Introduction
Authors
Abstract Classes
Summary
Bibliography
Toward Generic Software Tools for Computing
Optimal Policies in Nonlinear DSGE Models With
Occasionally Binding Constraints
Gary S. Anderson
[email protected]
Board of Governors of the Federal Reserve System
Division of Monetary Affairs
Society for Computational Economics
18th International Conference
Computing in Economics and Finance
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Outline
1
Introduction
2
Authors
3
Abstract Classes
4
Summary
5
Bibliography
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Outline
1
Introduction
2
Authors
3
Abstract Classes
4
Summary
5
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Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
DSGE Models with Occasionally Binding Constraints
Increased interest in solving models with constraints that
occasionally bind
multi-sector models with limits on inter-sectoral mobility of
factors
heterogeneous agent models with constraints on financial
assets available to agents
macro models with a zero lower bound on nominal interest
rates
Long history of researchers developing and applying a variety
of strategies
Challenging problems. Even without constraints must work
with approximations
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
An Example
As a concrete example(Christiano & Fisher, 2000) studied a
simple stochastic growth model with irreversible investment.
max E0
∞
X
β t U(ct )
t=0
with
ct + e kt+1 − (1 − δ)e kt ≤ f (kt , θt ) ≡ e (θt +αkt )
and gross investment non-negative
e kt+1 − (1 − δ)e kt ≥ 0
They provided several equation systems characterizing a
solution
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Policy and Lagrange Multiplier Functions
Find time invariant functions g , h such that given
Z
R(k, θ; g , h) = Uc (k, g (k, θ), θ) − h(k, θ) − β
m(g (k, θ), θ0 ; g , h)pθ (θ0 |θ)dθ0
then
m(k 0 , θ0 ; g , h) = Uc (k 0 , g (k 0 , θ0 , θ0 )[fk (k 0 , θ0 ) + 1 − δ] − h(k 0 , θ0 )(1 − δ) ≥ 0
R(k, θ; g , h) = 0
e
g (k,θ)
− (1 − δ)e k ≥ 0, h(k, θ) ≥ 0
h(k, θ)[e g (k,θ) − (1 − δ)e k ] = 0
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
A Parameterized Expectations Solution
The following is one of several PEA solutions in(Christiano &
Fisher, 2000). Find γ such that
R̄(k, θ; γ) = 0
where
R̄(k, θ; γ) = e
γ(k,θ)
Z
−
m(g (k, θ), θ0 ; g , h)pθ (θ0 |θ)dθ0
with g , h implicitly defined by
Uc (k, ḡ (K , θ), θ) = βe γ(k,θ)
with
(
g (k, θ) =
ḡ (k, θ)
log(1 − δ) + k
Anderson
if ḡ (k, θ) > log(1 − δ) + k
otherwise
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Some Authors Providing Down-loadable Code
(Haefke, 1998) FORTRAN, Gauss, MATLAB
(Maliar & Maliar, 2005a; Maliar & Maliar, 2005b)MATLAB
(Aruoba et al., 2006)FORTRAN90, MATLAB
(Carroll, 2006) Mathematica, MATLAB.
(Adam & Billi, 2006; Adam & Billi, 2007; Billi, 2007)MATLAB
(Nakov, 2008) MATLAB
(Hintermaier & Koeniger, 2010) MATLAB
(Fella, 2011) FORTRAN95
(Gordon, 2011) MATLAB
(Fernndez-Villaverde et al., 2012)FORTRAN90
(Iskhakov et al., 2012)MATLAB Presented yesterday at this
conference
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
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Abstract Classes
Summary
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Some Authors Providing Algorithms Only
(Marcet & Marshall, 1994)
(Krusell et al., 1997)
(Christiano & Fisher, 2000)
(Grüne & Semmler, 2004)
(Dennis, 2007)
(Benigno et al., 2009)
(Brumm & Grill, 2010)
(Judd et al., 2010)
(Marcet & Marimon, 2011)
(Malin et al., 2011)
(Ludwig & Schön, 2012)
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
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Generic Algorithm
Policy Function/ Lagrange Multiplier
Parameterized Expectations
1
Choose a function approximation method
2
Choose a metric for judging approximation quality
3
Guess parameters characterizing functions solving the
equation system
Use equation system to generate an improved guess
4
identify strategic ordering of subsets of the equation system to
facilitate solution
identify function evaluation points
compute expected values
solve for new parameters characterizing functions
5
If functions changed significantly repeat 4
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
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Abstract Classes
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Little Apparent Code Reuse
Very limited code reuse
Notable exception – several authors use the COMPECON
tools(Miranda & Fackler, 2002)
Down-loadable code typically written in very model specific
ways
Similarities in goals and methods hidden by idiosyncratic
model differences and coding conventions
Author’s with algorithmic innovations typically chose to build
a complete DSGE solution framework
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
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Abstract Classes
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Benefits from Interchangeable Components
Interchangeable components would facilitate experimentation
with alternative algorithmic designs
Instrumented with timers and memory monitors, could help
guide production code development
Design patterns(Gamma et al., 1995)
Template Method Define the skeleton of an algorithm in an
operation, deferring some steps to subclasses.
