Index

absolute summability, 348 autoregressive process adaptive mixture of t distributions, 272 vector (VAR), 365 adaptive radial-based direction sampling, auxiliary variable Gibbs sampling, 260 262 agent-based simulation, 57 backfitting, 166 AIC, 296 band-limited process, 340, 373 AIC criterion, 407 bandpass filter, 360 Akaike information criterion, 296 bandwidth, 326 aliasing, 338–340 basic structural model, 363, 365 all-pass model, 383 Bayes factor, 226 annual cycle, 361 Bayes’ theorem, 218 ant colonies, 88, 90 Bayesian estimation, 370 aperiodicity, 249 Bayesian estimation of VAR model, 294 AR(1) process, 207 benchmark, 61, 62, 65, 75, 76 ARCH test, 301 BFGS algorithm, 70 ARIMA model bias correction, 208 seasonal, 357, 364 bidirectional filter, 344 ARIMA process, 361 bilinear model, 380, 385, 391 ARMA model, 341, 377 block bootstrap, 199, 200 weak, 385 block-of-blocks bootstrap, 199, 207 ARMA process, 339, 348 block–block bootstrap, 200 artificial neural network, 85 Bonferroni inequality, 200 asymptotic convergence results, 99 bootstrap bagging, 164 asymptotic refinement, 187 bootstrap data generating process, 184, asymptotically pivotal test statistic, 187, 193 191 COPYRIGHTEDbootstrap MATERIAL DGP, 184, 193 autocovariance generating function, 341, bootstrap EDF, 184 347 bootstrap P -value, 184 partial-fraction decomposition of, double, 191 362 equal tail, 186 autocovariance matrix fast double, 192, 193 circulant, 350 symmetric, 186

Handbook of Computational Econometrics edited by D. A. Belsley, E. J. Kontoghiorghes © 2009 John Wiley & Sons, Ltd 488 INDEX bootstrap sample, 184 Atlas, 20 bootstrap test statistic, 184 Automatic Sequence Control bootstrap testing, 183 Calculator, Mark I, 1, 8 two-tailed test, 186 CDC 6400, 20 bootstrap-t, 172 CDC 6600, 20 Breusch–Godfrey test, 299 EDSAC, 1, 20 Breusch–Pagan test, 189 EDSAC 2, 20 Brookings Institution, 9, 24 EDVAC, 20 Bureau of the Census, 322, 331 ENIAC, 8 business cycle, 334 IBM 1130, 20 business software IBM 1620, 20 Excel, 7, 18, 23 IBM 3090, 20 Lotus 1–2-3, 3, 18 IBM 360, 20 VisiCalc, 3, 18 IBM 370, 20 Butterworth filter, 336, 343 IBM 650, 2, 9 IBM 7040, 20 CAC 40 stock index, 421 IBM 7090/94, 20 calculator formula, 56, 57 IBM Card Programmed Calculator, 2 canonical decomposition IBM PC, 21, 26 principle of, 362, 368 microcomputer, 3 causal filter, 353 PDP-10, 20 causality analysis, 305 RS/6000, 21 censored quantile regression, 86 Sun, 21 Census X-11 seasonal adjustment Tandy, 21, 26 method), 2 Titan, 20 Central Statistical Office, 322 UNIVAC, 2, 20 centrosymmetric matrix, 353 Victor 9000, 21 Chase Econometric Associates, 11, 16 conditional expectations, 351, 368 child, 90 conjugate prior, 223 Cholesky decomposition, 64 constructive methods, 87 Cholesky factorization, 353 continuous-time process, 340 Chow test, 301 contragrade method of filtering, 344 circulant autocovariance matrix, 350 convergence checks, 261 circulant matrix, 348, 359 convergence criterion, 60 clustered data, 197 convergence of heuristics, 99 cointegrated variables, 283 convergence of parameter estimates, 102 cointegrating rank, 287 convergence results, 97 cointegration convolution, 322 Johansen test, 297 circular, 324 computational econometrics, 5 linear, 324 computer programming languages Cramer–Wold´ factorization, 341, 344 Assembly, 4 critical value, 184 C, 4 crossover, 90, 91 Fortran, 4, 10 cumulation operator, 345 subroutine libraries, 10 curse of dimensionality, 166 computers cycle Apple, 21, 26 annual, 361 INDEX 489

