Index
A
Accuracy checks
aggregate uncertainty, 306
evaluation, 452
examples, 309
explicit solution, 452
implicit solution, 452
measurement techniques, 451
methods, 307
non-linear economic models, 528
powerful test, 310
R2 tests, 308
simple Markov equilibria models, 547–548
Active learning
algorithms and codes, 14
economic models, 2
Monte Carlo runs, 8
other applications, 30
parameter variances, 7
and passive learning, 7
showcase, 16
stochastic control, 6
Aggregate uncertainty
accuracy checks, 306
approximation property, 278
computational algorithms, 278
heterogeneous agents, 282
individual agents, 280
simulation procedures, 278
Algorithms, aggregate uncertainty
accuracy checks, compared, 313
explicit distribution, 286
histogram distribution, 289
individual functions, 283
individual policy rule and moments, 286
lower-order polynomials, 288
parameterized distribution, 284
policy functions, 282
reference moments/distribution, 285
separate individual policy rule, 289
American call option
delta for, 244, 249
in dividend absence, 241
European component for, 239
initial boundary condition, 232
integral equation for, 239
numerical results, 249
parameter values, 266
pricing accuracy, 268
stochastic differential equation (SDE) for, 226
under jump-diffusion process, 230, 253
Analytic method
error estimation, 173
for complicated models, 178, 184, 211
for higher-dimensional continuous time asset pricing model, 213
for multi-dimensional continuous time asset pricing model, 214
for ODE function, 210
for one-dimensional discrete time asset pricing models, 186
for 100th order polynomial approximation, 216
implementation, 181
log-linearized approximations, 186, 189
Approximate aggregation, representative agent model, 298
Approximating solution
asymmetric first-price auction, 70, 99
defined, 110
differential equation, 71
first-price auction problem, 77
initial value problem, 75
of integrals, 380
least square method, 78
Markov Chain, 387
nonproduct approaches, 353
numerical illustration, 373
polynomial coefficients, 364
by polynomials, 107, 111
projection method, 77, 110
spectoral methods, 77
symmetric auction, 95
Asset pricing problems
analytical solution, 163, 173, 218
continuous time problem, 163
discrete-time problem, 163
heterogeneous agent models, 161
higher-dimensional, 163, 199, 213
multi-dimensional, 213
one-dimensional, 163, 186, 207, 213
role of initial condition, 164
Asset-pricing model with financial frictions
equilibrium correspondence, 552
numerical approximation, 551
Asymmetric first-price auctions
bid preferences, 63
collusion or presence of coalitions, 58
equilibrium properties, 51, 84
investigating models, 40, 112
numerical strategies, 66
in ODEs, 67
procurement, 61
risk aversion, 55–56
speed and accuracy, 39
Auxiliary function, defined, 84
B
Basic value function iteration algorithm
HTCondor grid computing, 500
numerical methods, dynamic programming, 483
parallel dynamic programming, 502
Bayes-Nash equilibrium
approximate solutions, 38
auction games, 38
bid function, 44–45, 47, 62, 65
differential equations, 45, 52–53
symmetric derivations, 41
under monotonicity, 42
valuation distribution, 51
Bellman equation
dynamic programming, 480
high-dimension problems, 336
precomputation technique, 407
simple Markov equilibria, 527
Bernstein polynomials, 41, 110
Bid preferences, 38, 63, 65–66
Bidder’s distribution, 39–40
Boundary-value problems
numerical strategies, 40, 112
player’s valuation, 90–91
shape constraints, 110
shooting algorithm, 66
two-point, 49, 63, 66
Brownian motion, 225–226, 230, 253
Buroughs D825, 557
Business cycle predictions, calibration model, 535
C
C++, software GPGPU computing, 569
CDC 6600, 557
Change of variables
example, 417
numerical illustration, 419
parallel computation, 423
perturbation methods, 417, 462