Template Method lets subclasses redefine
certain steps of an algorithm without changing
the algorithm’s structure.
Strategy Define a family of algorithms, encapsulate each
one, and make them interchangeable. Strategy
lets the algorithm vary independently from
clients that use it.
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
Outline
1
Introduction
2
Authors
3
Abstract Classes
4
Summary
5
Bibliography
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
Outline
1
Introduction
2
Authors
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
Function Approximation
Approximation Metric
Evaluation Points
Expectations
3
Abstract Classes
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
(Christiano & Fisher, 2000)
The Generic Algorithm
Component
Function Approximation
Approximation Metric
Evaluation Points
Expectations
Implementation
Chebyshev Polynomials; Finite element piecewise linear
Approximation Parameter Fixed
Point; Galerkin and Collocation
variants of Weighted Residuals
Chebyshev Nodes
Gaussian quadrature, Monte Carlo
Integration
Implemented several PEA variants
Piecewise linear approximation methods for policy function
iteration (PFI)
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
Outline
1
Introduction
2
Authors
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
Function Approximation
Approximation Metric
Evaluation Points
Expectations
3
Abstract Classes
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
(Miranda & Fackler, 2002)
The authors provide down-loadable MATLAB code for solving
f [st , xt , Et h(st+1 , xt+1 )] = φt
where
st+1 = g (st , xt , t+1 )
and
a(st ) ≤ xt ≤ b(st ), xjt > aj (st ) ⇒ φjt ≤ 0, xjt < bj (st ) ⇒ φjt ≥ 0,
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
(Billi, 2011)
The Generic Algorithm
Component
Function Approximation
Approximation Metric
Evaluation Points
Expectations
Implementation
piecewise linear
Approximated Function at Nodes
Fixed Point; Collocation; Time Iteration
uniform grid
Gaussian quadrature
performs policy function iteration (PFI)
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
(Nakov, 2008)
The Generic Algorithm
Component
Function Approximation
Approximation Metric
Evaluation Points
Expectations
Implementation
piecewise linear
Approximation parameter
point
uniform grid
Gaussian Quadrature
fixed
performs policy function iteration (PFI)
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
Outline
1
Introduction
2
Authors
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
Function Approximation
Approximation Metric
Evaluation Points
Expectations
3
Abstract Classes
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
The Endogenous Grid Method (EGM)
The Generic Algorithm
Originally developed in (Carroll, 2006) – Mathematica and
MATLAB code available. Extended to perform value function
iteration by (Barillas & Fernandez-Villaverde, 2007) –
Fortran90 code available.
(Krueger & Ludwig, 2007; Rendahl, 2006) show that time
iteration, nesting EGM, is often applicable and useful
Applied to a model with occasionally binding constraints in
(Hintermaier & Koeniger, 2010) – MATLAB code
Extended to a class of non-concave problems in (Fella, 2011;
Iskhakov et al., 2012) – Fortran95 and MATLAB code
available.
