Handbook of Computational Economics, Volume 3

Front Cover......Page 1
Half Title......Page 2
Title Page......Page 4
Copyright......Page 5
Contents......Page 6
Contributors......Page 10
Acknowledgments......Page 12
Introduction to the Series......Page 13
Introduction for Volume 3 of the Handbook  of Computational Economics......Page 15
1 Learning About Learning in Dynamic Economic Models*......Page 19
1 Introduction......Page 20
2 The Framework......Page 21
3 What We Have Learned......Page 24
3.2 Rapid Decrease in Parameter Variances in the First Few Periods......Page 25
3.3 Nonconvexities......Page 26
3.4 Rankings......Page 27
3.5 Time-Varying Parameters......Page 28
4 What We Hope to Learn......Page 29
4.2 Efficiency......Page 30
4.5 Measurement Errors......Page 31
5 Algorithms and Codes......Page 32
6.1 Outline of the Beck and Wieland Model......Page 34
6.2 Constant Parameters......Page 35
6.3 Time-Varying Parameters Version......Page 38
7 Learning with Forward Looking Variables......Page 39
7.1 Extending the Framework......Page 40
7.2 An Example......Page 42
8 Other Applications of Active Learning......Page 48
9 Summary......Page 49
References......Page 50
2 On the Numerical Solution of Equilibria in Auction Models with Asymmetries within the Private-Values Paradigm......Page 55
1 Motivation and Introduction......Page 56
2.1 Notation......Page 58
2.2 Derivation of Symmetric Bayes-Nash Equilibrium......Page 59
2.3 Bidders from Two Different Urns......Page 62
2.4 General Model......Page 65
2.5 Special Case......Page 69
2.6.1 Risk Aversion......Page 73
2.6.2 Collusion or Presence of Coalitions......Page 76
2.6.3 Procurement......Page 79
2.6.4 Bid Preferences......Page 81
3.1 Shooting Algorithms......Page 84
3.2 Projection Methods......Page 95
4.1 Marshall et al. (1994)......Page 98
4.2 Bajari (2001)......Page 99
4.3 Fibich and Gavious (2003)......Page 102
4.4 Gayle and Richard (2008)......Page 103
4.5 Hubbard and Paarsch (2009)......Page 105
4.6 Fibich and Gavish (2011)......Page 108
4.7 Hubbard et al. (2013)......Page 109
5 Some Examples......Page 113
6.1 Comparisons of Relative Performance......Page 121
6.2 Potential Improvements......Page 126
References......Page 130
3 Analyzing Fiscal Policies  in a Heterogeneous-Agent Overlapping-Generations Economy......Page 135
1 Introduction......Page 136
2 Existing Literature......Page 137
3.1.1 The State Variables and the Government Policy......Page 140
3.1.3 Perfect Annuity Markets [Optional]......Page 141
3.1.5 The Household's Preference......Page 142
3.1.8 The Household's Decision Rules......Page 143
3.2 The Representative Firm......Page 144
3.2.2 A Small Open Economy [Optional]......Page 145
3.3.1 Accidental Bequests......Page 146
3.5 Social Welfare Measures......Page 147
3.5.2 Equivalent Variations......Page 148
3.5.3 Compensating Variations......Page 149
4.1 Solving the Household's Problem......Page 150
4.1.2 The Complementarity Problem......Page 151
4.2 Finding the Distribution of Households......Page 153
4.3.1 A Steady-State Equilibrium......Page 154
4.3.2 An Equilibrium Transition Path......Page 155
5.1 Demographics, Preference, and Technology Parameters......Page 156
5.2 Market Wage Processes......Page 159
5.3 The Government's Policy Functions......Page 161
6.1 Consumption Tax Reform......Page 162
6.1.1 The Computational Procedure......Page 163
6.1.2 The Long-Run Effect......Page 164
6.1.3 The Transition Effect......Page 165
6.1.4 The Welfare Effect......Page 166
