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Subject Index

accelerated failure time (AFT) model, 591–2coefficient interpretation, 606–7definition, 592leading examples, 585

accept-reject methods, 413–4, 445ACD. See average completed durationacronyms, 17AD estimator. See average derivativeadaptive estimator, 323, 328, 684adding-up constraints, 210additive model, 323, 327, 523additive random utility model (ARUM)

binary outcome models, 476–8generalized random utility models, 515–6identification, 504multinomial outcome models, 504–7nested logit model, 509, 526–7RPL model, 513welfare analysis in, 506–7

admissible estimator, 435AFT. See accelerated failure timeaggregated data

binary outcomes, 480–2cohort-level, 772nonlinear models, 482, 487multinomial outcomes, 513time-aggregated durations, 578, 600–3see also discrete-time duration data

AIC. See Akaike information criterionAID. See average interrupted durationAkaike information criterion (AIC), 278–9, 284, 624almost sure convergence, 947–8analog estimator, 135analogy principle, 135

and method of moments estimators, 167analysis of covariance, 733analysis of variance, 733Anscombe residual, 289

antithetic sampling, 408–9, 445applications with data

competing risks models, 658–62duration models, 603–8, 632–6IV estimation, 110–2kernel regression, 295–7, 300logit and probit models, 464–6, 486multinomial and nested logit models, 491–5, 511Poisson and negative binomial models, 671–4, 690panel fixed and random effects estimation, 708–15panel GMM linear estimation, 754–6panel nonlinear estimation, 792–5quantile regression, 88–90selection and two-part models, 553–6, 565survival function, 574–5, 582treatment evaluation estimation, 889–96see also data sets used in applications

Archimedean family, 654Arellano-Bond estimator, 765–6, 777

application, 754–6nonlinear models, 791unit roots, 768

ARMA. See autoregressive moving averageartificial nesting, 283ARUM. See additive random utility modelasymptotic distribution, 953–4

asymptotic efficiency, 954asymptotic normal distribution, 953definition, 74, 120, 953estimated asymptotic variance, 954of extremum estimators, 127–31of FGLS estimator, 82–3of FGNLS estimator, 156–7of first-differences estimator, 730–1of fixed effects estimator, 727–9of GMM estimator, 173–4, 182–3, 185–6, 194–5,

745–6of Hausman test statistic, 271–4

1006

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Cambridge University Press0521848059 - Microeconometrics: Methods and ApplicationsA. Colin Cameron and Pravin K. TrivediIndexMore information

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SUBJECT INDEX

of kernel density estimator, 301–2, 330–1of kernel regression estimator, 313, 331–3of LM test statistic, 235, 237–8of LR test statistic, 235, 237of m-estimators, 119–21of MD estimator, 292of ML estimator, 142–3of MM estimator, 134, 174of MSL estimator, 394–5of MSM estimator, 400–2of m-test statistics, 260, 263of NLS estimator, 152–4of NL2SLS estimator, 195–6of OIR test statistic, 181, 183of OLS estimator, 73–4, 80–1of panel GMM estimator, 745–6of quasi-ML estimator, 146of random effects estimator, 735of Wald test statistic, 226–8see also asymptotic theory

asymptotic efficiency, 954of optimal GMM, 177

asymptotic refinement, 359, 371–2by bootstrap, 256, 363–7, 371–2, 378–9definition, 359by Edgeworth expansion, 371–2by nested bootstrap, 374, 379

asymptotic theory definitions, 943–55asymptotic distribution, 953asymptotic variance, 954central limit theorems, 949–52consistency, 945convergence in distribution, 948–9convergence in probability, 944–7laws of large numbers, 947–8limit distribution, 948limit variance, 952–3stochastic order of magnitude, 954summary of definitions and theorems, 944

asymptotic variance, 74, 120, 954estimated asymptotic variance, 74, 954see also asymptotic distribution

asymptotically pivotal statistic, 359–60, 363–4, 366,372, 374, 379–80

ATE. See average treatment effectATET. See average treatment effect on the treatedattenuation bias, 903–5, 911, 915, 919–20attrition bias, 739, 800–1, 940augmented regression model, 429autocorrelation

in panel model errors, 705–8, 714–5, 722–5, 745–6dynamic panel models, 763–8, 791–2, 797–9,

806–7see also panel-robust inference

autoregressive moving average (ARMA) errorsdefinition, 159NLS estimator, 159panel data, 722–5, 729

auxiliary model, 404auxiliary regression

bootstrapping, 379, 382example, 241–3, 269–71Hausman test, 276, 718–9LM test, 240–1, 274m-test, 261–4, 544

available case analysis. See pairwise deletionaverage completed duration (ACD), 626average derivative (AD) estimator

definition, 326uses, 317, 483

average interrupted duration (AID), 626average selection bias, 868average squared error, 315average treatment effect (ATE), 33–4, 866–71

definition, 866difficulties estimating, 866local ATE, 883–6matching estimators, 871–8potential outcome model, 33–4selection on observables only, 868–9selection on unobservables, 868–71see also ATET; LATE; MTE

average treatment effect on the treated (ATET),866–78

application, 889–6definition, 866difficulties estimating, 866matching estimators, 871–8, 894–6selection on observables only, 868–9selection on unobservables, 868–71see also ATE; LATE; MTE

averaged data. See aggregated data

backward recurrence time, 626balanced bootstrap, 374balanced repeated replication, 855balancing condition, 864, 893–4bandwidth, 299, 307, 312bandwidth choice for kernel density estimator, 302–4

cross validation, 304example, 296–7optimal, 303, 306Silverman’s plug-in estimate, 304

bandwidth choice for kernel regression estimator,314–6

cross validation, 314–6example, 297, 316optimal, 314, 318plug-in estimate, 314

baseline hazard, 591in AFT model, 592identification in mixture models, 618–20in multiple spells models, 655–6in PH model, 591, 596–7, 601–2

Bayes factors, 456–8Bayes rule. See Bayes theorem

1007




SUBJECT INDEX

Bayes theorem, 421example, 422–4, 435–9

Bayesian central limit theorem, 433Bayesian information criterion (BIC), 278, 284

see also AICBayesian methods, 419–59

Bayes 1764 example, 458–9Bayesian approach, 420–35binary outcome models, 475compared to non-Bayesian, 164, 424–5, 432–41,

439–41count models, 687data augmentation, 454–5, 932–3, 935–9decision analysis, 434–5examples, 452–4hierarchical linear model, 847importance sampling, 443–5linear regression, 435–43, 449–50, 452–4Markov chain Monte Carlo simulation, 445–54,

935–9measurement error model, 915mixed linear model, 775model selection, 456–8multinomial outcome models, 514, 519panel data, 775, 809posterior distribution, 421, 430–4prior distribution, 425–30Tobit model, 563

BCA method. See bias-corrected and acceleratedbefore-after comparison

application, 890–1Berkson error model, 920Berkson’s minimum chi-square estimator, 480–1Berndt, Hall, Hall, and Hausman (BHHH) estimate,

138, 241, 395Berndt, Hall, Hall, and Hausman (BHHH) iterative

method, 343–4Bernoulli distribution, 140, 148, 468, 475, 483Bernstein-von Mises Theorem, 433, 459best linear unbiased predictor, 738, 776between estimator, 702, 736, 841

application, 710–3between-group variation, 709, 733between model, 702BFGS algorithm. See Boyden, Fletcher, Goldfarb, and

ShannonBHHH estimate. See Berndt, Hall, Hall, and HausmanBHHH method. See Berndt, Hall, Hall, and Hausmanbias-corrected and accelerated (BCA) bootstrap

method, 360biased sampling, 42–5, 626–7

see also sample selection; endogenous stratificationBIC. See Bayesian information criterionbinary endogenous variable, 562binary outcome models, 463–89

additive random utility model, 476–8aggregated data, 480–2alternative-invariant regressors, 478

alternative-varying regressors, 478choice-based samples, 478–9corrected score estimator, 916–8definition, 466example, 464–5identification, 476, 483index function model, 475–6marginal effects, 467, 470–1measurement error in dependent variable, 914measurement error in regressors, 919ML estimator, 468–9model misspecification, 472multiple imputation example, 937–8OLS estimator, 471panel data, 795–9semiparametric estimation, 482–6see also logit models; probit models

binding function, 404–5bivariate counts, 215, 685–7bivariate negative binomial distribution, 686–7bivariate ordered probit model, 523bivariate Poisson distribution, 686bivariate Poisson-lognormal mixture, 686bivariate probit model, 522–3bivariate sample selection model, 547–53

application, 553–5bounds, 566conditional mean, 548–50conditional variance, 549–50definition, 547Heckman two-step estimator, 550–1identification, 551, 565–6marginal effects, 552ML estimator, 548outcome equation, 547participation equation, 547semiparametric estimator, 565–6versus two-part model, 546, 552–3

Bonferroni test, 230bootstrap hypothesis tests

asymptotic refinement, 363–4, 366–7, 371–2,378–9

bootstrap critical value, 256, 363bootstrap p-value, 256, 363example, 366–8nonsymmetrical test, 363, 380power, 372–3symmetrical test, 363without asymptotic refinement, 363, 367–8,

378bootstrap methods, 357–83

asymptotic refinement, 359, 366–7bias estimate, 365bias-corrected estimator, 365, 368clustered data, 363, 377–8, 845confidence intervals, 364–5, 368consistency, 369–70critical value, 363

1008




SUBJECT INDEX

examples, 254–6, 366–8for functions of parameters, 363general algorithm, 360for GMM, 379–80heteroskedastic data, 363, 376–7introduction, 254–6for nonsmooth estimators, 373, 380–1number of bootstrap samples, 361–2panel data, 363, 377–8, 708, 746, 751p-value, 363recentering, 374, 379rescaling, 374sampling methods for, 360smoothness requirements, 370standard error estimate, 362, 366time series data, 381variance estimate, 362without asymptotic refinement, 358, 367–8see also bootstrap hypothesis tests

bounds identification, 29in measurement error models, 906–8

bounds in selection model, 566Boyden, Fletcher, Goldfarb, and Shannon (BFGS)

algorithm, 344

CAIC. See consistent Akaike information criterioncalibrated bootstrap, 374caliper matching, 874, 895canonical link function, 149, 469, 783case-control analysis, 479, 823causality, 18–38

examples, 69–70, 98Granger causality, 22identification frameworks and strategies,