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
(Barillas & Fernandez-Villaverde, 2007)
The Generic Algorithm
Component
Function Approximation
Approximation Metric
Evaluation Points
Expectations
Implementation
Piecewise linear
Approximated Function Fixed
Point at Nodes
uniform grid
Tauchen –41 discrete statesa
a
Adapted from code for (Ljungqvist & Sargent, 2004; Miranda & Fackler,
2002)
Extends EGM to value function iteration (VFI)
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
(Fella, 2011)
The Generic Algorithm
Component
Function Approximation
Approximation Metric
Evaluation Points
Expectations
a
Implementation
Piecewise linear
Approximated Function at Nodes
Fixed Point
7 Uniformly spaced points for discrete durable; double exponential
grid for assets
Tauchen – 49 discrete statesa
Adapted from code for (Barillas & Fernandez-Villaverde, 2007)
Extends the EGM to non-convex problems including discrete
state space
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
Outline
1
Introduction
2
Authors
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
Function Approximation
Approximation Metric
Evaluation Points
Expectations
3
Abstract Classes
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
(Judd et al., 2010)
The Generic Algorithm
Component
Function Approximation
Approximation Metric
Evaluation Points
Expectations
Implementation
Polynomial interpolation for subset
of endogenous state variables.
Approximated Function Fixed
Point at Nodes
endogenous solution domain determined by extent of ergodic set
non product monomial and one
point quadrature rules. Gaussian
quadrature for accuracy tests.
Cluster Grid approach chooses grid points endogenously based
on the extent of ergodic set
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
(Brumm & Grill, 2010)
The Generic Algorithm
Component
Function Approximation
Approximation Metric
Evaluation Points
Expectations
Implementation
Piecewise linear – Adaptive Simplicial Interpolation (ASI)
Approximated Function Fixed
Point at Nodes
Uniform grid augmented with
endogenously determined grids
points at function kinks
Discrete Markov Process
Adaptive Simplicial Interpolation endogenously places grid
points at “kinks” and uses Delaunay interpolation
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
(Fernndez-Villaverde et al., 2012)
The Generic Algorithm
Component
Function Approximation
Approximation Metric
Evaluation Points
Expectations
Implementation
Smolyak Projections and Interpolation; Complete polynomials
Selected Approximated Function
Value at Node points
Time Iteration guess at collocation
point. use them as t+1 functions
to compute time t functions, repeat till no change
Smolyak points
One of a number of authors using Smolyak points along with
complete polynomials
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
Outline
1
Introduction
2
Authors
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
Function Approximation
Approximation Metric
Evaluation Points
Expectations
3
Abstract Classes
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
Function Approximation Components
The Generic Algorithm
Chebyshev Polynomials(Christiano & Fisher, 2000)
Generic Polynomial interpolation(Judd et al., 2010)
Finite element piecewise linear(Christiano & Fisher, 2000;
Billi, 2011; Nakov, 2008; Fella, 2011; Barillas &
Fernandez-Villaverde, 2007)
Piecewise linear Adaptive Simplicial Interpolation
(ASI)(Brumm & Grill, 2010)
Smolyak Projections and Interpolation; Complete
polynomials(Fernndez-Villaverde et al., 2012)
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
Approximation Metric
The Generic Algorithm
Approximation Parameter Fixed Point Galerkin(Christiano &
Fisher, 2000)
Approximation Parameter Fixed Point Collocation (Fella,
2011)
Approximated Function at Nodes Fixed Point; Collocation
(Billi, 2011; Nakov, 2008; Barillas & Fernandez-Villaverde,
2007; Judd et al., 2010; Brumm & Grill, 2010)
Selected Approximated Function Value at Node
points(Fernndez-Villaverde et al., 2012)
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
Evaluation Points
The Generic Algorithm
Endogenous grid points(Carroll, 2006; Barillas &
Fernandez-Villaverde, 2007; Krueger & Ludwig, 2007;
Rendahl, 2006; Hintermaier & Koeniger, 2010; Fella, 2011)
Smolyak points(Fernndez-Villaverde et al., 2012)
Uniform grid(Billi, 2011; Nakov, 2008)
Uniform grid augmented with endogenously determined grids
points at function kinks(Brumm & Grill, 2010)
Chebyshev Points(Christiano & Fisher, 2000)
Cluster grid points determined by estimate of ergodic
set(Judd et al., 2010)
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Parametrized Expectations Algorithms (PEAs)
Miranda and Fackler
Endogenous Grid Method
Other Improved Grids
Common Components
Expectations
The Generic Algorithm
