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Numerical Methods in Economics
Numerical Methods in Economics
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1 Introduction
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1.1 What Economists Can Compute
1.2 Roles of Computation in Economic Analysis
1.3 Computation in Science
1.4 The Future of Computing
1.5 The Objectives and Nature of This Book
1.6 Basic Mathematics, Notation, and Terminology
1.7 Software and Supplemental Material
1.8 Further Reading
1.9 Exercises
2 Elementary Concepts
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2.1 Computer Arithmetic
2.2 Computer Processing and Algorithms
2.3 The Economics of Computation
2.4 Efficient Polynomial Evaluation
2.5 Efficient Computation of Derivatives
2.6 Direct versus Iterative Methods
2.7 Errors: The Central Problem of Numerical Mathematics
2.8 Making Innite Sequences Finite
2.9 Methods of Approximation
2.10 Evaluating the Errors in the Final Result
2.11 Computational Complexity
2.12 Further Reading and Summary
2.13 Exercises
3 Linear Equations and Iterative Methods
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3.1 Gaussian Elimination, LU Decomposition
3.2 Alternative Methods
3.3 Banded Sparse Matrix Methods
3.4 General Sparse Matrix Methods
3.5 Error Analysis
3.6 Iterative Methods
3.7 Operator Splitting Approach
3.8 Convergence of Iterative Schemes
3.9 Acceleration and Stabilization Methods
3.10 Calculating A^−1
3.11 Computing Ergodic Distributions
3.12 Overidentied Systems
3.13 Software
3.14 Summary and Further Reading
3.15 Exercises
4 Optimization
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4.1 One-Dimensional Minimization
4.2 Multidimensional Optimization: Comparison Methods
4.3 Newtons Method for Multivariate Problems
4.4 Direction Set Methods
4.5 Nonlinear Least Squares
4.6 Linear Programming
4.7 Constrained Nonlinear Optimization
4.8 Incentive Problems
4.9 Computing Nash Equilibrium
4.10 A Portfolio Problem
4.11 A Simple Econometric Example
4.12 A Dynamic Optimization Problem
4.13 Software
4.14 Further Reading and Summary
4.15 Exercises
5 Nonlinear Equations
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5.1 One-Dimensional Problems: Bisection
5.2 One-Dimensional Problems: Newtons Method
5.3 Special Methods for One-Dimensional Problems
5.4 Elementary Methods for Multivariate Nonlinear Equations
5.5 Newtons Method for Multivariate Equations
5.6 Methods That Enhance Global Convergence
5.7 Advantageous Transformations
5.8 A Simple Continuation Method
5.9 Homotopy Continuation Methods
5.10 A Simple CGE Problem
5.11 Software
5.12 Further Reading and Summary
5.13 Exercises
6 Approximation Methods
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6.1 Local Approximation Methods
6.2 Ordinary Regression as Approximation
6.3 Orthogonal Polynomials
6.4 Least-Squares Orthogonal Polynomial Approximation
6.5 Uniform Approximation
6.6 Interpolation
6.7 Approximation through Interpolation and Regression
6.8 Piecewise Polynomial Interpolation
6.9 Splines
6.10 Examples
6.11 Shape-Preserving Approximation
6.12 Multidimensional Approximation
6.13 Finite Element Approximations
6.14 Neural Networks
6.15 Further Reading and Summary
6.16 Exercises
7 Numerical Integration and Differentiation
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7.1 Newton-Cotes Formulas
7.2 Gaussian Formulas
7.3 Singular Integrals
7.4 Adaptive Quadrature
7.5 Multidimensional Quadrature
7.6 Example: Portfolio Problems
7.7 Numerical Differentiation
7.8 Software
7.9 Further Reading and Summary
7.10 Exercises
8 Monte Carlo and Simulation Methods
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8.1 Pseudorandom Number Generation
8.2 Monte Carlo Integration
8.3 Optimization by Stochastic Search
8.4 Stochastic Approximation
8.5 Standard Optimization Methods with Simulated Objectives
8.6 Further Reading and Summary
8.7 Exercises
9 Quasi-Monte Carlo Methods
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9.1 Equidistributed Sequences