35–3in linear regression model, 68–9in potential outcome models, 32–4, 862–5in simultaneous equations model, 26–7in single-equation model, 31and weighting, 820–1see also endogeneity

cdf. See cumulative distribution functioncensored least absolute deviations (CLAD) estimator,

564–5, 808censored models, 530–44, 579–80

conditional mean, 535count models, 680definitions, 532, 579–80examples, 530–1, 535ML estimator, 533–4semiparametric estimation, 563–5see also duration model; selection models; Tobit

models; truncated modelscensored normal regression model. See Tobit modelcensoring mechanisms, 532, 579–80

censoring from above, 532, 579censoring from below, 532, 579left censoring, 532, 579, 588

independent censoring, 580interval censoring, 579, 588noninformative censoring, 580random censoring, 579right censoring, 532, 579, 581, 589sample selection, 44–5, 547type 1 censoring, 579type 2 censoring, 580

census coefficient, 819central limit theorem (CLT), 949–2

Cramer linear transformation, 952Cramer-Wold device, 951definition, 950examples of use, 80, 130Liapounov CLT, 950Lindeberg-Levy CLT, 950multivariate, 951–2sample average, 949sampling scheme, 131, 950

CGF tests. See chi-square goodness-of-fitcharacteristic function, 370, 913, 950chatter, 394, 410Chebychev’s inequality, 946chi-square goodness-of-fit (CGF) tests, 266–7, 270–1,

474choice-based samples, 823

binary outcome models, 478–9see also endogenous stratification

Choleski decomposition, 416, 448CL model. See conditional logitCLAD estimator. See censored least absolute

deviationsClayton copula, 654CLT. See central limit theoremclustered data, 829–53

application, 848–53cluster bootstrap, 363, 377–8, 845cluster-robust inference, 707, 834, 842,

845cluster sampling, 41–2cluster-specific effects, 830–2, 837–45comparison to panel data, 831–2diagnostic tests, 841dummy variables model, 840fixed effects estimator, 840–1, 843–5hierarchical models, 845–8large clusters, 832nonlinear models, 841–5OLS estimator, 75, 833–7quasi-ML estimator, 150random effects estimator, 837–9, 843small clusters, 832see also panel data

cluster-robust standard errorsbootstrap, 363, 377–8, 845clustered data, 834, 842panel data, 706–7, 745–6, 789see also robust standard errors

1009




SUBJECT INDEX

cluster-specific fixed effects (CSFE) estimator,839–41, 843–4

application, 848–53between estimator, 840–1nonlinear models, 843–4within estimator, 140–1

cluster-specific fixed effects (CSFE) model, 831, 843cluster-specific random effects (CSRE) estimator,

837–9, 843–4application, 848–53

cluster-specific random effects (CSRE) model, 831,843–4

cluster variable, 707CM tests. See conditional momentcoefficient interpretation

in binary outcome models, 467, 473in competing risks model, 646in count model, 669in duration models, 606–7in misspecified linear model, 91–2in multinomial outcome models, 493–4, 501–3in nonlinear models, 122–4, 162–3in Tobit model, 541–2see also marginal effects

coherency condition, 562cohort-level data. See pseudo panelscointegration, 382, 767common parameters, 801compensating variation, 500–7, 512competing risks model (CRM), 642–8, 658–62

application, 658–62censoring, 642coefficient interpretation, 646definitions, 642–4dependent risks, 647–8exit route, 643identification, 646independent risks, 644–6ML estimator, 644–5proportional hazards, 645–6spell duration, 643with unobserved heterogeneity, 647, 659

complementary log-log model, 466–7, 603complete case analysis. See listwise deletioncomplex surveys, 41–2, 814–6, 853–6composition methods, 415computational difficulties, 350–2concentration parameter, 109conditional analysis, 717conditional expectations, 955–6conditional independence assumption, 23, 863, 865

definition, 863for participation, 863given propensity score, 865selection on observables only, 868unconfoundedness, 863

conditional likelihood, 139–40, 824panel models, 731–2, 782–3, 796–9, 805

conditional logit (CL) model, 500–3, 524–5application, 491–4definition, 500fixed effects binary logit, 797, 844marginal effects, 493, 501–3, 525ML estimator, 501from ARUM, 505see also multinomial outcome models

conditional ML estimator, 731–2, 782–3, 796–9, 805,824

conditional moment (CM) tests, 264–5, 267–9, 319consistent CM test, 268in duration models, 632example, 269–71in Tobit model, 544see also m-tests

conditional meansquared error loss, 67–9

conditional modestep loss, 68

condition number, 350conditional quantile

asymmetric absolute loss, 68confidence intervals, 231–2, 316, 364–5, 368consistent Akaike information criterion (CAIC), 278consistent test statistic, 248consistency

definition, 945of extremum estimators, 125–7, 132–3of GMM estimator, 173–4, 182of m-estimator, 132–3of ML estimator, 142, 146–50of NLS estimator, 155of OLS estimator, 73, 80strong consistency, 947weak consistency, 947see also asymptotic distribution; identification;

pseudo-true valueconstant coefficients model. See pooled modelcontagion, 612contamination bias, 903–4contemporaneous exogeneity assumption, 748–9, 752,

781continuous mapping theorem, 949control function approach, 37control function estimator, 869–70, 890control group, 49conventions, 16–17convergence criteria, 339–40, 458convergence in distribution, 948–9

continuous mapping theorem, 949definition, 948limit distribution, 948transformation theorem, 949vector random variables, 949see also central limit theorem

convergence in probability, 944–7alternative modes of convergence, 945

1010




SUBJECT INDEX

consistency, 945definition, 945probability limit, 945Slutsky’s theorem, 945uniform convergence, 126, 301vector random variables, 945see also law of large numbers

copulas, 216, 651–5count example, 687definition, 651–2dependence parameter, 653–4leading examples, 654ML estimator, 655survival copulas, 652

correlated random effects model, 719, 786counterfactual, 32, 555, 861, 871

see also potential outcome modelcount data, 665

examples, 665heteroskedasticity, 665right-skewness, 665see also count models

count models, 665–93censored, 680application, 671–4, 690endogenous regressors, 683, 687–9endogenous sampling, 823finite mixture models, 678–9hurdle models, 680–1measurement error in dependent variable, 915measurement error in regressors, 915–8mixture models, 675–7multivariate, 685–7OLS estimator, 684negative binomial model, 675–7NLS estimator, 684panel data, 792–5, 802–8Poisson model, 666–74sample selection, 680semiparametric regression, 684–5truncated, 679–80zero-inflated, 681

covariance matrix. See variance matrixcovariance structures, 177, 379, 753, 766–7covariates. See regressorsCox CRM model. See competing risksCox PH model. See proportional hazardsCox-Snell residual, 289, 631, 633–6CPS. See Current Population SurveyCramer linear transformation, 952Cramer-Rao lower bound, 143, 954

see also semiparametric efficiency boundCramer’s theorem, 949Cramer-Wold device, 130, 951CRM. See competing risks modelcross-equation parameter restrictions, 210cross-section data, 47cross-validation, 304, 314–6, 318, 321

CSFE estimator. See cluster-specific fixed effectsCSRE. See cluster-specific random effectscumulant, 370cumulative distribution function (cdf ), 576cumulative hazard function

definition, 577–8in competing risks model, 644–5as diagnostic tool, 631–2in likelihood function, 588Nelson-Aalen estimator, 582–4, 605–6, 662in proportional hazards model, 590

Current Population Survey (CPS), 58, 814–5curse of dimensionality

in Bayesian methods, 419–20multivariate kernel density estimator, 306multivariate kernel regression estimator, 319high-dimensional integrals, 393

data augmentation, 454–5, 932imputation step, 455, 932for missing data, 932–8prediction step, 455, 933regression example, 933

data-generating process (dgp), 72–3, 124misspecified, 90, 132

data mining, 285–6data sets. See microdatadata sets used in applications

Current Population Survey Displaced WorkersSupplement (McCall), 603–8, 632–6, 658–62

fishing-mode choice data (Kling and Herriges),463–6, 486, 491–5

National Longitudinal Survey (Kling), 110–2National Supported Work demonstration project

(Dehejia and Wahba), 889–95Panel Survey of Income Dynamics cross-section

sample, 295–7, 300Panel Survey of Income Dynamics panel sample

(Ziliak), 708–15, 754–6patents-R&D panel data (Hausman, Hall, and

Griliches), 792–5Rand Health Insurance Experiment expenditures,

553–6, 565Rand Health Insurance Experiment medical doctor

contacts, 671–4, 692strike duration data (Kennan), 574–5, 582Vietnam World Bank Livings Standards Survey,

88–90, 848–53see also applications with data

data structures, 39–62data sources, 58–9handling microdata, 59–61natural experiments, 54–8observational data, 40–8social experiments, 48–54

data summary approach to regression, 820Davidon, Fletcher, and Powell (DFP) algorithm, 344,

350–1

1011




SUBJECT INDEX

decomposition of variance, 955–6degenerate distribution, 948degrees-of-freedom adjustment, 75, 102, 138, 185–6,

278, 841delta method, 231–2

bootstrap alternative, 363density kernel, 421density-weighted average derivative (DWAD)

estimator, 326dependent variable, 71descriptive approach to regression, 820deviance, 149, 244deviance residual, 289, 291DFP algorithm. See Davidon, Fletcher, and Powell

algorithmdgp. See data-generating processdiagnostic tests. See specification testsDID estimator. See differences-in-differencesdifferences-in-differences (DID) estimator, 55–7,