Cholesky decomposition, correlated shocks, 387
Closed-form solutions, 84, 88
Cluster grid algorithms
e-distinguishable set (EDS) technique, 374
Component-wise splitting (CS) methods
American call price, 269, 273
pricing accuracy, 268
Computational algorithm, OLG model
equilibrium transition path, 137
for households, 132, 150
marginal value functions, 135
for steady-state equilibrium, 136
Computational hardware, advance in, 558
Computational techniques
numerical illustration, 397
parallel methods, 422
serial vs parallel methods, 423
Computer algebra packages, MATHEMATICA, 600
Consumption growth
approximation methods, 168
dividend growth and, 187, 208
in multi-dimensional asset pricing model, 213
and price-dividend function, 208
random shocks in, 180, 208, 213
Wachter model, 214
Consumption tax reforms, OLG model
computational procedure, 145
long-run effect, 146
transition effect, 147
welfare effect, 148
Control variables, 2, 5, 7, 11–12
Conventional value function iteration, 398–399
shortcomings, 399
CPU memory, 561
Crank-Nicolson scheme
accuracy checking, 249
for American call price, 250
delta estimation, 250–251
with method of lines, 231, 249–251
numerical integration method, 249–250
for option price, 251, 273
price estimation, 250
in PSOR method, 268
splitting method, 262
CUDA 5, recent developments, 594
CUDA C, Software, 572
Cumulative distribution, 40–42, 55–56, 81, 97
D
Degree reverse lexicographic ordering, 606
Dependent tasks, applications, 424
Derivative-free optimization methods, nonlinear equation, 390
Deterministic economy, OLG model
no lifetime heterogeneity, 119
with heterogeneity, 120
Differential equations, defined, 48–50
Dual control (DC)
active control solution, 7
adaptive control method, 17
average value approach, 10
in BW model, 16–17, 20
computational efficiency, 9
measurement errors, 12–13
Monte Carlo runs, 11, 20
number of runs, 10
parameter estimation, 7
rankings, 9
time-varying parameters, 14
Dynamic economic model
different parameter vectors, 328
numerical methods, 332
Dynamic programming
accuracy check, 403
Bellman Equation, 443, 480
Chebyshev interpolation, 498
computational challenges, 481
defined, 480
endogenous grid method (EGM), 400
envelope condition method (ECM), 401, 443
Epstein-Zin preferences, 511
FOCs, 443
high dimensions, 398, 459
numerical illustration, 404, 483
optimal bond allocation errors, 499
parallelization, 500
stochastic model, 481
theoretical challenges, 481
Dynamic programming (DP) algorithm
contraction properties, 527, 538
convergence properties, 526, 553
error bounds, 529
numerical applications, 538, 548
E
E-distinguishable set (EDS) technique
cluster grid algorithms, 374
computational details, 443
construction, 376
GSSA, 441
steps, 441
Eigen library, 570
Endogenous grid method (EGM)
alternative conditions, 402
dynamic programming methods, 400
labor supply elasticity, 402
Envelope condition method (ECM)
alternative conditions, 402
derivatives on value function, 447
dynamic programming methods, 401, 459
iterates on value function, 446
labor supply elasticity, 402
Equilibrium law of motion, for state variables, 375
Ergodic set, other grids, 378
Estimation methods
rules for, 536
simulated moments estimator (SME), 536
target function, 536
Euler equation
FOCs, 340
global methods, 434
for high-dimensions, 337
initial age of consumption, 120
intertemporal choice conditions, 392
non-linear economic models, 535
parameter values, 529
parameterized expectations algorithm (PEA), 366
piecewise-continuous functions, 540
precomputation technique, 406
simple Markov equilibria, 526, 529
Euler method, auction market
defined, 76
implicit and explicit, 71–75
Expected optimal feedback (EOF)
active perturbation, 7, 10
criterion value, 17, 20
DualI software, 16
probability density function, 19