9.2 Low Discrepancy Methods
9.3 Fourier Analytic Methods
9.4 The Method of Good Lattice Points
9.5 Estimating Quasi-Monte Carlo Errors
9.6 Acceleration Methods and qMC Schemes
9.7 Further Reading and Summary
9.8 Exercises
About the Book
About the Book – 2nd Edition
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5 Nonlinear Equations – 2nd Edition
5.1 One-Dimensional Problems: Bisection 2nd Edition
5.2 One-Dimensional Problems: Newtons Method 2nd Edition
5.3 Special Methods for One-Dimensional Problems 2nd Edition
5.4 Elementary Methods for Multivariate Nonlinear Equations – 2nd Edition
5.5 Newtons Method for Multivariate Equations – 2nd Edition
5.6 Methods That Enhance Global Convergence – 2nd Edition
5.7 Advantageous Transformations – 2nd Edition
5.8 A Simple Continuation Method – 2nd Edition
5.9 Homotopy Continuation Methods – 2nd Edition
5.10 A Simple CGE Problem – 2nd Edition
5.11 Software – 2nd Edition
5.12 Further Reading and Summary – 2nd Edition
5.13 Exercises – 2nd Edition
1 Introduction – 2nd Edition
1.1 What Economists Can Compute – 2nd Edition
1.2 Roles of Computation in Economic Analysis – 2nd Edition
1.3 Computation in Science – 2nd Edition
1.4 The Future of Computing – 2nd Edition
1.5 The Objectives and Nature of This Book — 2nd Edition
1.6 Basic Mathematics, Notation, and Terminology — 2nd Edition
1.7 Software and Supplemental Material – 2nd Edition
1.8 Further Reading — 2nd Edition
1.9 Exercises — 2nd Edition
10 Numerical Quadrature – 2nd Edition
11 Monte Carlo and quasi-Monte Carlo – 2nd Edition
13 Projection Methods for Functional Equations – 2nd Edition
13.1 An Ordinary Differential Equation Example
13.2 A Partial Differential Equation Example – 2nd Edition
13.3 General Projection Method – 2nd Edition
13.4 Boundary Value Problems- 2nd Edition
13.5 Continuous-Time Growth Model- 2nd Edition
13.6 Computing Conditional Expectations – 2nd Edition
13.7 Further Reading and Summary – 2nd Edition
13.8 Exercises – 2nd Edition
13 Projection Methods- 2nd Edition
15 Perturbation Methods in Euclidean Spaces – 2nd Edition
16 Perturbation Methods in Function Spaces – 2nd Edition
17 Asymptotic Methods – 2nd Edition
17.1 Bifurcation Methods – 2nd Edition
17.2 Portfolio Choices for Small Risks – 2nd Edition
17.3 Gauge Functions and Asymptotic Expansions – 2nd Edition
17.4 Method of Undetermined Gauges – 2nd Edition
17.5 Asympotic Expansions of Integrals – 2nd Edition
17.6 Hybrid Perturbation-Projection Methods – 2nd Edition
17.7 Further Reading and Summary – 2nd Edition
17.8 Exercises – 2nd Edition
18 Perfect Foresight Models – 2nd Edition
19 Rational Expectations Models – 2nd Edition
2 Elementary Aspects of Numerical Analysis 2nd Edition
20 Dynamic Games – 2nd Edition
21 Supergames – 2nd Edition
22 Numerical Algebraic Geometry – 2nd Edition
3 Linear Equations – 2nd Edition
4 Unconstrained Optimization – 2nd Edition
6 Constrained Optimization – 2nd Edition
7 Nonlinear Complementarity Problems – 2nd Edition
8 Classical Approximation Methods – 2nd Edition
9 Modern Approximation Methods – 2nd Edition
12 Finite-Difference Methods – 2nd Edition
12.1 Classification of Ordinary Differential Equations – 2nd Edition
12.2 Solution of Linear Dynamic Systems – 2nd Edition
12.3 Finite-Difference Methods for IVPs – 2nd Edition
12.4 Economic Examples of IVPs – 2nd Edition
12.5 Boundary Value Problems for ODEs: Shooting – 2nd Edition
12.6 Finite-Horizon Optimal Control Problems – 2nd Edition
12.7 Innite-Horizon Optimal Control and Shooting – 2nd Edition
12.8 Integral Equations – 2nd Edition
12.9 Further Reading and Summary – 2nd Edition
12.10 Exercises – 2nd Edition
14 Numerical Dynamic Programming — 2nd Edition
14.1 Discrete Time Dynamic Programming Problems – 2nd Edition
14.2 Continuous Time Dynamic Programming Problems – 2nd Edition
14.3 Finite-State Methods — 2nd Edition
14.4 Acceleration Methods for Infinite-Horizon Problems – 2nd Edition
14.5 Discretization Methods for Continuous Problems — 2nd Edition
14.6 Methods for Solving Linear-Quadratic Problems — 2nd Edition