768–70, 878–9application, 890–1consistency, 770definition, 768introduction, 55–7natural experiments, 878with controls, 878–9without controls, 878

direct regression, 906disaggregated data

contrasted with aggregated data, 5–10discrete factor models, 678

see also finite mixture modelsdiscrete outcomes. See binary outcomes; counts;

multinomial outcomesdiscrete-time duration data, 577–8, 600–3

cumulative hazard function, 578discrete-time proportional hazards, 600–3gamma heterogeneity, 620hazard function, 578logit model, 602ML estimator, 601nonparametric estimation, 581–4probit model, 602survivor function, 578

dissimilarity parameter, 509disturbance term. See error termdouble bootstrap, 374dummy endogenous variable model, 557dummy variable estimator, 784–5, 800, 805, 840

see also LSDV estimatorduration data, 573–664

different types, 626, 641duration models, 573–664

accelerated failure time, 591–2applications, 574–5, 583, 589, 603–8, 632–6,

658–62censoring, 579–82, 587–9, 595, 642competing risks, 642–8, 658–62

cumulative hazard function, 577–8discrete time, 577–8, 600–3generalized residual, 631hazard function, 576, 578key concepts, 576–8mixture models, 613–25ML estimator, 587–9multiple spells, 655–8multivariate, 648–55nonparametric estimators, 580–4OLS estimator, 590–1panel data, 801–2parametric models, 584–91proportional hazards, 592–7risk set, 581, 594semiparametric estimation, 594–600, 610–2specification tests, 628–32survivor function, 576, 578time-varying regressors, 597–600see also proportional hazards model

DWAD estimator. See density-weighted averagederivative

dynamic panel models, 763–8, 791–2, 797–9,806–7

Arellano-Bond estimator, 765–6binary outcome models, 806–7count models, 806–7covariance structures, 766–7inconsistency of standard estimators, 764–5initial conditions, 764–5IV estimators, 764–5linear models, 763–8MD estimator, 767nonlinear models, 791–2, 797–9, 806–7nonstationary data, 767–8transformed ML estimator, 766true state dependence, 763–4unobserved heterogeneity, 764weak exogeneity, 749

EDF bootstrap. See empirical distribution functionbootstrap

Edgeworth expansions, 370–1efficient score, 141Eicker-White robust standard errors, 74–5, 80–1, 112,

137, 164, 175see also heteroskedasticity robust-standard errors

EM algorithm see expectation maximizationempirical Bayes method, 442empirical distribution function (EDF) bootstrap, 360

see also paired bootstrapempirical likelihood, 203–6empirical likelihood bootstrap, 379–80encompassing principle, 283endogeneity

definition, 92due to endogenous stratification, 78, 824–5Hausman test for, 271–2, 275–6

1012




SUBJECT INDEX

identification frameworks and strategies, 35–7see also endogenous regressors; exogeneity

endogenous regressors, 78binary, 557, 562in count models, 683–4, 687–9in discrete outcome models, 473in duration models, 598dummy, 557, 562inconsistency of OLS, 95–6in linear panel models, 744–63in linear simultaneous equations model, 23–30in nonlinear panel models, 792in potential outcome model, 30–3returns-to-schooling example, 69–70in selection models, 559–62in single-equation models, 30see also GMM estimator; IV estimator

endogenous sampling, 42–5, 78, 822–9, 856consistent estimation, 827–9leading examples, 823see also censored models; endogenous

stratification; sample selection modelsendogenous stratification, 820, 826–7, 856equation-by-equation OLS, 210equicorrelated errors, 701, 722–4, 804equidispersion, 668, 670error components model. See RE modelerror components SEM, 762error components SUR model, 762error components 2SLS estimator, 760error components 3SLS estimator, 762error term, 71, 168

additive, 168nonadditive, 168

errors-in-variables. See measurement errorestimated asymptotic variance, 954

see also asymptotic distributionestimated prediction error. See cross-validationestimating equations estimator, 13–5

asymptotic distribution, 134–5, 174clustered data, 842computation, 339definition, 134generalized, 134, 790, 794, 804variance matrix estimation, 137–9weighted, 829see also MM estimator

Euler conditions, 171, 749exact identification. See just identificationexchangeable errors, 701, 804exhaustive sampling, 815–6exogeneity, 22–3

conditional independence, 23Granger causality, 22of instrument, 106overidentifying restrictions test for, 277panel data assumptions, 700, 748–52, 754,

781

strong exogeneity, 22weak exogeneity, 22

exogenous sampling, 42–3exogenous stratified sampling, 42, 78, 814–5, 820,

825, 856exogenous regressor. See exogeneityexpectation maximization (EM) algorithm, 345–7

for data imputation, 930–2E (Expectation) step, 346for finite mixture model, 623–5M (Maximization) step, 346compared to NR algorithm, 625

expected elapsed duration, 626experimental data, 48–58

control group, 49natural experiments, 54–8social experiments, 48–54treatment group, 49

explanatory variables. See regressorsexponential conditional mean, 124, 155, 669

coefficient interpretation, 124, 162–3, 669exponential distribution, 140, 584–6

for generalized (Cox-Snell) residual, 631exponential family density, 427

conjugate prior for, 427–8see also linear exponential family

exponential-gamma regression model, 616,633–4

exponential-IG regression model, 634exponential regression model

application with censored data, 606–8, 633example with uncensored data, 159–63

extreme value distribution. See type 1 extreme valueextremum estimator, 124–39

asymptotic distribution, 127–31consistency, 125–7definition, 125formal proofs, 130–2informal approach, 132–3statistical inference, 135–9variance matrix estimation, 137–9

factor analysis, 650factor loadings, 517, 650–1, 689factor model, 517, 648, 686Fairlee-Gumble-Morgenstern copula, 654fast simulated annealing (FSA) method, 347–8FD estimator. See first-differencesFE estimator. See fixed effectsfeasible generalized least squares (FGLS) estimator,

81–3asymptotic distribution, 82definition, 82example, 84–5in fixed effects model, 729in mixed linear model, 775nonlinear, 155–8in pooled model, 720–1

1013




SUBJECT INDEX

feasible generalized least squares (cont.)in random effects model, 705, 734–6, 738, 837–9,

849–51as sequential two-step m-estimator, 201systems FGLS, 208–9

feasible generalized nonlinear least squares (FGNLS)estimator, 155–8

asymptotic distribution, 156definition, 156example, 159–63as optimal GMM estimator, 180–1systems FGNLS, 217

FGLS estimator. See feasible generalized least squaresFGNLS estimator. See feasible generalized nonlinear

least squaresFIML estimator. See full information maximum

likelihoodfinite mixture models, 621–5

counts, 678–9definition, 622EM algorithm, 623–5latent class interpretation, 623number of components, 624–5panel data, 786see also mixture models

finite-sample biasof GMM estimator, 177of IV estimator, 108–12of tests, 250–4, 262

finite-sample correction termfor sampling without replacement, 817

first-differences (FD) estimator, 704–5, 729–31application, 710–11, 714asymptotic distribution, 730–1compared to FE estimator, 731consistency, 730, 764definition, 704–5, 730IV estimator, 758

first-differences (FD) model, 704, 729–31, 758first-differences (FD) transformation, 783–4fixed effects (FE) estimator, 704, 726–9, 756–9,

781–5, 791–2application, 710–3, 792–5asymptotic distribution, 727–9binary outcome models, 796–9clustered data, 839–41compared to DID estimator, 768compared to FD estimator, 729as conditional ML estimator, 732consistency, 727, 764, 781–2, 784–5count models, 802–8definition, 704, 726, 781–4duration models, 802dynamic models, 764–6, 791–2, 797–9, 806–7as FGLS estimator, 729Hausman test for, 717–9identification, 702incidental parameters, 704, 726

inconsistency, 764, 781–2, 784–5IV estimators, 758as LSDV estimator, 733multinomial outcome models, 798selection models, 801Tobit model, 800versus random effects, 701–2, 715–9, 788

fixed effects (FE) model, 704, 726–33, 756–9, 781–5,791–2

cohort-level, 772clustered data, 831, 843definition, 700, 726dynamic models, 764–6, 791–2, 797–9, 806–7endogenous regressors, 756–9identification, 702incidental parameters, 704, 726marginal effects, 702nonlinear models, 781–5, 796–808, 791time-varying regressors, 702versus random effects, 701–2, 715–9, 788see also fixed effects estimators

fixed coefficient, 846fixed design. See fixed in repeated samplesfixed in repeated samples, 76–7

bootstrap sampling method, 360in kernel regression, 312Liapounov CLT, 951Markov LLN, 948Monte Carlo sampling method, 251

fixed regressors. See fixed in repeated samplesflexible parametric models

count models, 674–5hazard models, 592selection models, 563

flow sampling, 44, 626forward orthogonal deviations IV estimator, 759forward orthogonal deviations model, 759forward recurrence time, 626Fourier flexible functional form, 321frailty, 612, 662

see also unobserved heterogeneityFrank copula, 654Frechet bounds, 653–4frequentist approach, 421–2, 424, 439–40FSA method. See fast simulated annealingfull conditional distributions, 431

see also Gibbs samplerfull information maximum likelihood (FIML)

estimator, 214nested logit model, 510–2nonlinear models, 219

functional approachto measurement error, 901

functional form misspecification, 91–2diagnostics for, 272–3, 277–8

gamma distribution, 585–6, 614gamma function, 586

1014




SUBJECT INDEX

Gaussian quadrature, 389–90, 393, 809Gauss-Hermite quadrature, 389–90Gauss-Laguerre quadrature, 389–90Gauss-Legendre quadrature, 389–90

Gauss-Newton (GN) algorithm, 345example, 348

GEE estimator. See generalized estimating equationsgeneral to specific tests, 285generalized additive model, 323, 327generalized cross-validation, 315generalized estimating equations (GEE) estimator,