seasonal, 361 user characteristics, 38 secular, 361 econometric methodologies general to specific, 12 data augmentation, 258 LSE, 12 Data Resources, Inc., 11, 16 econometric modeling languages, 25 data window, 327, 342 econometric programming languages, 27 DCDFLIB, 71 econometric software dead space, 334 AREMOS, 16, 20, 27, 39 decision making, 430 AUTOBOX, 16, 20, 24 delay operator, 346 AUTOREG, 16, 27 density estimation, 154 B34S, 16, 20, 24, 27 density forecast, 412 Betahat, 20 dependent data, 198 BRAP, 24 DIEHARD, 67, 73 Brookings Model Project programs, difference operator, 344, 354, 360 363 9 seasonal, 358 CEF, 12, 16 differential evolution, 85, 88, 91 Damsel, 11, 15, 27 diffuse prior distribution, 370 definition of, 5 digital signal processing, 322 EasyReg, 20 direct sampling, 234 ECON, 9 Dirichlet kernel, 339 EPS, 11, 15, 25, 27 discontinuous methods, 94 EViews, 20, 311 discrete Fourier transform, 333 FP, 16, 20, 24, 25 discrete-time process, 340 Gauss, 20, 27, 39 divide-and-conquer principle, 167, 255 GIVE, 12 double bootstrap, 190, 201, 204 Gremlin, 11 P -value, 191 Gretl, 20 drifting random walk, 364 IDIOM, 16, 20, 24 JMulTi, 311 econometric computing, 5 LIMDEP, 16, 20, 24, 27 breadth, 22 MASSAGER, 12 constructive commands, 30 MicroFit, 16, 20, 24 depth, 22, 24 MicroTSP, 20 developer characteristics, 38 ModelEasy+, 16, 20 directive commands, 30 MODLER, 2, 10, 11, 15, 18, 20, 25, human interface, 15, 26, 37 27, 39 interactive, 2 Mosaic, 20, 25 isolated development, 27 NIMODEL, 12 mainframes, 17 Ox, 20, 27, 39 matrix inversion, 33 PcGive, 12, 16, 20, 24, 37, 311 microcomputer, 18 RATS, 16, 20, 24 minicomputers, 17 REG-X, 12, 16 motivating forces, 17 SHAZAM, 16, 20, 24 network-resident, 2 Soritec, 16, 20 roundoff error, 31, 32 Stata, 20 software design, 35 SYMSIS, 12 textbook treatment of, 14 Tiny TROLL, 18 490 INDEX econometric software (continued) bandpass, 360 TROLL, 10, 11, 13, 20, 25, 39 bidirectional, 344 TSP, 9, 11, 13, 16, 20 Butterworth, 336, 343 WinSolve, 16, 39 causal, 353 WYSEA, 16, 20, 39 contragrade method, 344 XSIM, 11, 15, 25, 27 electronic, 321 economic database systems, 2, 12, 16 finite impulse response (FIR), 343 EM algorithm, 420 frequency response of, 336 empirical distribution function (EDF), gain of, 336 184 Henderson, 329, 331, 336 empirical likelihood, 153 highpass, 321, 325, 360 end-of-sample problem, 329, 331 ideal, 366 Epanechnikov kernel, 157 infinite impulse response (IIR), 343 epiconvergence, 125 Kalman, 365, 368, 372 equal-tail bootstrap P -value, 186 lowpass, 321, 325, 355 equilibration algorithms, 443 phase effect of, 336 equilibrium modeling, 433 symmetric, 336 error in rejection probability (ERP), 187 Wiener–Kolmogorov, 341, 342, 359, errors in variables, 154 362, 372 estimation methods wrapped, 350 autoregressive corrections, 27 finite impulse response (FIR) filter, 343 full information maximum likelihood, 2, 24 finite precision, 57 limited information maximum first-order moving average process likelihood, 2, 9 MA(1), 364 ordinary least squares, 9, 10 fixed point problems, 438 seemingly unrelated regression fixed regressor bootstrap, 202, 207 equations, 2, 23 fixed-interval smoothing algorithm, 371, stepwise regression, 10 372 three-stage least squares, 2, 23 forecast combination, 415 two-stage least squares, 9, 10, 23, 27 forecast error variance decomposition, two-stage least squares), 2 310 weighted least squares, 10 forecasting VAR process, 303 EViews, 311 forward-shift operator, 346 evolution strategies, 85 Fourier analysis, 332 evolutionary algorithms, 88 Fourier coefficients, 332 EXPAR model, 379, 400, 405 Fourier integral transform, 338 expectation maximization, 86 Fourier matrix, 349 expected frequency curve, 157 Fourier synthesis, 339 expected value optimization, 145 Fourier transform discrete, 333, 348, 349 fast double bootstrap (FDB), 192, 193, fast (FFT), 333 204, 207, 209 integral transform, 338 fast Fourier transform (FFT), 333 frequency mixed-radix, 359 fundamental, 332 filter