14.7 Continuous Methods for Continuous-State Problems — 2nd Edition
14.8 Parametric Approximations and Simulation Methods — 2nd Edition
14.9 Shape-Preserving Methods — 2nd Edition
14.10 Continuous-Time Problems – 2nd Edition
14.11 Further Readings and Summary – 2nd Edition
14.12 Exercises — 2nd Edition
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10 Finite-Difference Methods
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10.1 Classification of Ordinary Differential Equations
10.2 Solution of Linear Dynamic Systems
10.3 Finite-Difference Methods for IVPs
10.4 Economic Examples of IVPs
10.5 Boundary Value Problems for ODEs: Shooting
10.6 Finite-Horizon Optimal Control Problems
10.7 Innite-Horizon Optimal Control and Shooting
10.8 Integral Equations
10.9 Further Reading and Summary
10.10 Exercises
11 Projection Methods for Functional Equations
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11.1 An Ordinary Differential Equation Example
11.2 A Partial Differential Equation Example
11.3 General Projection Method
11.4 Boundary Value Problems
11.5 Continuous-Time Growth Model
11.6 Computing Conditional Expectations
11.7 Further Reading and Summary
11.8 Exercises
12 Numerical Dynamic Programming
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12.1 Discrete Time Dynamic Programming Problems
12.2 Continuous Time Dynamic Programming Problems
12.3 Finite-State Methods
12.4 Acceleration Methods for Infinite-Horizon Problems
12.5 Discretization Methods for Continuous Problems
12.6 Methods for Solving Linear-Quadratic Problems
12.7 Continuous Methods for Continuous-State Problems
12.8 Parametric Approximations and Simulation Methods
12.9 Shape-Preserving Methods
12.10 Continuous-Time Problems
12.11 Further Readings and Summary
12.12 Exercises
13 Regular Perturbations of Simple System
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13.1 The Mathematics of Regular Perturbation Methods
13.2 Comparative Statics
13.3 Perturbing an IVP
13.4 Perturbing a BVP: Comparative Dynamics
13.5 Continuous-Time, Deterministic Control with One State and One
13.6 Stochastic Control
13.7 Perturbing Discrete-Time Systems
13.8 Perturbing Jump Process Control Problems
13.9 Global Quality Test of Asymptotic Approximation
13.10 Exercises
14 Regular Perturbations in Multidimensional System
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14.1 Multidimensional Comparative Statics and Tensor Notation
14.2 Linearization of Multidimensional Dynamic System
14.3 Locally Asymptotically Stable Multidimensional Control
14.4 Perturbations of Discrete-Time Problems
14.5 Multisector, Stochastic Growth
14.6 Further Reading and Summary
14.7 Exercises
15 Advanced Asymptotic Methods
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15.1 Bifurcation Methods
15.2 Portfolio Choices for Small Risks
15.3 Gauge Functions and Asymptotic Expansions
15.4 Method of Undetermined Gauges
15.5 Asymptotic Expansions of Integrals
15.6 Hybrid Perturbation-Projection Methods
15.7 Further Reading and Summary
15.8 Exercises
16 Solution Methods for Perfect Foresight Models
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16.1 A Simple, Autonomous Overlapping Generations Model
16.2 Equilibrium in OLG Models: Time Domain Methods
16.3 Fair-Taylor Method
16.4 Recursive Models and Dynamic Iteration Methods
16.5 Recursive Models with Nonlinear Equation Methods
16.6 Accuracy Measures
16.7 Tax and Monetary Policy in Dynamic Economies
16.8 Recursive Solution of an OLG Model
16.9 “Consistent” Capital Income Taxation
16.10 Further Reading and Summary
16.11 Exercises
17 Solving Rational Expectations Models
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17.1 The Lucas Asset Pricing Model
17.2 Monetary Equilibrium
17.3 Information and Asset Markets
17.4 Commodity Storage Models
17.5 A Simple Stochastic Dynamic Growth Model
17.6 Fixed-Point Iteration
17.7 Time Iteration
17.8 Generalizations
17.9 Further Reading and Summary
17.10 Exercises
4.6 Linear Programming