790, 794, 804, 809generalized extreme value (GEV) distribution, 508

see also nested logit modelgeneralized information matrix equality, 142, 145, 264generalized inverse, 261generalized IV estimator, 187generalized least squares (GLS) estimator, 81–5

asymptotic distribution, 82definition, 82as efficient GMM, 179example, 84–5nonlinear, 155–8

generalized linear models (GLMs), 149–50, 155count data, 683conditional ML estimator, 783GEE estimator, 791quasi-ML estimator, 149–50see also LEF models

generalized method of moments (GMM) estimator,166–222

asymptotic distribution, 173–4, 182–3based on additional moment restrictions, 169,

178–9based on moment conditions from economic theory,

171based on optimal conditional moment, 179–80bootstrap for, 379–80computation, 339definition, 173endogenous counts, 683–4, 687–9with endogenous stratification, 827with exogenous stratification, 823–4examples, 167–71, 178–9finite-sample bias, 177identification, 173, 182linear IV, 183–92linear systems, 211–2nonlinear IV, 192–9one-step GMM estimator, 187, 196, 746, 755optimal GMM, 176optimal moment condition, 179–81, 188optimal weighting matrix, 175–6panel data, 744–66, 789–90, 792practical considerations, 219–20test based on, 245two-step, 176, 187, 746, 755variance matrix estimation, 174–5

weak instruments, 177–8see also panel GMM estimator

generalized nonlinear least squares (GNLS) estimator.See feasible generalized nonlinear least squares

generalized partially linear model, 323generalized random utility models, 515–6generalized residual, 289–90

in duration models, 631in LM test, 239–40plots of, 633–6

generalized Tobit model, 548generalized Weibull distribution, 584–6genetic algorithms, 341GEV distribution. See generalized extreme valueGeweke, Hajivassiliou, Keane (GHK) simulator,

407–8for MNP model, 518

GHK simulator. See Geweke, Hajivassiliou, Keanesimulator

Gibbs sampler, 448–50data augmentation, 454–5, 933example, 452–4in latent variable models, 514, 519, 563see also Markov chain Monte Carlo

GLMs. See generalized linear modelsGLS estimator. See generalized least squaresGMM estimator. See generalized method of momentsGN algorithm. See Gauss-NewtonGNLS estimator. See feasible generalized nonlinear

least squaresGompertz distribution, 585–6Gompertz regression model, 606–8gradient methods, 337–48

see also iterative methodsGranger causality, 22grid search methods, 337, 351grouped data. See aggregated data

Halton sequences, 409–10Hausman test, 271–4

applications, 719, 850–1asymptotic distribution, 272auxiliary regressions, 273bootstrap, 378computation, 272–3, 378, 717–9definition, 271–2for endogeneity, 271–2, 275–6for fixed effects, 717–9, 737, 788, 839for multinomial logit model, 503power, 273–4robust versions, 273, 378, 718–9

Hausman-Taylor IV estimator, 761Hausman-Taylor model, 760–2Hawthorne effect, 53hazard function

baseline in PH model, 591cumulative hazard, 577–8, 582–4definition, 576, 578

1015




SUBJECT INDEX

hazard function (cont.)in mixture models, 616–8multivariate, 649nonparametric estimator, 581, 583parametric examples, 585piecewise constant, 591see also duration models

Health and Retirement Study (HRS), 58Heckit estimator. See Heckman two-step estimatorHeckman two-step estimator

application, 554in Roy model, 556in selection model, 550–1semiparametric estimator, 565–6in Tobit model, 543, 567–8

Hessian matrixestimate, 137Newton-Raphson algorithm, 341–2singular, 350–1

heterogeneous treatment effects, 882, 885–7IV estimator, 886–7LATE estimator, 885RD design, 882

heterogeneitywithin-cell, 480see also unobserved heterogeneity

heteroskedastic errorsadaptive estimation, 323, 328conditional heteroskedasticity, 78definition, 78in GLMs, 149–50in linear model, 84–5, 94–5multiplicative, 84–5, 86–7in nonlinear model, 157–63residuals, 289–90tests for, 241, 267, 275Tobit MLE inconsistency, 538working matrix for, 82–3, 156–8

heteroskedasticity-robust standard errorsbootstrap, 379–80clustered data, 834example, 84–5for extremum estimator, 137, 164intuition, 81for NLS estimator, 155, 164for OLS estimator, 74–5, 80–1, 112panel data, 705for WLS estimator, 83see also robust standard errors

hierarchical linear models (HLMs), 845–8Bayesian analysis, 847clustered data, 845coefficient types, 846–7individual-specific effects, 848mixed linear models, 774–6, 847panel data, 847–8random coefficients model, 847two-level model, 846

hierarchical models, 429Bayesian analysis, 441–2, 447, 450, 514see also hierarchical linear models

histogram, 298see also kernel density estimator

HLM. See hierarchical linear modelhot deck imputation, 929, 940HRS. See Health and Retirement StudyHuber-White robust standard errors, 137, 144, 146

see also robust standard errorshurdle model, 680–1, 690

see also two-part modelhyperparameters, 428, 847hypothesis tests, 223–58

based on extremum estimator, 224–33based on ML estimator, 233–43based on GMM estimator, 245based on m-estimator, 244bootstrap, 254–6, 363–8, 372–3, 378–9for common misspecifications, 274–7, 670–1examples, 236, 241–3, 252–4, 254–6, 372–3induced test, 230joint versus separate, 230–1, 285, 629–30power, 247–50, 253–4size, 246–7, 251–3see also LM tests; LR test; Wald tests, m-tests

identificationin additive random utility models, 504in binary outcome models, 476, 483bounds identification, 29definitions, 29–31in fixed effects model, 702of GMM estimator, 173, 182just identification, 31, 214in linear regression model, 71–2in measurement error models, 905–14in mixture models, 618–20in multinomial probit model, 517in natural experiments, 57–8observational equivalence, 29order condition, 31, 213over identification, 31, 214rank condition, 31in sample selection model, 551, 565, 566set identification, 29in simultaneous equations model, 29–31, 213–4in single-index models, 325and singular Hessian, 351weak identification, 100see also identification strategies

identification strategies, 36–7control function approach, 37exogenization, 36incidental parameter elimination, 36–7instrumental variables, 37matching, 37reweighting, 37

1016




SUBJECT INDEX

identified reduced form, 36IG distribution. See inverse-Gaussianignorable missingness, 927

estimator consistency if MCAR, 927estimator inconsistency if MAR only, 927problems if nonignorable, 940weak exogeneity, 927

ignorability assumption, 863see also conditional independence assumption

importance sampling, 407–8, 443–5, 518accelerated, 409GHK simulator, 407–8importance sampling density, 444importance sampling estimator, 444importance weight, 445target density, 444

imputation methods, 928–39data augmentation, 454–5, 932–4example, 936–8hot deck imputation, 929listwise deletion, 928mean imputation, 928–9multiple imputation, 934–5pairwise deletion, 928regression-based imputation, 930–2

imputation (I) step, 455, 932IM test. See information matrix testIMSE. See integrated mean squared errorincidental parameters, 36

clustered data FE model, 832, 840, 844panel data FE model, 704, 726, 781–2, 805

inclusive value, 510–1incomplete gamma function, 586incomplete panels. See unbalanced panelsindependence of irrelevant alternatives, 503, 505, 527independent variables. See regressorsindependently-weighted IV estimator, 192independently-weighted optimal GMM estimator, 177index function model

binary outcome model, 475–6, 482–3bivariate probit model, 522–3ordered multinomial model, 519–20Tobit model, 536see also single-index model

indicator function, 298indirect inference, 404–5individual-specific effects model

additive, 780binary outcome models, 795–6cluster-specific effects, 830count models, 802–3definitions, 700, 780duration models, 802multiplicative, 780, 793one-way, 700parametric, 780selection models, 801single-index, 780

Tobit models, 800–1two-way, 738see also FE models; RE models

induced test, 230information criteria, 278–9, 283–4

Akaike, 278–9, 284, 624Bayesian, 278, 284consistent Akaike, 278Kullback-Liebler, 147, 169, 278, 280Schwarz, 278, 284

information matrix, 142block-diagonal, 144, 240, 329

information matrix equality, 141–2, 145generalized, 142, 145see also BHHH estimate; OPG version

information matrix (IM) test, 265–6bootstrap, 378computation, 261–2, 378definition, 265example, 270power, 267

instrumental variables (IV) estimatoralternative estimators, 190–2application, 110–2definition, 100–1example, 102–3finite-sample bias, 108–12, 191–2, 196identification, 100, 105–7independently-weighted IV estimator, 192jackknife IV estimator, 192LIML estimator, 191, 214in linear model, 98–112, 183–92, 211–2linear IV as GMM estimator, 170, 186local average treatment effects estimator, 883–9in measurement error models, 908–10, 912–3in natural experiments, 54–5in nonlinear models, 192–9in panel models, 764–5, 757–61quantile regression, 190in selection models, 559split-sample estimator, 191–2systems IV estimator, 211–2, 218–9in treatment effects models, 883–9two-stage IV estimator, 102, 187two-stage least squares estimator, 101–2, 187–91Wald estimator, 98–9see also GMM estimator; panel GMM estimator

instrumentsdefinition, 96–7, 100examples, 97–8by exclusion restriction, 106by functional form restriction, 106invalid, 100, 105–7optimal, 180for panel data, 750–1, 754–6relevance, 108weak, 100, 104–12, 177–8, 191–2, 196, 751–2, 756see also instrumental variables estimator

1017




SUBJECT INDEX

integrated hazard function. See cumulative hazardfunction

integrated mean squared error (IMSE), 303integrated squared error (ISE), 302, 314interval data models

definition, 532–3, 579ML estimator, 534–5

interruption bias, 626intraclass correlation, 816, 831, 835–8inverse-Gaussian (IG) distribution, 614–5, 677inverse law of probability, 421inverse-Mills ratio, 540–1, 553–4inverse transformation method, 409, 412–3inverse-Wishart distribution, 443, 453, 514irrelevant regressors, 93ISE. See integrated squared erroriterated bootstrap, 374iterative methods, 337–48

