Takes value 0 (default) or 1.

### Simulation¶

SIMU_RESOLVE

Whether resolving the system of equations in simulations. Takes value of 0 or 1 (default). Only one of SIMU_RESOLVE and SIMU_INTERP can take value one.

SIMU_INTERP

Whether directly interpolating the policy and state transition functions in simulations. Takes value of 0 (default) or 1. Only one of SIMU_RESOLVE and SIMU_INTERP can take value one.

SimuSeed

The seed for random number generators in simulations. Default: 0823.

EnforceSimuStateInbound

Whether enforcing endogenous states are inbound for each period, after interpolating state transition functions. Effective only when SIMU_INTERP = 1.

num_samples

Number of sample paths. Default: 1

num_periods

Number of periods of each sample path. Default: 1000

### Function approximations¶

USE_SPLINE

Whether to use multi-dimensional linear interpolations or cubic splines for function approximations. Takes value of 0 or 1 (default).

Only one of USE_SPLINE, USE_PCHIP, USE_ASG can be set to one.

USE_PCHIP

Whether to use shape-preserving cubic interpolations for function approximations. Takes value of 0 (default) or 1.

Only one of USE_SPLINE, USE_PCHIP, USE_ASG can be set to one.

USE_ASG

Whether to use the adaptive sparse grid method for function approximations. Takes value of 0 (default) or 1.

Only one of USE_SPLINE, USE_PCHIP, USE_ASG can be set to one.

INTERP_ORDER

Takes value of 2 (default) or 4. Only effective if USE_SPLINE=1. INTERP_ORDER=2 corresponds to linear interpolation and INTERP_ORDER=4 corresponds to cubic splines with natural end conditions.

EXTRAP_ORDER

The order of extrapolations when extrapolating a spline. Takes value of 2 (default) or 4. Only effective if USE_SPLINE=1 and INTERP_ORDER=4.

AsgMaxLevel

The maximum level used in the adaptive sparse grid method. Default 10. Only effective if USE_ASG=1.

AsgMinLevel

The minimum level used in the adaptive sparse grid method. Default 4. Only effective if USE_ASG=1.

AsgThreshold

The tolerance for refinement in the adaptive sparse grid method. Default 1e-2. Only effective if USE_ASG=1.

AsgOutputMaxLevel

The max level used in the adaptive sparse grid method for var_output. Default 10. Only effective if USE_ASG=1.

AsgOutputThreshold

The tolerance for refinement in the adaptive sparse grid method for var_output. Default MIN(1e-2, AsgThreshold). Only effective if USE_ASG=1.

AsgFixGrid

Whether fix the adaptive grid to the one passed in struct WarmUp. Takes value of 0 (default) or 1.

### Equation Solver¶

TolSol

Absolute residual tolerance in equation solving. Default: 1e-8

SolMaxIter

Maximum number of function evaluations in equation solving. Default: 200

### Miscellaneous¶

NumThreads

Number of (OpenMP) threads in policy iterations and simulations. Default: the number of cores (via Matlab function feature('numcores'))

### DEBUG¶

GNDSGE_DEBUG_EVAL_ONLY

Only evaluate the system of equations instead of solving it.

IterSaveAll

Save all variables in the workspace after policy iterations.