6.2.1 The Computational Procedure......Page 168
6.2.2 The Long-Run Effect......Page 170
6.2.3 The Transition Effect......Page 171
6.2.4 The Welfare Effect......Page 172
7 Concluding Remarks......Page 174
References......Page 177
1 Introduction......Page 179
2 Discrete Time Portfolio Decision Making......Page 182
2.1 Investor's Problem......Page 183
2.2 A Survey of Approximation Methods......Page 186
2.3 Polynomial Methods......Page 188
3 Discrete Time Asset Pricing......Page 189
3.1 Analytic Method......Page 191
3.2 More Complicated Models......Page 196
3.3 Alternative Asset Pricing Models......Page 202
3.4 A Survey of Log-Linearized Approximations......Page 204
3.5 Heterogeneous Agents......Page 207
4 Continuous Time Portfolio Decision Problem......Page 213
4.1 Optimal Investment Decisions in Continuous Time......Page 215
4.2 Solutions......Page 220
5 Continuous Time Asset Pricing......Page 223
5.1 One-Dimensional Asset Pricing Model Under Infinite Horizon......Page 225
5.1.1 Campbell and Cochrane (1999) in Continuous Time......Page 226
5.1.2 Initial Conditions: Empirical Approach......Page 227
5.2 Multi-Dimensional Asset Pricing Models......Page 231
5.3 Stochastic Differential Utility: Choosing Initial Conditions......Page 233
6 Conclusion......Page 236
References......Page 237
1 General Introduction......Page 243
2 The Problem Statement—In the Case of Stochastic Volatility and Poisson Jump Dynamics......Page 244
3.1 Introduction......Page 248
3.2 Problem Statement—Merton's Model......Page 250
3.3 Jamshidian's Representation......Page 251
3.4 Limit of the Early Exercise Boundary at Expiry......Page 259
3.5 American Call with Log-Normal Jumps......Page 261
3.5.1 Delta for the American Call......Page 262
3.6 Properties of the Free Boundary at Expiry......Page 263
3.7 Numerical Implementation......Page 265
3.8 Numerical Results......Page 267
4 American Call Options under Jump-Diffusion and Stochastic Volatility Processes......Page 271
4.1 Numerical Solution Using the Method of Lines......Page 272
4.2 Numerical Solution Using the Componentwise Splitting Method......Page 276
4.3 Numerical Solution Using Finite Difference Method with PSOR......Page 281
4.4 Numerical Results......Page 284
5 Conclusion......Page 291
References......Page 292
6 Solving and Simulating Models with Heterogeneous Agents and Aggregate Uncertainty......Page 295
1 Introduction......Page 296
2 Example Economy......Page 297
3.1.1 Obtain Aggregate Policy Functions from Simulation......Page 300
3.1.2 Obtain Aggregate and Individual Policy Functions Through Simulation......Page 301
3.1.3 Obtain Aggregates by Integrating over a Parameterized Distribution......Page 302
3.1.4 Obtain Aggregates by Explicit Aggregation......Page 304
3.2 Perturbation Approaches......Page 307
3.2.1 Perturbation Around Scalar Steady State Values......Page 308
3.2.2 Perturbation Around the Steady State Cross-Sectional Distribution......Page 312
4 Models with Nontrivial Market Clearing......Page 315
6 Simulation with a Continuum of Agents......Page 316
6.1 Grid Method I: Calculation of Inverse Required......Page 319
6.2 Grid Method II: No Calculation of Inverse Required......Page 321
6.3 Simulating Using Smooth Density Approximations......Page 322
7 Accuracy......Page 324
8 Comparison......Page 331
9 Other Types of Heterogeneity......Page 335
A. Explicit Aggregation and Perturbation Techniques......Page 336
Acknowledgments......Page 340
References......Page 341
7 Numerical Methods for Large-Scale Dynamic Economic Models......Page 343