BFGS, 344BHHH, 343–4convergence criteria, 339–40DFP, 344, 350–1expectation maximization, 345–7, 623–5, 930–2fast simulated annealing, 347–8Gauss-Newton, 345, 348line search, 338Newton-Raphson, 338–9, 341–3, 348numerical derivatives, 340simulated annealing, 347starting values, 340, 351step size adjustment, 338

IV estimator. See instrumental variables

jackknife, 374–6bias estimate, 375bias-corrected estimator, 375example, 376IV estimator, 192standard error estimate, 375, 855

Jensen’s inequality, 956jittered data, 290joint duration distributions, 648–55

copulas, 651–5mixtures, 650–1multivariate hazard function, 649multivariate survivor function, 649–50

joint limits, 767joint versus separate tests, 230–1, 285, 629–30just identification, 31, 100, 173

Kaplan-Meier (KM) estimator, 581–3application, 575, 583, 604–5for baseline hazard, 596–7confidence bands for, 583definition, 581tied data, 582

kernel density estimator, 298–306alternatives to, 306

application, 296–7, 300asymptotic distribution, 301–2, 330–1bandwidth choice, 302–4bias, 301, 330–1confidence interval for, 305consistency, 300convergence rate, 302definition, 299derivative estimator, 305examples, 252–3, 367–8multivariate, 305–6Nadaraya-Watson kernel regression estimator, 312optimal bandwidth, 303optimal kernel, 303variance, 301, 331

kernel functions, 299–300comparison, 300definition, 299higher-order, 299, 306, 313leading examples, 300optimal for density estimation, 303properties, 299

kernel matching, 875, 895–6kernel regression estimator, 311–9

alternatives to, 319–22asymptotic distribution, 313, 331–3bandwidth choice, 314–6bias, 313, 331–2bootstrap confidence interval for, 380–1boundary problems, 309, 320–1conditional moment estimator, 317–8confidence interval for, 316consistency, 313convergence rate, 314definition, 312derivative estimator, 317introduction to nonparametric regression, 307–11multivariate, 318–9optimal bandwidth, 314optimal kernel, 314undersmoothing, 380variance, 301, 331see also nonparametric regression

Khinchine’s theorem, 948KLIC. See Kullback-Liebler information criterionKM estimator. See Kaplan-Meierk-NN estimator. See nearest neighbors estimatorKolmogorov LLN, 80, 111, 947Kolmogorov test, 267Kullback-Liebler information criterion (KLIC), 147,

169, 278, 280

LAD estimator. See least absolute deviationsLagrange multiplier (LM) test

asymptotic distribution, 235, 237–8based on GMM-estimator, 245based on m-estimator, 244bootstrap, 379

1018




SUBJECT INDEX

comparison with LR and Wald tests, 238–9computation, 239–41, 256, 274definition, 234–5examples, 236, 241–3for heteroskedasticity, 241, 267, 275in duration models, 632interpretation, 239–40for omitted variables, 274OPG version, 240–1for random effects, 737, 841score test, 234–5in Tobit model, 544for unobserved heterogeneity, 630, 636see also hypothesis tests

Laplace approximation, 390Laplace distribution, 178, 541Laplace transform, 577LATE estimator. See local average treatment effectslatent class model, 622

see finite mixture modelslatent variable, 475, 532latent variable models

additive random utility model, 476–8, 504–7binary outcomes, 475–8endogenous, 560–1ordered multinomial model, 519–20see also censored models; truncated models

law of iterated expectations, 955law of large numbers (LLN), 947–8

definition, 947examples of use, 80, 129Khinchine’s theorem, 948Kolmogorov LLN, 947Markov LLN, 948sampling schemes, 131, 948strong law, 947weak law, 947

least absolute deviations (LAD) estimatorapplication, 88–90asymptotic distribution, 88binary outcome models, 484bootstrap, 381censored LAD, 564–5, 808definition, 87two-stage LAD, 190see also quantile regression

least-squares dummy variable (LSDV) estimator, 704,732–3, 840

least-squares dummy variable (LSDV) model, 704,732, 840

least squares (LS) estimatorsclustered data, 833–7feasible generalized LS, 81–3, 155–8generalized LS, 81–5, 155–8linear, 70–85nonlinear LS, 150–9ordinary LS, 70–81panel data, 211, 702–3, 720–5

systems of equations, 207–8, 211, 217see also FGLS; FGNLS; OLS; NLS

leave-one-out estimate, 192, 304, 315, 375LEF. See linear exponential familylength-biased sampling, 43–4, 626Liapounov CLT, 80, 131, 950likelihood-based hypothesis tests, 233–43

comparisons of, 235–6, 238–9definitions, 234–5examples, 236–7, 241–3see also LM tests; LR tests; Wald tests

likelihood function, 139–41conditional likelihood function, 139, 731–2, 824definition, 139joint, 19, 824–7leading examples, 140–1marginal, 432, 595partial, 594–6

likelihood principle, 139, 420, 433likelihood ratio (LR) test

asymptotic distribution, 235, 237based on GMM-estimator, 245based on m-estimator, 244comparison with LM and Wald tests, 238–9definition, 234examples, 236, 241–3nonnested models, 279–83quasi-LR test statistic, 244uniformly most powerful test, 237see also hypothesis tests

LIML estimator. See limited information maximumlikelihood

limit distribution, 948see also asymptotic distribution

limit variance matrix, 952–3definition, 952replacement by consistent estimate, 952sandwich form, 953

limited information maximum likelihood (LIML)estimator, 191, 214

Lindeberg-Levy CLT, 80, 131, 950line search, 338linear exponential family (LEF) models, 147–9

conjugate priors, 427–8conditional ML estimator, 782consistency, 148leading examples, 148pseudo-R2, 288residuals, 289–90tests based on, 240, 268, 274–5see also generalized linear models

linear panel estimators, 695–778application, 708–15, 725Arellano-Bond estimator, 764–5between estimator, 703covariance estimator, 733conditional ML estimator, 731–2differences-in-differences estimator, 768–70

1019




SUBJECT INDEX

linear panel estimators (cont.)error components 2SLS estimator, 760error components 3SLS estimator, 762first differences estimator, 704–5, 729–31first differences IV estimator, 758fixed effects estimator, 704, 726–9fixed effects IV estimators, 757–9forward orthogonal deviations IV estimator, 759Hausman-Taylor IV estimator, 761LSDV estimator, 704, 732–3MD estimator, 753, 76–7panel bootstrap, 708, 377–8, 708, 746, 751panel GMM estimators, 744–68panel-robust inference, 705–8, 722, 745–6, 751pooled OLS estimator, 702–3, 720–5random effects estimator, 705, 734–6random effects IV estimator, 759–60within estimator, 704, 726–9within IV estimator, 758

linear panel models, 695–778analysis-of-covariance model, 733application, 708–15, 725between model, 702dynamic models, 763–8endogenous regressors, 744–63first differences model, 704, 730, 758fixed effects model, 700–2, 726–34, 757–9fixed versus random effects, 701–2, 715–9forward orthogonal deviations model, 759Hausman-Taylor model, 760–2incidental parameters problem, 704, 726individual dummies, 699individual-specific effects model, 700LSDV model, 704, 732minimum distance estimator, 753, 766–7mean-differenced model, 758measurement error, 739, 905mixed linear models, 774–6pooled model, 699, 720–5random effects differenced model, 760–1random effects model, 700–2, 734–6, 759–60residual analysis, 714–5strong exogeneity, 700, 749–50, 752time dummies, 699time-invariant regressors, 702, 749–51time-varying regressors, 702, 749–51two-way effects model, 738unbalanced data, 739weak exogeneity, 749, 752, 758within model, 704, 758see also linear panel estimators

linear probability model, 466–7linear programming methods, 341linear regression model

definition, 16–17, 70–1linear systems of equations, 207–14

panel data models as, 211seemingly unrelated regressions, 209–10

simultaneous equations, 22–31, 213–4systems FGLS estimator, 208systems GLS estimator, 208systems GMM estimator, 208systems ML estimator, 214systems OLS estimator, 211systems 2SLS estimator, 212

linearization method, 855link function, 149, 469, 783listwise deletion, 60, 928

consistency under MCAR, 928example, 936–8inconsistency under MAR only, 928

Living Standards Measurement Study (LSMS), 59,88–90, 848–53

LLN. See law of large numbersLM test. See Lagrange multiplier testlocal alternative hypotheses, 238, 247–8, 254local average treatment effects (LATE) estimator,

883–9assumptions, 884–5comparison with IV estimator, 885definition, 884heterogeneous treatment effect, 885monotonicity assumption, 885selection on unobservables, 883Wald estimator, 886see also ATE; ATET; MTE

local linear regression estimator, 320–1, 333local polynomial regression estimator, 320–1local running average estimator, 308, 320local weighted average estimator, 307–8logistic distribution, 476–7logistic regression. See logit modellogit model, 469–70

application, 464–5as ARUM, 477, 486–7clustered data, 844definition, 469for discrete-time duration data, 602GLM, 149imputation example, 937–9index function model, 476marginal effects, 470measurement error example, 919ML estimator, 468–9multinomial logit, 494–5, 500–3, 525nested logit, 509–12, 526–7ordered logit, 520panel data, 795–9probit model comparison, 471–3random parameters logit, 512–6see also binary outcome models

log-likelihood function. See likelihood functionlength-biased sampling, 43–4

log-logistic distribution, 585–6, 592log-normal distribution, 585–6, 592log-normal model, 533, 545–6