ranking, 9–10
time-varying parameters, 10
F
First-order condition for bidder
defined, 82, 86, 92
First-order conditions (FOCs)
dynamic programming, 443
Euler equations, 340
general equilibrium asset pricing model, 588
intertemporal choice conditions, 391
FLOPS (Floating Point Operations Per Second), 558, 592
Fourier transforms
American call pricing, 231
Duhamel’s principle and, 230
for European options, 230
IPDE for, 233–235, 273
G
General equilibrium asset pricing model
first-order conditions, 588
results, 591
solution, 589
trade volume, 587
Generalized stochastic simulation algorithm (GSSA)
accuracy checks, 457
running time, 457
stochastic growth model, 367, 456
Government, OLG model, 128
accident bequests, 128
GPGPU computing
algorithmic design, 562
in economics, 559
hardware architecture, 560
MATLAB, 567
recent developments, 593
simple example, 565
value function iteration, 580
Graphical processing units (GPUs)
callable libraries, within kernels, 594
in entertainment industry, 558
low cost and low power consumption in, 558
memory, 561–562
OpenMP/MPI, 562
scaling, 562
Grid method I
calculating information, 301
end-of-period distribution, 302
next period’s beginning-of-period distribution, 303
simulation procedure, 301
Grid method II
advantage, 303
end-of-period distribution, 303
next period’s beginning-of-period distribution, 304
Gröbner bases
advantages, 602
algebra systems, 611
Buchberger algorithm, 609–610
comprehensive, 619
defined, 601, 604–605
disadvantages, 602
economic applications, 616, 625
elimination theorem, 608
homotopy methods, compared with, 649
manifold of solutions, 625
MATHEMATICA, 611
parameterized solutions, 618, 620
polynomial solutions, 614–615
in SINGULAR polynomial order, 611–613
symbolic methods, 649
triangular form, 600
under lexicographic monomial order, 604
H
Hardware architecture, GPGPU computing
process, 560
Heterogeneous agents
accuracy factor, 306
aggregate uncertainty, 282
constrained simulation, 299
individual policy rules, 298
other types, 317
time paths, 299
High dimensions
accuracy check, 366, 466
Bellman equation, 336
computational techniques, 331, 466
conventional projection methods, 347
dynamic programming methods, 398, 459
ECM method, 443
Euler equation, 337
feasibility problems, 328, 333, 388
Gaussian product rule, 334
impractical application, 381
intertemporal choice conditions, 335, 397
nonproduct techniques, 344
numerical analysis, 332, 387, 431
optimization methods, 346
PEA feasibility, 363
perturbation methods, 332–333
practical solution, 466
Smolyak elements, 359, 363
stochastic growth model, 364, 431
trade-off, 466
value function iteration (VFI), 398
Homotopy methods, all solution
advanced features, 639
advantages, 602, 646
application methods, 645
basis ingredients, 633
BERTINIX software, 645, 647
computer implementation, 641–642
defined, 634
disadvantages, 602
first-order predictor, 634
Gröbner basis, comparison, 649
individual elements, 638
other software packages, 646
parameter continuation, 647–648
parameterized solutions, 618, 644
polynomial solution, 601
solution paths, 635, 637, 643
Households
capital, 126
in closed economy, 127
computational algorithm, 132
decision rules, 125
distribution, 126, 135
government policy, 122
heterogeneous nature of, 122
income tax function, 125
labor supply, 126
natural borrowing constraints (optional), 124
optimization problems, 123
perfect annuity markets (optional), 123
preference, 124
problem solving, 132
representative firm, 126
in small open economy (optional), 127
social security system, 125
state variables, 122
Hybrid perturbation methods, 419
benefits, 463
computational model, 448
description, 420
numerical illustration, 421
I
Independent tasks, applications, 423
Individual agents
exogenous driving processes, 281
firm problems, 280