Nyquist, 334, 337, 340 auxiliary, 328 seasonal, 366 INDEX 491 frequency domain, 322, 326, 332, 335, hybrid heuristics, 93 338, 345, 359 hybrid metaheuristics, 93 frequency response function, 336 frequency response of filter, 336 ideal filter, 366 fundamental frequency, 332 importance sampling, 239 impulse response, 63, 64 gain of filter, 336 impulse response analysis, 306 GARCH, 58, 60–62, 65, 68, 75, 76 indicator function, 184 GARCH model, 85, 378, 387, 414 infinite impulse response (IIR) filter, 343 generalized additive model, 166 initial conditions generalized instrumental variables, 197, for Kalman filter, 370 198 for Wiener–Kolmogorov filter, 355, generalized inverse, 359 360 generating function integrated moving average model autocovariance, 341 IMA(2, 1), 364 genetic algorithms, 85, 88, 90, 99 integrated variable, 282 geometric ergodicity, 399 interior point algorithm, 122 Gibbs algorithm, 417 Internet, 7 Gibbs sampling, 255 ARPANET, 3 with data augmentation, 258 inversion method, 235 global optimization, 137 irreducibility, 249 Granger-causality, 305 iterated bootstrap, 190 graph theory, 430 graphical user interface, 26 jackknife, 153 greedy algorithms, 87 Jacobian, 69 griddy Gibbs sampling, 257 JMulTi, 311 gross domestic product (GDP), 334, 361 Johansen test for cointegration, 297 guided search, 94 join-point problem, 200 JSTOR, 5, 6 Hamilton filter, 419 J test, 203 Hannan–Quinn criterion, 296 Harvard University, 8, 24 Kalman filter, 365, 368, 372 hat matrix, 195 initial conditions, 370 Henderson filter, 329, 331, 336 Kalman gain, 368 Hessian, 69, 70 kernel density estimators, 156 heteroskedasticity, 174 kernel function, 324 heteroskedasticity-robust test, 204, kernel regression estimators, 161 206 kernel smoothing, 325 heteroskedasticity, tests for, 188 Koenker test, 188, 189 heuristic, 83 heuristic optimization, 87 lag operator, 346 hidden Markov model, 86 finite-sample version, 346 hidden Markov model (HMM), 380 Lagrange multiplier test, 389–395 highest posterior density region, 225 Breusch–Godfrey form, 391 highpass filter, 321, 325, 360 Laurent expansion, 344 hill climbing, 88 Laurent polynomial, 346 HQ criterion, 296 least median of squares estimator, 112 492 INDEX likelihood function, 218, 370 Massachusetts Institute of Technology, 24 likelihood ratio test, 389–395 Center for Computational Research linear congruential random number in Economics and generator, 234 Management Science, 2, 11 linear process, 382 mathematical and statistical programming linearity testing, 385 languages LM test, 299 GAMS, 39 local polynomial regression, 326, 345 Mathematica, 39 local search methods, 87 Matlab, 7, 39 logit model, 240, 253 R, 39 low-level hybridization, 95 S-Plus, 39 lowpass filter, 321, 325, 355 matrix LSTAR model, 86 centrosymmetric, 353 circulant, 348, 359 machine interface, 23 orthonormal, 348 macroeconometric model solution persymmetric, 352 techniques spectral factorization, 349 Gauss–Seidel, 13 Toeplitz, 346 Jacobi, 13 unitary, 349 model-consistent, 13 matrix inversion lemma, 352, 356 Newton, 13 maximized Monte Carlo test, 189 rational expectations, 13 maximum likelihood estimation, 70 macroeconometric models, 11 MCMC methods, 416 Brookings Quarterly Econometric measurement equation, 365 Model of the United States, memetic algorithm, 96 9 memoryless methods, 94 Cambridge growth model, 12 merit function, 123 Candide, 12 metaheuristics, 93 first computer solution of, 2 Metropolis–Hastings algorithm, 250, 416, first simultaneous solution of, 2 417 HM Treasury, 12 Metropolis–Hastings within Gibbs Klein–Goldberger, 2 method, 257 LINK Project, 12 minimax, 121 Liverpool model, 12 minimum mean square error estimation, London Business School, 12 342, 351, 372 National Institute of Economic and minimum volume ellipsoid, 167 Social Research, 12 misspecification tests Southampton model, 12 Durbin–Watson, 10 TIM, 12 nominal confusion, 34 Wharton model, 9, 18 mixed-radix fast Fourier transform, 359 marginal likelihood, 243 mixing coefficient marginal significance level, 184 beta, 398 Markov chain, 249 strong, 398 Markov chain Monte Carlo, 249 mixture of t distributions, 272 Markov switching ARCH model, 418 model selection, 85, 110, 225 Markov switching model, 380, 397, money supply of USA, 365 418 Monte Carlo test, 185 INDEX 493