1020




SUBJECT INDEX

log-odds ratio, 470, 472log-sum, 510log-Weibull distribution. See type 1 extreme valuelong panel, 723–5, 767longitudinal data. See panel dataloss function, 66–69

absolute error, 67asymmetric expected error, 67Bayesian decision analysis, 434–5expected, 66KLIC, 68, 147, 168, 278–9squared error, 67–9, 156step, 67–8

Lowess regression estimator, 320–1application, 297, 309–10, 712–5

LR test. See likelihood ratio testLS estimators. See least squaresLSDV. See least-squares dummy variableLSMS. See Living Standards Measurement Study

MAR. See missing at randommarginal analysis of panel data, 717, 787marginal effects, 122–4

in binary outcome models, 466–5, 467, 470–1calculus method, 123computing, 122–4definition, 122example, 162–3finite-difference method, 123in fixed effects model, 702, 788in multinomial models, 493–4, 501–3, 519–23, 525population-weighted, 821in sample selection models, 552in single-index models, 123in Tobit model, 541–2see also coefficient interpretation

marginal likelihood, 432, 595marginal treatment effects (MTE) estimator, 886market-level data, 482, 513Markov chain Monte Carlo (MCMC) methods,

445–54convergence, 449, 458in data augmentation, 933examples, 452–4, 512, 687, 936–9Gibbs sampler, 448–50, 514, 519, 563Metropolis algorithm, 450–1Metropolis-Hastings algorithm, 451–2, 512

Markov LLN, 77, 131, 948Marshall-Olkin method, 649–51, 686matching assumption, 864

see also overlap assumptionmatching estimators, 871–8, 889–96

application, 889–96assumptions, 863–5ATE matching estimator, 877ATET matching estimator, 874, 877, 894–6balancing condition, 893caliper matching, 874

counterfactuals, 871exact matching, 872, 891inexact matching, 873interval matching, 875–6kernel matching, 875, 895–6nearest-neighbor matching, 875, 894–6propensity score matching, 873–8, 892radius matching, 876, 895–6selection on observables only, 871stratification matching, 875–6, 893–6variance computation, 877–8, 895

maximum empirical likelihood (MEL) estimator, 206maximum likelihood (ML) estimator, 139–46

asymptotic distribution, 142–3conditional ML estimator, 731–2, 782–3, 796–9consistency, 142, 824definition, 141endogenous stratification, 824–7example, 143–4exogenous stratification, 824MSL estimator, 393–8quasi-ML estimator, 146–50regularity conditions, 141, 145–6restricted, 233unrestricted, 233variance matrix estimation, 144weighted ML estimator, 828see also quasi-ML estimator

maximum rank correlation estimator, 485maximum score estimator, 341, 381, 483–4, 800maximum simulated likelihood (MSL) estimator,

393–8asymptotic distribution, 394–5bias-adjusted MSL, 396–7compared to MSM, 402–3count model examples, 677–8, 687, 689definition, 394example, 397–8multinomial probit model, 518number of simulations, 396random parameters logit model, 522

MCAR. See missing completely at randomMD estimator. See minimum distance estimatormean-differenced estimator, 783, 805–6mean-differenced model, 758, 783mean imputation, 928, 936–8mean integrated squared error (MISE), 303, 314mean-scaling estimator, 783, 805–6mean-square convergence, 946mean substitution. See mean imputationmeasurement error

in cohort-level data, 772–3in dependent variable, 913–4in microdata, 46, 60in panel data, 739, 905in regressors, 899–922see also measurement error model estimators;

measurement error models

1021




SUBJECT INDEX

measurement error model estimators, 899–922attenuation bias, 903–5, 911, 915, 919–20bounds identification, 906–8corrected score estimator, 916–8IV estimator, 908–10, 912–3linear models, 900–11nonlinear models, 911–20OLS estimator inconsistency, 902–4using additional moment restrictions, 909–10using instruments, 908–9using known measurement error variance, 902–3,

910using replicated data, 910–1, 913using validation sample, 911

measurement error models, 899–922attenuation bias, 903–5, 911, 915, 919–20classical measurement error model, 901–2dependent variable measured with error, 913–4examples, 919–20identification, 905–14linear models, 900–11multiple regressors, 904nonclassical measurement error, 904, 920nonlinear models, 911–20panel models, 905scalar regressor, 903serial correlation, 909variance inflation, 904, 916see also measurement error model estimators

median regression. See LAD estimatorMEL. See maximum empirical likelihoodm-estimator, 118–22

asymptotic distribution, 120clustered data, 842–3definition, 118–9sequential two-step, 200–2simulated m-estimator, 398–9tests based on, 244, 263–4weighted m-estimator, 829, 856see also extremum estimators

method of moments (MM) estimatorasymptotic distribution, 134, 174definition, 172examples, 167see also estimating equations estimator; GMM

estimatormethod of scoring, 343, 348method of simulated moments (MSM) estimator,

399–404asymptotic distribution, 400–2compared to MSL, 402–3definition, 400example, 403MNP model, 497, 518number of simulations, 399

method of simulated scores (MSS) estimatorfor MNP model, 519

method of steepest ascent, 344

Metropolis algorithm, 450–1Metropolis-Hastings algorithm, 451–2, 512microdata sets, 58–61

handling, 59–61leading examples, 58–9

microeconometrics overview, 1–17midpoint rule, 388, 391–2minimum chi-square estimator, 203

see also Berkson’s minimum chi-square estimatorminimum distance (MD) estimator, 202–3, 753, 766–7

asymptotic distribution, 202bootstrap for, 379–80covariance structures, 766–7definition, 202equally-weighted, 202generalized, 222indirect inference, 404–5OIR test, 203optimal, 202, 753panel data, 753, 766–7relation to GMM, 203, 753

misclassification, 914MISE. See mean integrated squared errormissing at random (MAR), 926–7

definition, 926and ignorable missingness, 927, 932relation to MCAR, 927

missing completely at random (MCAR),926–7

definition, 927and ignorable missingness, 927relation to MCAR, 927

missing data, 923–41deletion methods, 928examples, 924ignorable assumption, 927imputation with models, 929–41imputation without models, 928–9MAR assumption, 926–7MCAR assumption, 927nonignorable missingness, 927, 940see also imputation methods

misspecification tests. See specification testsmixed estimator, 439–41mixed linear model, 774–6

Bayesian methods, 775FGLS estimator, 775fixed parameters, 774ML estimator, 776random parameters, 774restricted ML estimator, 776nonstationary panel data, 767–8prediction, 776see also hierarchical linear model

mixed logit model, 500–3example, 495definition, 500see also RPL model

1022




SUBJECT INDEX

mixed proportional hazards (MPH) model,611–25

Weibull-gamma mixture, 615see also mixture models

mixture hazard function, 616–8mixture models, 611–25

application, 623–6counts, 675–9durations, 611–25identification, 618–20MSL estimator, 393–8, 687multinomial outcomes, 515–6multiplicative heterogeneity, 613specification tests, 628–32see also finite mixture models; unobserved

heterogeneityML estimator. See maximum likelihoodMM estimator. See method of momentsMNL estimator. See multinomial logitMNP estimator. See multinomial probitmodel diagnostics, 287–91

binary outcome models, 473–4duration models, 628–32example, 290–1multinomial outcome models, 499pseudo-R2 measures, 287–9, 291residual analysis, 289–91see also model selection methods

model misspecification, 90–4see also endogeneity; functional form

misspecification; heterogeneity; omitted values;pseudo-true value

model selection methodsBayesian, 456–8nested models, 278–81nonnested models, 278–84order of testing, 285see also model diagnostics; specification tests

moment-based simulation estimators,398–404

see MSL estimator; MSM estimatormoment-based tests. See m-testsmoment matching. See indirect inferenceMonte Carlo integration, 391–2

direct, 391example, 392importance sampling, 407, 443–5simulators, 393–4, 406–10see also quadrature

Monte Carlo studies, 250–4example, 251–4

moving average estimator, 308moving blocks bootstrap, 373, 381MPH model. See mixed proportional hazardsMSL estimator. See maximum simulated likelihoodMSM estimator. See method of simulated momentsMSS estimator. See method of simulated scoresMTE. See marginal treatment effects

m-tests, 260–71asymptotic distribution, 260, 263auxiliary regressions, 261–3bootstrap, 261, 379chi-square goodness of fit, 266–7, 270–1,

474conditional moment test, 264–5, 267–9, 319CM test interpretation, 268computation, 261–3definition, 260Hausman test, 271–4, 717–9information matrix tests, 265–6, 270outer-product-of-the-gradient form, 262overidentifying restrictions test, 181, 183, 267,

747power, 268rank, 261

multicollinearity, 350–1in multinomial probit model, 517in panel model, 752in sample selection model, 542, 551

multilevel models. See hierarchical modelsmultinomial logit (MNL) model, 500–3, 525

application, 494–5as additive random utility model, 505definition, 500marginal effects, 494, 501–3, 525ML estimator, 501panel data, 798see also multinomial outcome models

multinomial outcome models, 490–528application, 491–5alternative-invariant regressors, 498alternative-varying regressors, 497conditional logit, 500–3, 524–5definition, 496–7identification, 504index function model, 519–20marginal effects, 501–3, 524–5mixed logit, 500–3ML estimator, 496, 501multinomial logit, 500–3, 525multinomial probit, 516–9ordered models, 519–20OLS estimator, 471panel data, 798random parameters logit, 512–6random utility model, 504–7semiparametric estimation, 523–4

multinomial probit (MNP) model, 516–9Bayesian Methods, 519definition, 516–7identification, 517ML estimator, 518MSL estimator, 518MSM estimator, 518MSS estimator, 518see also multinomial outcome models