1 Introduction......Page 345
2 Literature Review......Page 350
3 The Chapter at a Glance......Page 358
4 Nonproduct Approaches to Representing, Approximating, and Interpolating Functions......Page 371
4.1 Smolyak's (1963) Sparse Grid Method......Page 372
4.1.1 How Does the Smolyak Method Work?......Page 373
4.1.2 The Automated Smolyak Method......Page 378
4.2 Generalized Stochastic Simulation Algorithm......Page 381
4.2.3 Finding the Polynomial Coefficients by Way of Regression......Page 382
4.2.4 Advantages of the Stochastic Simulation Approach......Page 383
4.2.5 Marcet's (1988) Parameterized Expectations Algorithm......Page 384
4.2.6 Generalized Stochastic Simulation Algorithm by Judd et al. (2011b)......Page 385
4.2.7 Numerical Illustration of the Importance of the Approximating Function and Fitting Method......Page 391
4.3 ε -Distinguishable Set and Cluster Grid Algorithms......Page 392
4.3.1 Eliminating Simulated Points Outside the High-Probability Set......Page 393
4.3.2 Constructing an ε-Distinguishable Set of Points......Page 394
4.3.3 Other Grids on the Ergodic Set......Page 396
4.3.4 Numerical Illustration of the Accuracy of the Smolyak Method......Page 397
5.1 Gauss-Hermite Product Quadrature Rules......Page 398
5.2 Monomial Rules......Page 400
5.2.2 Monomial Rule M2 with 2N2+1 Nodes......Page 401
5.3 Monte Carlo Integration Method......Page 402
5.4 Quasi-Monte Carlo Integration Methods......Page 403
5.5 Nonparametric Kernel-Density Methods......Page 404
5.7 Correlated Shocks and Cholesky Decomposition......Page 405
5.8.1 Monte Carlo Integration......Page 406
5.8.2 Nonparametric Kernel-Density Integration Method......Page 407
6 Derivative-Free Optimization Methods......Page 408
6.1.1 Intratemporal Choice FOCs......Page 409
6.2 The Intratemporal Choice Quantities......Page 410
6.3 The Intertemporal Choice Functions......Page 412
6.4 Coordination Between the Intratemporal and Intertemporal Choices......Page 414
6.5 Numerical Illustration of the Importance of Coordination......Page 415
7.1 Conventional Value Function Iteration......Page 416
7.2.1 Endogenous Grid Method......Page 418
7.2.2 Envelope Condition Method......Page 419
7.2.4 EGM and ECM in a Model with Elastic Labor Supply......Page 420
7.3 Increasing the Accuracy of Dynamic Programming Methods......Page 421
7.4 Numerical Illustration of Dynamic Programming Methods......Page 422
8.1.1 Precomputation of Expectations for Polynomial Functions......Page 423
8.1.2 Precomputation of the Expectation in the Euler Equation......Page 424
8.1.3 Precomputation of the Expectation in the Bellman Equation......Page 425
8.1.4 Relation of Precomputation of Integrals to the Literature......Page 426
8.2 Precomputation of Intratemporal Choice Manifolds......Page 427
8.3 Precomputation of Aggregate Decision Rules......Page 429
9.1 Plain Perturbation Method......Page 430
9.2 Advantages and Shortcomings of Perturbation Methods......Page 434
9.3.1 An Example of the Change of Variables Technique......Page 435
9.4 Hybrid of Local and Global Solutions......Page 437
9.4.1 Description of the Hybrid Method......Page 438
9.5 Numerical Instability of High-Order Perturbation Solutions in Simulation......Page 439
10 Parallel Computation......Page 440
10.1.1 Applications with Independent Tasks......Page 441
10.1.3 Speedup and Efficiency of Parallelization......Page 442
10.2 Parallel Computation on a Desktop Using MATLAB......Page 443
10.2.1 Numerical Example of GPU Computation Using MATLAB......Page 445