Monty Hall problem, 220 outliers, 86 moving average process, 347 first-order MA(1), 364 pairs bootstrap, 193, 195, 197, 199, 205 integrated IMA(2, 1), 364 pairwise bootstrap, 193 moving block bootstrap, 199, 207 parameter estimation, 85 multi-agent methods, 94 partial-fraction decomposition of multiple tests, 200 autocovariance generating multivariate regression model, 197 function, 362 mutation, 90 particle swarm optimization, 92 PcGive, 311 Nash equilibria, 431 percentile bootstrap, 173 nearest neighbors, 161 periodic extension, 332, 366 neighborhood definition, 104 periodogram, 334, 366 Nelder–Mead simplex algorithm, 85 persymmetric matrix, 352 networks, 429 phase effect of filter, 336 communications, 429 pivotal test statistic, 185, 186 dynamic transportation, 470, 474 point-optimal test, 202, 203 economic equilibrium problems, 432 polynomial economics, 432 cubic, 329 energy, 429 Laurent, 346 equilibrium problems, 431, 432 polynomial algebra, 322, 346 logistical, 429 polynomial regression, 327, 345 market equilibrium problems, 432, polynomial trend, 326 433, 436, 438 population, 90, 96 physical, 429 population-based methods, 88, 94 spatial equilibrium problems, 432 portfolio optimization, 96 supply chain, 432, 478, 479 portmanteau test, 298, 386 telecommunications, 431 posterior density, 218 transportation, 429, 443, 446 posterior odds, 226 non-informative prior, 224 power spectrum, 334 non-negative polynomials, 138 prediction error, 154 non-nested hypothesis test, 203 prepivoting, 190 nonlinear forecast, 409 Princeton University, 24 nonlinear least squares, 69, 70 prior density, 218 nonlinear least squares estimator, 401 probit model, 258 nonlinear regression model, 195, 197, projected dynamical systems, 432, 203 464–466, 468–470, 472, 475 nonparametric density estimators, 156 pseudo-random numbers, 234 nonparametric regression, 155 pseudo-spectrum, 362 normality test, 300 P test, 203 NP-completeness, 84 P -value, 184 numerical standard error, 245 Nyquist frequency, 334, 337, 340 QR decomposition, 57 quadratic module, 139 observation equation, 368 quantile regression, 163 optimization heuristic, 83 quantum mechanics, 340 orthonormal matrix, 348 quasi-maximum likelihood estimator, 401 494 INDEX quasi-maximum likelihood estimator, semi-infinite programming, 123 (continued) serial correlation, 198 asymptotic normality, 404 SETAR model, 379, 392, 409, 411, 417 consistency, 402 Shannon–Nyquist sampling theorem, 326, 338, 339 R software, 155 sieve bootstrap, 198 Rademacher distribution, 196 signal extraction, 345 random number generator, 67, 72, 73, signal processing 75 digital, 322 linear congruential, 234 simplex method, 430 random walk with drift, 364 simulated annealing, 85, 88 recursive residual bootstrap, 195, 207, simulation-based testing, 183 209 simultaneous equations model, 197 reduced form of time-series model, 361 sinc function, 326, 339 reduced form VAR model, 288 single agent methods, 94 reduced rank regression, 292 singular-value decomposition, 57 regression model smoothers, 160 multivariate, 197 smoothing algorithm, 365, 368, 371 nonlinear, 195, 197, 203 fixed-interval, 371, 372 rejection probability function (RPF), smoothing parameter, 337 