1023




SUBJECT INDEX

multiple duration spells, 655–8fixed effects, 656lagged duration dependence, 657ML estimator, 658random effects, 657recurrent spells, 655

multiple imputation, 934–9estimator, 934examples, 935–9relative efficiency, 935variance of estimator, 934–5

multiple treatments, 860multiplicative errorsmultistage surveys, 41–2, 814–6, 853–6

variance estimation, 853multivariate data

binary outcomes, 521–3counts, 685–7durations, 640–64see also systems of equations

multivariate-t distribution, 442

NA estimator. See Nelson-AalenNational Longitudinal Survey (NLS), 58, 110–2National Longitudinal Survey of Youth (NLSY),

58–9National Supported Work (NSW) demonstration

project, 889–95natural conjugate pair, 427–8natural experiments, 32, 54–8

definition, 54differences-in-differences estimator, 55–7, 768–70,

878–9examples, 54exogenous variation, 54–5identification, 57–8instrumental variables, 54–5regression discontinuity design, 879–83

ncp. See noncentrality parameternearest neighbors (k-NN) estimator, 319–20

definition, 319example, 308–9symmetrized, 308, 320see also nonparametric regression

nearest-neighbor matching, 875, 894–6negative binomial distribution, 675negative binomial model, 675–7

application, 690bivariate, 215, 686–7hurdle model, 681ML estimator, 677MSL estimator, 677–8NB1 variant, 676NB2 variant, 676panel data, 804, 806

negative hypergeometric distribution, 806neglected heterogeneity. See unobserved

heterogeneity

Nelson-Aalen (NA) estimator, 582–4application, 605–6, 662confidence bands for, 584definition, 582tied data, 582

nested bootstrap, 374, 379nested logit model, 507–12, 526–7

from ARUM, 526–7definition 510–1different versions of, 511–2example, 511GEV model, 508, 526ML estimator, 510sequential estimator, 510welfare analysis, 510see also multinomial models

nested models 278, 281see also nonnested models

neural network models, 322Newey-West robust standard errors, 137, 175,

723definition, 175see also robust standard errors

Newton-Raphson (NR) method, 341–3examples, 338–9, 348

NLFIML estimator. See nonlinear full-informationmaximum likelihood

NLS estimator. See nonlinear least squaresNLSY. See National Longitudinal Survey of YouthNL2SLS estimator. See nonlinear two-stage least

squaresNL3SLS estimator. See nonlinear three-stage least

squaresnoise-to-signal ratio, 903noncentral chi-square distribution, 248noncentrality parameter (ncp), 248nonclassical measurement error, 904, 920nongradient methods, 337, 341, 347–8nonignorable missingness, 927, 940

attrition bias due to, 940selection bias due to, 927, 932, 940

nonlinear estimatorscoefficient interpretation, 122–4extremum estimatorm-estimator, 118–22GMM estimator, 166–222ML estimator, 139–46NLS estimator, 150–9overview, 117–22panel models, 779–810

nonlinear full-information maximum likelihood(NLFIML) estimator, 219

nonlinear GMM estimator, 192–9asymptotic distribution, 194–5definition, 194–5example, 197–8, 199, 688instrument choice, 196NL2SLS estimator, 196

1024




SUBJECT INDEX

optimal, 195panel data, 789–90

nonlinear in parameters, 27nonlinear in variables, 27nonlinear IV estimator. See nonlinear GMMnonlinear least squares (NLS) estimator, 150–9

asymptotic distribution, 152–4consistency, 152–3definition, 151example, 155, 159–64time series, 158–9variance matrix estimation, 154–5

nonlinear panel estimators, 779–810application, 792–5conditional ML estimator, 781–2, 805dummy variable estimator, 784–5, 800, 805first-differences estimator, 783–4fixed effects estimator, 783–5, 794, 796–802, 805–8GEE estimator, 790, 794, 804mean-differenced estimator, 783, 805–6mean-scaling estimator, 783, 805–6ML estimator, 785–6NLS estimator, 787, 794panel GMM estimator, 789–90panel-robust inference, 788–91quadrature, 785–6, 796, 800quasi-differenced estimator, 783–4quasi-ML estimator, 791random effects estimator, 785–6, 794–6, 800–1,

803–4selection models, 801semiparametric, 808

nonlinear panel models, 779–810application, 792–5binary outcome models, 795–6conditional mean models, 780–1count models, 792–5, 802–6dynamic models, 791–2, 797–9, 806–7endogenous regressors, 792exogeneity assumptions, 781finite mixture models, 786fixed effects models, 781–5, 791–2fixed versus random effects, 788incidental parameters problem, 781–2, 805individual-specific effects models, 780–1parametric models, 780, 782–3, 785–7, 792pooled models, 787, 794random effects models, 785–6, 792selection models, 801semiparametric, 808Tobit models, 800–1transition models, 801–2

nonlinear regression model, 151additive error, 168, 193, 217nonadditive error, 168, 193, 218

nonlinear systems of equations, 214–9additive errors, 217copulas, 651–5

mixtures, 650–1ML estimator, 215–6NLFIML estimator, 219NL3SLS estimator, 219nonadditive errors, 217–8nonlinear panel model, 216nonlinear SUR model, 216quasi-ML estimator, 150seemingly unrelated regressions, 216simultaneous equations, 219systems FGNLS estimator, 217systems GMM estimator, 219systems IV estimator, 218–9systems MM estimator, 218systems NLS estimator, 217

nonlinear three-stage least squares (NL3SLS)estimator, 219

nonlinear two-stage least squares (NL2SLS) estimatorasymptotic distribution, 195–6definition, 195–6example, 199see also nonlinear GMM estimator

nonnested modelsCox LR test, 279–80definition, 278example, 283–4information criteria comparison, 278–9overlapping, 281strictly nonnested, 281Vuong LR test, 280–3

nonparametric bootstrap. See paired bootstrapnonparametric density estimation. See kernel density

estimatornonparametric maximum likelihood (NPML)

estimator, 622nonparametric regression, 307–22

convergence rate, 311, 314kernel, 311–9local linear, 320local weighted average, 307–8Lowess, 320nearest-neighbors, 308–9, 319–20series, 321statistical inference intuition, 309–11test against parametric model, 319see also semiparametric regression

nonrandomly varying coefficient, 846normal copula, 654normal distribution, 140

truncated moments, 540, 566–7normal limit product rule. See Cramer linear

transformationNPML estimator. See nonparametric maximum

likelihoodNR method. See Newton-Raphson methodNSW demonstration project. See National Supported

Worknuisance parameters. See incidental parameters

1025




SUBJECT INDEX

numerical derivatives, 340, 350numerical integration. See quadrature

observational data, 40–8, 814–7biased samples, 42–5clustering, 42identification strategies, 36–7measurement error, 46missing data, 46population, 40sample attrition, 47sampling methods, 40–4, 815–7sampling units, 41, 815sampling without replacement, 816–7survey methods, 41–2, 814–7survey nonresponse, 45–6types of data, 47–8

observational equivalence, 29odds ratio, 470

see also posterior odds ratioOIR test. See overidentifying restrictions testOLS estimator. See ordinary least squaresomitted variables bias, 92–3, 700, 716

LM tests for, 274one-step GMM estimator, 187, 196

panel, 746, 755see also two-stage least squares

one-way individual-specific effects model. Seeindividual-specific effects model

on-site sampling, 43, 823optimal Bayesian estimator, 434optimal GMM estimator, 176, 179–81, 187, 195

compared to 2SLS, 187–8optimal MD estimator, 202, 753OPG. See outer-product of the gradientOrbit model, 914order of magnitude, 954ordered logit model, 520, 682ordered multinomial models, 519–20ordered probit model, 520, 535ordinary least squares (OLS) estimator, 70–81

asymptotic distribution, 73–4, 80–1bias in standard errors with clustering, 836–7binary data, 471clustered data, 833–7coefficient interpretation in misspecified model,

91–2consistency 72, 80definition, 71example, 84–5finite-sample distribution, 79heteroskedasticity-robust standard errors, 74–5, 81identification, 71–2inconsistency, 91, 95–6inefficiency, 80nonlinear, 150–9panel data, 702–3, 720–5see also least squares estimators

orthogonal polynomials, 321, 329, 390definition 390

orthogonal regression approach, 920orthonormal polynomials, 321, 329, 390outcome equation, 547, 867outer product (OP) estimate, 138, 241, 395outer-product of the gradient (OPG) version

LM test, 240–1m-test, 262–4small-sample performance, 262

overdispersion, 670–1, 674–6, 690measurement error, 915–6panel data, 794, 806tests for, 671

overidentification, 31, 100, 173, 176, 379–80, 747see also GMM estimator

overidentifying restrictions (OIR) testasymptotic distribution, 181, 183bootstrap, 379–80definition, 181, 267, 277panel data, 747, 756

overlap assumption, 864, 871in RD design, 881

oversampling, 41, 478–9, 814, 872

paired bootstrap, 360, 366–8, 376, 378pairwise deletion, 928

biased standard errors, 928panel attrition, 739, 801panel bootstrap, 377, 707, 746, 751, 789panel data, 47panel data models and estimators, 695–810

comparison to clustered data, 831–2see also linear panel; nonlinear panelpanel GMM estimators, 744–68, 789–90

application, 754–6Arellano-Bond estimator, 765–6asymptotic distribution, 745–6bootstrap, 389–90compared to MD estimator, 753computation, 751–2definition, 745efficiency, 747, 756exogeneity assumptions, 748–52instruments, 744, 747–51IV estimators for FE model, 757–9IV estimators for RE model, 759–60just-identified, 745nonlinear, 789–90OIR test, 747, 756one-step GMM estimator, 746, 755overidentified, 7452SLS estimator, 746, 755two-step GMM estimator, 746, 755variance matrix estimation, 751

panel GMM model, 744–66application, 754–6dynamic, 763–6

1026




SUBJECT INDEX

with individual-specific effects, 750–62without individual-specific effects, 744–53see also panel GMM estimators

panel IV estimators. See panel GMM estimatorspanel-robust statistical inference, 377, 705–7, 722,