government’s role, 281
level of consumption, 280
Initial boundary value problem (I-BVP), 216, 218
Initial conditions
differential equation, 207
divided growth volatility, 214
financial ratios, 209
linear ODE version, 210, 215–216
parameter values, 216
PDE solution, 213
stochastic differential utility, 215
Integral transform approach, 226, 230, 253
Integrals
approximating functions, 380
Gauss-Hermite product quadrature rules, 380
Monte Carlo integration method, 384
precomputation technique, 405
Integration method, numerical illustration, 388
Intel Phi, recent developments, 595
Interpolating functions
nonproduct approaches, 353
repeated elements, 363
Intertemporal choice conditions
coordination with intratemporal choice, 396
equilibrium condition, 391
equilibrium separation from intratemporal, 436
Euler method, 392
first-order conditions (FOCs), 391
high dimensions, 335, 397
precomputation technique, 405
quantity parameters, 392
vector parameters, 394
Intratemporal choice manifolds
accuracy check, 449
iteration-on-allocation, 449
precomputation methods, 409
separation from intertemporal methods, 445
Inverse bid functions
asymmetric auctions, 95
boundary condition, 48, 57, 60
estimation routine, 39
numerical strategies, 40, 63, 66, 81
theoretical models, 101
vector parameters, 87
Iteration-on-allocation
numerical challenges, 451
partial convergence, 451
stochastic growth model, 450
IVP solution
asset prices behavior, 207
boundary condition, 216
derivation calculation, 202
Gaussian probability density function, 203
investor’s lifetime utility, 201
linear procedure, 202
non-homogenous linear, 204
J
JEDC project
cluster grid algorithm (CGA), 434
generalized stochastic simulation algorithm (GSSA), 439
monomial rule Galerkin method
(MRGAL), 434
participating methods, 432
perturbation methods (PER), 432
Smolyak method (SMOL-MKK), 434
stochastic simulation algorithm (SSA), 434
Jump-diffusion dynamics
American call price, 273
European call option, 237
Merton’s model, 230
method of lines, 253
option pricing, 229–230
K
Kernel-Density methods, 386
L
Left-boundary condition, defined, 48
Lévy processes, 225
Lexicographical ordering, 606
M
Markov Chain, approximation
function, 387
MATHEMATICA
computer algebra packages, 600
Gröbner bases, 611
Reals option, 618
triangular decomposition, 617
MATLAB application
GPGPU computing, 567
GPU computation, 427
for large-scale, 465
numerical examples, 427
parallel methods, 425
Message Passing Interface (MPI), 558
Method of lines (MOL)
accuracy checks, 273
American call option, 249, 269
convergence pattern in, 266
Crank-Nicolson scheme, 250–251
delta estimation, 251
for numerical solutions, 254
integro-partial differential equation, 273
parameter values, 266, 268
pricing estimation, 250–251
Monomial rules
M2 with 2N2 +1 nodes, 383
multidimensional hypercubes, 382
Monte Carlo integration method
for active learning, 8
for dual control (DC), 11, 20
high-dimensional problems, 388
Integrals, 384
for multidimensional hypercubes, 385
for optimal feedback (OF) method, 20
Multi-country model
approximating function, 454
projection methods, 453
Multidimensional hypercubes
monomial rules, 382
Quasi-Monte Carlo methods, 385
sparse grid method (Smolyak), 354
Multiple-Instruction Multiple-Data (MIMD), 557
Multistage portfolio optimization, 496
application, 496
numerical methods, 486, 509
rational spline function, 497
transaction costs, 507
N
Neoclassical stochastic growth model, 337
Non-linear economic models
accuracy, 528
data-based estimators, 534
Euler equations, 535
Feller property, 532
invariant distribution, 530, 532
simulation-based estimation, 536
Non-optimal economies
accuracy properties, 546
computational algorithm, compared, 550
numerical implementation, 545
overlapping generation model, 549
recursive methods, 538