187 spatial price, 430, 431, 444, 454–458, rejection sampling, 237 460, 461 relative numerical efficiency, 246 dynamic, 475, 476 resampling, 193, 198, 199 spectral density function, 350, 359 residual bootstrap, 195, 204, 205, 207, pseudo, 362 209 spectral factorization of matrix, 349 residual-based block bootstrap, 200 splines, 162 restarts, 106 STAMP program, 367 roots of unity, 349 state-space representation, 363, 365 rounding error, 57–59, 62, 67, 71 stationarity running-interval smoother, 163 second order, 395 strict, 395 sampling theorem, 326, 338, 339 statistical software Savage–Dickey density ratio, 228 BMD, 10 sawtooth function, 366 OMNITAB, 10 Schwarz criterion, 296 SAS, 12 score test, 389–395 SPSS, 12 search with memory usage, 94 Statistics Quiz, 65, 66 seasonal ARIMA model, 357, 364 StRD, 67–69 seasonal component of data sequence, structural change, 201 357 structural form VAR model, 288 seasonal cycle, 361 structural time-series model, 361, 363 seasonal differencing operator, 358 structural VAR, 306 seasonal frequency, 366 structural VEC model, 308 seasonal summation operator, 358, 364 subgradient, 129 secular cycle, 361 subsampling, 341 semi-definite programming, 138 sum-of-squares representations, 138 INDEX 495 summability University of Chicago, 24 absolute, 348 University of London, 20 summation operator London School of Economics and seasonal, 358, 364 Political Science, 12, 24 supF statistic, 202, 204, 205 University of Michigan, 2 supremum test statistic, 393 University of Minnesota, 24 switching regression, 85 University of Pennsylvania, 2, 8, 20, 24 symmetric bootstrap P -value, 186 University of Warwick Symposium on Large-Scale Digital ESRC Macroeconomic Modelling Calculating Machinery, 1 Bureau, 12 University of Wisconsin, 24 tabu list, 89 unobserved components, 351, 363, 368 tabu search, 88, 89 updating equation of Kalman filter, 369 test statistic asymptotically pivotal, 187, 191 VAR model pivotal, 185, 186 Bayesian estimation, 294 tests for heteroskedasticity, 188 reduced form, 288 tests for structural change, 201 structural form, 288 TESTU01, 67, 73, 75 VAR models, 85 threshold accepting, 85, 89, 99, 113 VAR order selection, 295 threshold autoregressive models, 85 VAR process, 285 threshold methods, 88 variance decomposition, 63 threshold sequence, 89, 104 variational inequalities, 432, 435, 438, threshold vector error correction models, 448–450, 456, 458, 460, 461 85 VEC model, 286 time domain, 322, 346 VEC models, 85 time-reversibility, 250 VECM, 286 Toeplitz matrix, 346 vector autoregression, 62–64, 75 top Lyapunov exponent, 396 vector autoregressive process, 285, 365 traffic assignment, 443 vector error correction model, 286 trajectory methods, 88, 94 volatility forecast, 414 TRAMO–SEATS program, 331, 344, von Neumann, John 363, 367, 368, 373 transition equation, 368 first computer program, 5 trend of data sequence, 325, 326, 329, first stored-program computer, 5 334, 336, 344, 354, 355, 361 game theory, 5 trend/cycle component of data sequence, 357 Wald test, 389–395, 406 trigonometric basis, 332 wavelet analysis, 322, 340 trigonometric identity, 332 wavelets, 154 two-stage least squares, 197, 198 wavepacket, 340 Wharton Econometric Forecasting unguided search, 94 Associates, 11, 16 unitary matrix, 349 white noise University of Auckland, 24 strong, 382 University of Cambridge, 1, 20, 24 testing, 386–388 Department of Applied Economics, weak, 382 1, 2, 12 white noise process, 340 496 INDEX white noise process, (continued) wild bootstrap, 196, 197, 204, 205 band-limited, 340 worst case analysis, 121 Wiener process, 340 worst case strategy, 122 Wiener–Kolmogorov filter, 341, 342, wrapped filter, 350 359, 362, 372 initial conditions, 355, 360 z transform, 322, 345, 348