746, 751, 788–90for Hausman test, 718

Panel Study in Income Dynamics (PSID), 58, 889parametric bootstrap, 360Pareto distribution

of the first kind, 609of the second kind, 616

partial additive model, 323partial equilibrium analysis, 53, 862, 972

see also SUTVApartial F-statistic, 105, 109, 111partial likelihood estimator, 594–6partial ML estimator, 140partial R-squared, 104–5, 111partially linear model, 323–5, 327, 565, 684participation equation, 547, 551Pearson chi-square goodness-of-fit test, 266Pearson residual, 289, 291peer-effects model, 832percentile, 86percentile method, 364–5, 367–8percentile-t method, 364, 366–7PH model. See proportional hazardspiecewise constant hazard model, 591Pitman drift, 248PML estimator. See pseudo-ML estimatorPoisson distribution, 668Poisson-gamma mixture, 675Poisson-IG mixture, 677Poisson regression model, 666–74

application, 671–4, 690, 792–5, 850–3asymptotic distribution of estimators, 668–9bivariate, 686censored MLE, 535with clustered data, 844, 850–3coefficient interpretation, 669definition, 668equidispersion, 668example, 117–8, 121–2LEF density, 148measurement error, 915–8mixtures, 675–9ML estimator, 668overdispersion, 670–1panel data, 792–5, 802–6quasi-ML estimator, 668–9, 682–3truncated MLE, 535underdispersion, 671zero-truncated, 680see also count models

polynomial baseline hazard, 591, 636pooled cross-section time series model. See pooled

model

pooled estimators, 702–3, 720–5application, 710–2, 725FGLS estimator, 720–1GEE estimator, 790, 794NLS estimator, 794OLS estimator, 211, 702–3, 720–5WLS estimator, 702–3, 721

pooled model, 699, 720–5, 787–8pooling tests, 737population-averaged model. See pooled modelpopulation moment conditions

for estimation, 172for testing, 260see also GMM estimator; MM estimator; m-tests

posterior distribution, 421, 430–4asymptotic behavior, 432–4conditional posterior, 431definition, 421expected posterior loss, 434expected posterior risk, 434full conditional distribution, 431highest posterior density interval, 431highest posterior density region, 431marginal posterior, 430observed-data posterior, 930posterior density interval, 431posterior mean, 423, 434posterior mode, 433posterior moments, 430posterior precision, 423see also Bayesian methods

posterior odds ratio, 456posterior (P) step, 455, 933potential outcome model, 30–4, 861–5

see also treatment effects; treatment evaluationpower of tests, 247–50, 253–4

bootstrapped tests, 372–3conditional moment test, 267–9example, 253–4Hausman test, 273–4local alternative hypotheses, 247–8uniformly most powerful test, 237Wald tests, 248–50

precision parameter, 423predetermined instruments. See weak exogeneityprediction, 66–70

best linear, 70conditional, 66error, 66–70in linear panel models, 738in mixed linear model, 774–6optimal, 66–70rotation groups, 814in structural model, 28weighted, 821

pretest estimator, 285primary sampling units (PSUs), 41, 815,

845–55

1027




SUBJECT INDEX

prior distribution, 425–30conjugate prior, 427definition, 420Dickey’s prior, 439diffuse prior, 426flat prior, 426hierarchical priors, 428–9, 441–2improper prior, 426informative prior, 437–9Jeffreys’ prior, 426noninformative prior, 425, 435–7normal-gamma prior, 437sensitivity analysis for, 429–30see also Bayesian methods

probit model, 470–71application, 465–6as additive random utility model, 477bivariate probit, 522–3bootstrap example, 254–6definition, 470discrete-time duration data, 602as GLM, 149index function model, 476logit model comparison, 471–3marginal effects, 467, 471ML estimator, 470Monte Carlo study example, 251–4multinomial probit, 516–9ordered probit, 520, 535panel data, 795–6simultaneous equations probit, 523, 560–1see also binary outcome models

probit selection equation, 548product copula, 654product integral, 578product rule, 949

see also Cramer linear transformationprogram evaluation. See treatment evaluationprojection pursuit model, 323propensity score, 864–5

application, 893–4balancing condition, 864, 893–4conditional independence assumption, 865definition, 864matching, 873–8, 892see also treatment evaluation

proportional hazards (PH) model, 592–7application, 605–7baseline survivor function estimator, 596–7coefficient interpretation, 606–7competing risks model, 645–6definition, 591discrete-time model, 600–3leading examples, 585mixed PH, 611–25panel data, 802partial likelihood estimator, 594–6

pseudo-ML estimator (PML). See quasi-ML estimator

pseudo panels, 771–3cohort, 771cohort fixed effects, 772–3measurement error, 772–3

pseudo-random number generators, 410–6, 957–9accept-reject methods, 413–4composition methods, 415inverse transformation method, 413leading distributions, 957–9multivariate normal, 416transformation method, 413uniform variates, 412see also MCMC methods

pseudo R-squared measuresfor binary outcome models, 473–4definitions, 287–9example, 290–1for multinomial outcome models, 499

pseudo-true value, 94, 132, 146, 281PSID. See Panel Study in Income DynamicsPSUs. See primary sampling unitspure exogenous sampling, 825p-value, 226, 229, 234, 286, 363

quadrature, 388–90Gaussian, 389–90multidimensional, 393in nonlinear panel models, 785–6, 796, 800see also Monte Carlo integration

qualititative response models. See binary outcomes,multinomial outcomes

quantile, 86–7quantile regression, 85–90

application, 88–90asymmetric absolute loss, 68, 85asymptotic distribution, 88bootstrap, 381computation, 341definition, 87IV estimator, 190multiplicative heteroskedasticity, 86–7

quasi-difference, 783–4quasi-experiment. See natural experimentquasi-maximum likelihood (QML) estimator, 146–50

asymptotic distribution, 146in binary outcome models, 469in clustered models, 842–3definition, 146in LEF, 147–9with multivariate dependent variable, 150in nonlinear systems, 216in panel models, 768, 786in Poisson model, 668–9, 682–3

quasi-random numbers. See pseudo-random numbersQML estimator. See quasi-ML estimator

random assignment, 49–50, 862see also sampling schemes

1028




SUBJECT INDEX

random coefficients model, 94, 385, 774–6, 786see also hierarchical models

random effects (RE) estimator, 705, 734–6, 759–62,785–6

application, 710–1, 725asymptotic distribution, 735clustered data, 837–9, 843–4consistency, 699, 764definition, 705, 734error components 2SLS estimator, 760error components 3SLS estimator, 762FGLS estimator, 734–6GEE estimator, 790, 794, 804Hausman test, 717–9incidental parameters, 704, 726IV estimators, 759–60ML estimator, 736, 785–6, 794–7, 800–1, 803–4NLS estimator, 787, 794quasi-ML estimator, 791two-way effects model, 738versus fixed effects, 701–2, 715–9

random effects (RE) model, 700–2, 734–6, 759–62,785–6

binary outcome models, 795–6Chamberlain model, 719, 786clustered data, 831, 843–4count models, 794, 803–4definition, 700, 734dynamic models, 792duration models, 801–2endogenous regressors, 756–7, 759–62Mundlak model, 719nonlinear models, 785–6selection models, 801Tobit model, 800–1two-way effects model, 738versus random effects, 701–2, 715–9see also hierarchical models; random effects

estimatorrandom number generators. See pseudo-random

numbersrandom parameters logit (RPL) model, 512–6

Bayesian methods, 514definition, 513ML estimator, 513–4

random parameters model. See random coefficientsmodel

random utility models. See ARUMrandomization bias, 53, 867randomized experiment, 50–3

National Supported Work demonstration project,889

randomized trials, 49–53randomly varying coefficient, 847–8rank condition for identification, 31, 182, 214rank-ordered logit model, 521rank-ordered probit model, 521raw residual, 289, 291

RD design. See regression discontinuity designreceiver operators characteristics (ROC) curve, 474reduced form, 21, 25, 213

see also structural modelRE estimator. See random effectsregression-based imputation, 930–2

EM algorithm, 932nonignorable missingness, 932

regression discontinuity (RD) design, 879–83fuzzy RD design, 882heterogeneous treatment effects, 882RD estimator, 882–3sharp RD design, 880–1treatment assignment mechanism, 879–81

regressors, 71alternative-varying, 478, 497–8endogenous, 23–33fixed, 76–7irrelevant, 93omitted, 92–3stochastic, 77time-varying, 597–600, 702, 749–51see also endogenous regressors

regularity conditions for ML, 141–2, 151–6relative risk, 470, 503reliability ratio, 903renewal function, 626renewal process, 626, 638repeated cross section data, 47, 770–3

see also differences-in-differencesrepeated measures. See panel datareplicated data, 910–1, 913RESET test, 277–8residual analysis

definitions, 289–90duration data, 633–6example, 290–1panel data, 714–5small-sample correction, 289

residual bootstrap, 361response-based sampling, 43restricted ML estimator, 233, 776revealed preference data, 498, 516ridge regression estimator, 440Robinson difference estimator, 324–5, 565robust sandwich variance matrix estimate. See

sandwich variance matrixrobust standard errors

bootstrap, 362–3, 376–8Eicker-White, 74–5, 80–1, 112, 137for extremum estimator, 137–9Huber-White, 137, 144, 146Newey-West, 137, 175, 723see also cluster-robust; heteroskedasticity-robust;

panel-robust; systems-robustROC curve. See receiver operators characteristics

curverotating panels, 739

1029




SUBJECT INDEX

Roy model, 555–7, 562definition, 556dummy endogenous variable, 557Heckman two-step estimator, 556ML estimator, 556panel semiparametric estimation, 808as treatment effects model, 867

RPL model. See random parameters logitR-squared, 287

pseudo, 287–9uncentered, 241, 263

running mean estimator, 308

SA method. See simulated annealingsample attrition, 47sample moment conditions

see population moment conditionssample selection bias, 44–5sample weights, 817–21, 853–6

see also weightingsampling schemes

assumptions fo

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