10.3 Parallel Computation on Supercomputers......Page 446
10.3.1 Numerical Example of Parallel Computation Using a Blacklight Supercomputer......Page 448
11.1 The Model......Page 449
11.2 Methods Participating in the JEDC Project......Page 450
11.3 Global Euler Equation Methods......Page 452
11.3.2 Separating the Intertemporal and Intratemporal Choices......Page 454
11.3.3 Smolyak Method with Iteration-on-Allocation and FPI......Page 455
11.3.4 Generalized Stochastic Simulation Algorithm......Page 457
11.3.5 ε-Distingishable Set Algorithm......Page 459
11.4.1 Bellman Equation, FOCs, and Envelope Condition......Page 461
11.4.2 Separating the Intertemporal and Intratemporal Choices......Page 463
11.4.3 Envelope Condition Method Iterating on Value Function......Page 464
11.4.4 Envelope Condition Method Solving for Derivatives of Value Function......Page 465
11.5 Hybrid of Local and Global Solutions......Page 466
11.6 Solving for Consumption and Labor Using Iteration-on-Allocation......Page 467
11.7 Accuracy Measures......Page 469
11.8 Explicit Versus Implicit Solutions......Page 470
12.1 Projection Methods......Page 471
12.2 Generalized Stochastic Simulation Methods......Page 474
12.3 Dynamic Programming Methods......Page 477
12.4 Local Solution Methods......Page 479
12.4.1 The Importance of the Change of Variables......Page 480
12.4.2 The Benefits of Hybrid Solutions......Page 481
12.5 Speeding up Computations in MATLAB......Page 483
12.6 Practical Recommendations About Solving High-Dimensional Problems: Summary......Page 484
Acknowledgments......Page 487
References......Page 488
8 Advances in Numerical Dynamic Programming and New Applications......Page 497
1 Introduction......Page 498
2 Theoretical Challenges......Page 499
3.1 Outline of the Basic Value Function Iteration Algorithm......Page 501
3.2 Typical Applications......Page 502
3.2.1 Optimal Growth Example......Page 503
3.2.2 Multistage Portfolio Optimization Example......Page 504
4.2 Numerical Integration......Page 506
4.3 Approximation......Page 507
4.3.2 Multidimensional Complete Chebyshev Polynomial Approximation......Page 508
4.3.3 Shape-Preserving Chebyshev Interpolation......Page 509
4.3.4 Shape-Preserving Hermite Interpolation......Page 511
5.1 Application in Optimal Growth Problems......Page 512
5.2 Application in Multistage Portfolio Optimization Example......Page 514
5.2.1 Numerical Results of Shape-Preserving Rational Spline Hermite Interpolation......Page 515
5.2.2 Other Shape-preserving Methods......Page 517
6.1 The Variety of Parallel Programming Architectures......Page 518
6.2 Parallel Dynamic Programming......Page 520
6.3 Application to Stochastic Optimal Growth Models......Page 522
6.3.2 Numerical Example......Page 523
6.3.3 Parallelization Results......Page 524
7 Dynamic Portfolio Optimization with Transaction Costs......Page 525
7.1 Numerical Example......Page 527
8.1 A Stochastic IAM with Epstein-Zin Preferences......Page 528
8.2 Dynamic Programming with Epstein-Zin Preferences......Page 529
8.3 Numerical Examples......Page 530
9 Conclusions......Page 532
References......Page 533
9 Analysis of Numerical Errors......Page 535
1 Introduction......Page 536
2 Dynamic Stochastic Economies......Page 538
2.1 A Growth Model with Taxes......Page 539
2.2 An Asset-Pricing Model with Financial Frictions......Page 540
2.3 An Overlapping Generations Economy......Page 541
3.1 Numerical Approximations of Equilibrium Functions......Page 543