Discretized Tauchen Matrix Multiplication(Fella, 2011;
Barillas & Fernandez-Villaverde, 2007)
Gaussian Quadrature(Christiano & Fisher, 2000; Billi, 2011;
Nakov, 2008)
Non product monomial and one point quadrature rules(Judd
et al., 2010)
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Outline
1
Introduction
2
Authors
3
Abstract Classes
4
Summary
5
Bibliography
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
A General Framework
Policy Function/ Lagrange Multiplier
Parameterized Expectations
The Generic Algorithm
Following(Hintermaier & Koeniger, 2010)
First Order Conditions
F(x− , y− , λ− , x0 , y0 , λ0 , x+ , y+ , λ+ , b) = 0
Equality Constraints
Q(x− , y− , x0 , y0 , x+ , y+ , b) = 0
Occasionally Binding Constraints
O(x− , y− , x0 , y0 , x+ , y+ , b) ≥ 0
Complementary Slackness Conditions
O(x− , y− , x0 , y0 , x+ , y+ , b)Λ(x− , y− , λ− , x0 , y0 , λ0 , x+ , y+ , λ+ , b) = 0
with
Λ(x− , y− , λ− , x0 , y0 , λ0 , x+ , y+ , λ+ , b) ≥ 0
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
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Abstract Classes
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Abstract Classes for Dynamic Models with OBCs
General Framework
The Generic Algorithm
Object oriented implementation of the algorithms requires
committing to collection of inter-operating classes
Existing code provides guidance for designing classes
Program data lead to fields
Program operations lead to methods
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
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Abstract Classes
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Abstract Classes for Dynamic Models with OBCs
General Framework
The Generic Algorithm
Variable
Relation
StateVariable
NonStateVariable
LagrangeMultiplier
Equation
Inequality
System
Grid
EquationSystem
InequalitySystem
BooleanFunction
ApproximateFunction
Report
ValueFunction
PolicyFunction
CPUUsageReport
MemoryUsageReport
Expression
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
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Abstract Classes
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Bibliography
Grid Abstract Class Tentative Example
Grid Fields
evaluationPts
variableSpecs
Grid Methods
getEvaluationPts() returns list of points – abstract
evaluateAtPts( Function ff) returns a list of points – abstract
pointsWhereTrue( BooleanFunction qq) returns a list of points
– default implemented
cpuUsageReport() returns a CPUUsageReport – abstract
memoryReport() returns a MemoryReport –abstract
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Outline
1
Introduction
2
Authors
3
Abstract Classes
4
Summary
5
Bibliography
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Summary
Two decades of solving dynamic models with occasionally
binding constraints
Dozens of proposed solution algorithms
Broadly similar structure
Composed from a few common components
But generally incompatible implementations
Existing code and algorithms could provide guide to good API
Synergy would be enhanced if components were more
interchangeable
Experimentation in prototyping would be useful
Interoperability could facilitate experimentation and algorithm
design
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
Authors
Abstract Classes
Summary
Bibliography
Outline
1
Introduction
2
Authors
3
Abstract Classes
4
Summary
5
Bibliography
Anderson
Generic Tools – Occasionally Binding Constraints
Introduction
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Bibliography
Bibliography I
Adam, Klaus, & Billi, Roberto M. 2006.
Optimal monetary policy under commitment with a zero
bound on nominal interest rates.
Journal of money, credit and banking, 38(7), 1877–1905.
Adam, Klaus, & Billi, Roberto M. 2007.
Discretionary monetary policy and the zero lower bound on
nominal interest rates.
Journal of monetary economics, 54(3), 728–752.
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Bibliography II
Aruoba, S. Boragan, Fernandez-Villaverde, Jesus, &
Rubio-Ramirez, Juan F. 2006.
Comparing solution methods for dynamic equilibrium
economies.
Journal of economic dynamics and control, 30(12),
2477–2508.
Barillas, Francisco, & Fernandez-Villaverde, Jesus.
2007.
A generalization of the endogenous grid method.
Journal of economic dynamics and control, 31(8), 2698–2712.
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Bibliography III
Benigno, Gianluca, Chen, Huigang, Otrok,
Christopher, Rebucci, Alessandro, & Young,
Eric R. 2009 (April).
Optimal policy with occasionally binding credit constraints.
First Draft Nov 2007.
Billi, Roberto M. 2007 (April).
Optimal inflation for the u.s.
Research Working Paper RWP 07-03. Federal Reserve Bank of
Kansas City.
Billi, Roberto M. 2011.
Optimal inflation for the us economy.
American economic journal: Macroeconomics, 3(3), 29–52.