short-run equilibrium condition, 545
simulation problems, 539
taxes, growth model, 539
theoretical algorithm, 544
Nonparametric Kernel-Density Methods, 386
accuracy checks, 389
Nontrivial market clearing
individual problems, 297
numerical solutions, 297
Numerical errors
computational algorithm, 552
contraction property, 518, 527
DP algorithm, 527, 529
estimation process, 518
non-linear economic models, 518
numerical solutions, 542
propogating effects, 518
simple Markov equilibria, 525
worst-case error bounds, 530
Numerical integration, dynamic programming
approximation, 489
Chebyshev polynomials, 490–491
Gauss-Hermite quadrature, 489
shape-preserving Hermite Interpolation, 493
Numerical methods, dynamic programming
basic value function iteration algorithm, 483
multistage portfolio optimization example, 486
optimal growth, example, 485
Numerical tools, dynamic programming
optimization, 488
smooth functions, 488
value function approximation, 488
NVIDIA Kepler, recent developments, 594
O
OpenACC, recent developments, 595
OpenMP/MPI, graphical processing units (GPUs), 562
Optimal Feedback (OF) method
average value compared, 10
constant parameters, 17
DualI software, 16
Monte Carlo experiments, 20
parameter matrix, 9, 19
probability density function, 19
rankings, 12–13
relative performance of, 20
time-varying parameters, 14
two-level grid search, 17
Optimal growth problems
finite-horizon models, 494
infinite-horizon models, 496
SNOPT application, 496
Option pricing
partial differential equation (PDE), 230
under jump-diffusion dynamics, 230, 232
under stochastic volatility, 229, 258
Overlapping-generations economy (OLG model)
aggregate risk, 121
blackbox optimizers, 118
equilibrium dynamics, 540
fiscal policy analysis, 118
lifecycle consumers, 119
macroeconomic variables, 121
recursive competitive equilibrium, 129
social welfare measures, 129
standard Euler-equation methods, 156
stochastic sequences, 523
stylized model economy, 122
P
Parallel computing systems
classification, 557
Parallel dynamic programming
HTCondor MW system, 502
optimal growth problem, 506
value function iteration, 502
Parallel methods
computational techniques, 422
MATLAB application, 425
numerical applications, 430
supercomputers, 428
Parallel programming architectures
basic approaches, 500
grid computing, 501
high-throughput computing (HTC), 500
HTCondor Master-Worker (MW), 501
Parallelization, speedup efficiency, 424
Parameterized expectations algorithm (PEA)
Euler equation, 366
high dimension problems, 363
Passive learning, 2, 7
Perturbation methods
accuracy checks, 313, 417
advantages, 290, 416
aggregation constraints, 292
explicit aggregation, compared, 293
high dimensions, 332, 416
hybrid, 419
individual policy rules, 294
local solution, 461
numerical instability, 421
plain techniques, 412–413
scalar steady state values, 290
second-order approximation, 293
shortcomings, 416
steady state cross-sectional distribution, 294
Polynomial coefficients
approximating functions, 364
way of regression, 364
Polynomial equation
Markov-switching DSGE model, 600
multiple equilibria, 600
Polynomial methods
for analytic functions, 162–163, 173
Chebyshev nodes, 170
error analysis, 162, 213, 216
integral approximation, 169
low-order, 186–187
Monte Carlo simulations, 193
9th order, 187
orthogonal, 170, 176, 194
precomputation technique, 405–406
Portfolio decision
analytic methods, 218
continuous time problem, 163–164, 195
discrete time method, 162
initial conditions, 202
investor’s problem, 165
for non-linear integral equation, 185–186
pricing models, 191
surveys on, 166
Potential bidder, 38, 40, 58
Precomputation methods
aggregate decision rules, 411
intratemporal choice manifolds, 409
numerical illustration, 409
Probability density function, 41–42, 55–56, 58, 81, 91