3.1.1 Methods Based on the Euler Equations......Page 544
3.1.2 Dynamic Programming......Page 545
3.2.1 Accuracy of Equilibrium Functions......Page 546
3.2.2 Accuracy of the Simulated Moments......Page 548
3.3 Calibration, Estimation, and Testing......Page 551
3.3.1 Calibration......Page 553
3.3.2 Simulation-Based Estimation......Page 554
4 Recursive Methods for Non-optimal Economies......Page 556
4.1.1 A Growth Model with Taxes......Page 557
4.1.2 An Overlapping Generations Economy......Page 558
4.2 Numerical Solution of Non-Optimal Economies......Page 560
4.2.1 The Theoretical Algorithm......Page 562
4.3 Simulated Statistics......Page 563
5.1 Accuracy for Models with Simple Markov Equilibria......Page 565
5.2.1 An Overlapping Generations Model......Page 567
5.2.2 Asset-Pricing Models with Market Frictions......Page 569
6 Concluding Remarks......Page 571
References......Page 572
1 Introduction......Page 575
2.1.1 Processing Hardware......Page 578
2.1.2 Memory......Page 579
2.2 Algorithmic Design......Page 580
2.3 Software......Page 582
3 A Simple GPGPU Example......Page 583
3.1 Matlab......Page 585
3.2 C++......Page 587
3.3 CUDA C......Page 590
3.4 Thrust......Page 595
4.1 Model......Page 598
4.3 Results......Page 600
5 Example: A General Equilibrium Asset Pricing Model with Heterogeneous Beliefs......Page 604
5.1 Model......Page 605
5.1.1 First-Order Conditions......Page 606
5.2 Solution......Page 607
5.3 Results......Page 609
6 The Road Ahead......Page 611
6.1 NVIDIA Kepler and CUDA 5......Page 612
6.3 OpenACC......Page 613
7 Conclusion......Page 614
References......Page 615
11 Computing All Solutions to Polynomial Equations in Economics......Page 617
1 Introduction......Page 618
2 Gröbner Bases and Polynomial Equations......Page 620
2.1 What Is a Gröbner Basis? A Brief Introduction......Page 621
2.1.1 A Formal Definition of Gröbner Bases......Page 623
2.1.2 Elimination Ideals and the Shape Lemma......Page 626
2.1.3 Buchberger's Algorithm......Page 627
2.1.4 Computing Gröbner Bases with Computer Algebra Systems......Page 629
2.2.1 Root Count for Univariate Polynomials......Page 632
2.2.2 Sufficient Conditions for the Shape Lemma......Page 633
2.2.3 What If the Shape Lemma Does Not Apply?......Page 634
2.2.4 Finding All Solutions with Computer Algebra Systems......Page 635
2.2.5 Parameterized Gröbner Bases......Page 636
2.2.6 Parameterized Shape Lemma with Computer Algebra Systems......Page 638
3.1 A Bertrand Game......Page 640
3.1.2 Solving the System with SINGULAR......Page 642
3.2 A Simple Walrasian Exchange Economy......Page 643
3.2.1 Polynomial System and Equilibria......Page 644
3.2.2 Finding All Equilibria with SINGULAR......Page 645
4.1 Brief Introduction to All-Solutions Homotopies......Page 649
4.1.1 Mathematical Background......Page 650
4.1.2 All Roots of Univariate Polynomials......Page 651
4.1.3 Multivariate Systems of Polynomial Equations......Page 655
4.2.1 Homogenization and Projective Space......Page 657
4.2.2 The m-Homogeneous Bezout Number......Page 661
4.2.3 Parameter Continuation Homotopy......Page 662
5.1.1 BERTINI......Page 663
5.2.1 Solving the Bertrand Pricing Game with BERTINI......Page 664
5.2.2 Application of Parameter Continuation......Page 665
5.3 Walrasian Exchange Economy......Page 666
5.4 Homotopy Continuation Compared to Gröbner Basis......Page 667
References......Page 669
A......Page 671
C......Page 672
E......Page 673
H......Page 674
I......Page 675
M......Page 676
P......Page 677
S......Page 678
T......Page 679
V......Page 680