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Abstract Classes
Summary
Bibliography
Bibliography IV
Brumm, Johannes, & Grill, Michael. 2010 (July).
Computing equilibria in dynamic models with occasionally
binding constraints.
Tech. rept. 95. University of Mannheim Center for Doctoral
Studies in Economics.
Carroll, Christopher D. 2006.
The method of endogenous gridpoints for solving dynamic
stochastic optimization problems.
Economics letters, 91(3), 312–320.
Anderson
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Authors
Abstract Classes
Summary
Bibliography
Bibliography V
Christiano, Lawrence J., & Fisher, Jonas D. M. 2000.
Algorithms for solving dynamic models with occasionally
binding constraints.
Journal of economic dynamics and control, 24(8), 1179–1232.
Dennis, Richard. 2007.
Optimal policy in rational expectations models: New solution
algorithms.
Macroeconomic dynamics, 11(01), 31–55.
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Abstract Classes
Summary
Bibliography
Bibliography VI
Fella, Giulio. 2011 (May).
A generalized endogenous grid method for non-concave
problems.
Working Papers 677. Queen Mary, University of London,
School of Economics and Finance.
Fernndez-Villaverde, Jess, Gordon, Grey,
Guerron-Quintana, Pablo A., & Rubio-Ramrez,
Juan Francisco. 2012 (May).
Nonlinear adventures at the zero lower bound.
CEPR Discussion Papers 8972. C.E.P.R. Discussion Papers.
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Abstract Classes
Summary
Bibliography
Bibliography VII
Gamma, Erich, Helm, Richard, Johnson, Ralph E., &
Vlissides, John. 1995.
Design patterns: Elements of reusable object-oriented
software.
Reading, MA: Addison-Wesley.
Gordon, Grey. 2011 (June).
Computing dynamic heterogeneous-agent economies:
Tracking the distribution.
PIER Working Paper Archive 11-018. Penn Institute for
Economic Research, Department of Economics, University of
Pennsylvania.
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Authors
Abstract Classes
Summary
Bibliography
Bibliography VIII
Grüne, Lars, & Semmler, Willi. 2004.
Using dynamic programming with adaptive grid scheme for
optimal control problems in economics.
Journal of economic dynamics and control, 28(12),
2427–2456.
Haefke, Christian. 1998 (November).
Projections parameterized expectations algorithms (matlab).
QM&RBC Codes, Quantitative Macroeconomics & Real
Business Cycles.
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Authors
Abstract Classes
Summary
Bibliography
Bibliography IX
Hintermaier, Thomas, & Koeniger, Winfried. 2010.
The method of endogenous gridpoints with occasionally
binding constraints among endogenous variables.
Journal of economic dynamics and control, 34(10),
2074–2088.
Iskhakov, Fedor, Rust, John, & Schjerning, Bertel.
2012.
A generalized endogenous grid method for discrete-continuous
choice.
SCE Computation in Economics and Finance.
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Summary
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Bibliography X
Judd, Kenneth L., Maliar, Lilia, & Maliar, Serguei.
2010 (May).
A cluster-grid projection method: Solving problems with high
dimensionality.
Working Paper 15965. National Bureau of Economic Research.
Krueger, Dirk, & Ludwig, Alexander. 2007.
On the consequences of demographic change for rates of
returns to capital, and the distribution of wealth and welfare.
Journal of monetary economics, 54(1), 49–87.
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Abstract Classes
Summary
Bibliography
Bibliography XI
Krusell, Per, Quadrini, Vincenzo, & Rios-Rull,
Jose-Victor. 1997.
Politico-economic equilibrium and economic growth.
Journal of economic dynamics and control, 21(1), 243–272.
Ljungqvist, Lars, & Sargent, Thomas J. 2004.
Recursive macroeconomic theory.
The MIT Press.
Ludwig, Alexander, & Schön, Matthias. 2012 (May).
Endogenous grid methods in higher dimensions: Delaunay
interpolation and hybrid methods.
CMR, University of Cologne, Germany.
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Abstract Classes
Summary
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Bibliography XII
Maliar, Lilia, & Maliar, Serguei. 2005a.
Parameterized expectations algorithm: How to solve for labor
easily.
Computational economics, 25(3), 269–274.
Maliar, Lilia, & Maliar, Serguei. 2005b.
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