Projected successive over-relaxation (PSOR)
benchmark solution, 226
Crank-Nicolson scheme, 268
negative correlation, 269
numerical solution, 263
price estimation, 269
R
Real business cycle (RBC) model, value function
iteration (VFI), 580
Recent developments, GPGPU computing
CUDA 5, 594
Intel Phi, 595
NVIDIA Kepler, 594
OpenACC, 595
Runge-Kutta methods
convergence rate, 74
numerical analysis, 82
truncation error, 73
S
Second-price auctions, 39, 86–87, 95
Shape-preservation
Hermite interpolation, 493–494
linear approximation, 494
multistage portfolio optimization, 496
other methods, 499
rational spline function, 497
Simple Markov equilibria
Bellman’s equation, 527
Euler equations, 526, 529
for non-optimal economies, 538
numerical errors, 525
optimality principle, 527
second welfare theorem, 527
Simulated moments estimator (SME), 537
Simulation procedures
aggregate uncertainty, 278
algorithms, 282
calibration, 533
compared, 306
data-based estimators, 534
grid method I, 301
parameter estimation, 533
Single-Instruction Multiple-Data (SIMD), 557, 563
SINGULAR
finding all equilibria with, 627
solving system with, 624
SMOL-JMMV algorithm
computational techniques, 439
Smolyak elements, 437
steps, 437
Smolyak elements
computational details, 447
and FPI, 437
high dimensions, 359, 363
with iteration-on-allocation, 437
numerical illustrations, 379
SMOL-JMMV algorithm, 437
Smooth density approximations
implementation, 282
procedure, 304
time series, 304
Social security, OLG model
computational procedure, 150
long-run effect, 152
transition effect, 153
welfare effect, 154
Social welfare measures
compensating variations, 131
equivalent variations, 130
veil of ignorance, 130
Software GPGPU computing
C++, 569
CUDA C, 572
MATLAB, 567
thrust, 577
Sparse Grid Method (Smolyak)
automated, 360
multidimensional hypercubes, 354
two-dimensional example, 355
Stochastic growth model
accuracy checks, 456
advantages, 365
asset-pricing model with financial frictions, 522
competitive equilibrium, 521–524
DSICE model, 510
dynamic programming, 481
generalized algorithm, 363
generalized stochastic simulation algorithm (GSSA), 367
high dimensions, 364, 431
human activities, impact on climate, 510
ill-conditioned problem, 367
iteration-on-allocation, 450
multidimensional problem, 505
numeric example, 505
optimal carbon tax parameters, 512
optimal growth model, 504
in overlapping generations, 523
precomputation methods, 408
quantitative properties, 520
simulated points, 364
stationary distributions, 530
tax models, 521
Stochastic volatility
American call options, 269, 253
European options, 230
free boundary, impact on, 266
in Merton’s model, 232
integral transform approach, 226
integro-partial differential equation, 273
method of lines, 254, 266
option pricing, 229
Supercomputers
numerical example, 430
parallel methods, 428
Symmetric auction, 58, 76, 95
T
Taylor polynomial approximation, 163, 168–169, 175, 210, 213, 215, 218
Taylor’s methods, auction market
defined, 72
implicit and explicit, 72
inverse-bid function, 71–72
truncation error, 73
Theoretical challenges, dynamic programming
curse of dimensionality, 481
infinite-horizon problems, 482
numerical value function, 482
Theoretical model, auction market
Bayes-Nash equilibrium, 41
extensions, 55
first-price auction, 47
left-boundary condition, 47
notation, 40
right-boundary condition, 47
special case, 51
two-bidder case, 44
Thrust, software, 577
Two-point boundary-value problems, 40, 49, 63, 66–67, 112
U
Uncertainty, idiosyncratic risk, 121
US economy
demographics, 138
government policy, 143
market wage processes, 141
policy experiments, 144
preference, 138
technology parameters, 138
V
Value function iteration (VFI)
model, 580–581
results, 582
solution, 582
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