COPYRIGHTED MATERIAL I NDEX

Transcription

1 I NDEX adaptive expectations model autoregressive form distributed lag form estimation examples partial adjustment addition matrices 41 probability 13 ADF see augmented Dickey Fuller test adjusted coefficient of multiple determination adjustment lags 520, AIC see Akaike s information criterion Akaike s FPE criterion 503, 505 Akaike s information criterion (AIC) 420, , , , 557 Almon lag see polynomial lags Amemiya s criterion 420, , analysis of covariance analysis of variance concept multiple regression , test for stability , ARCH see autoregressive conditional heteroskedasticity ARIMA see autoregressive integrated moving average ARMA see autoregressive moving average artificial regression asymptotic properties consistency efficiency 23, unbiasedness 23, variance augmented Dickey Fuller (ADF) test , 572 autocorrelation autocorrelated errors autocorrelation function 483, 484 bootstrap resampling methods 608 coefficient of multiple determination 242, dummy variables effect of AR(1) errors on OLS estimation exercises grid-search procedures iterative procedures levels vs first differences omitted variables partial 505 principles solutions tests autocovariance function 483 autoregression adaptive expectations model estimation effect on OLS estimates estimation 492 order determination 261 time series models autoregressive conditional heteroskedasticity (ARCH) models , 404 autoregressive integrated moving average (ARIMA) model , 560 Box Jenkins model 497 estimation 493 autoregressive moving average (ARMA) model Box Jenkins model errors 261 estimation residuals 494 auxiliary regressions , 162 COPYRIGHTED MATERIAL Bayes theorem model selection Bayesian inference Bayesian information criterion, (BIC)

2 [ 622 ] I NDEX Bayesian vector autoregression (BVAR) 554, 570 Berenblut Webb test , 591 Bernt, Hall, Hall and Hausman (BHHH) method best linear unbiased estimators (BLUE) 23, 68, best linear unbiased scalar (BLUS) residuals , 407 BHHH see Bernt, Hall, Hall and Hausman method bias 23, 24 25, BIC see Bayesian information criterion bivariate normal distributions block-recursive models 385 BLUE see best linear unbiased estimators BLUS see best linear unbiased scalar residuals BM test bootstrap resampling methods 601, 603, 604 applications 608 autocorrelation 608 bootstrap-t method 606 cointegration tests 608 confidence intervals data 607 direct method of generation 607 heteroskedasticity 608 hypotheses testing residuals 607 unit root tests 608 bounded influence estimation , 557 Box Cox test , 442 Box Jenkins model 564 forecasting , 551 seasonality 502 summary assessment time-series analysis trend elimination 500 Breusch and Pagan test , 218, 590 BVAR see Bayesian vector autoregression CADF see cross-sectionally augmented Dickey-Fuller test causality 386, 387, causation regression analysis 75 variables 61 censored normal regression model 344 CES see constant elasticity of substitution characteristic roots characteristic vectors chi-squared distributions 19, 20 idempotent matrices and 55 Chow test , 325, 326, 537 CIPS statistic 562 classical statistical inference CN see condition number Cobb Douglas production functions 228, Cobweb model Cochrane Orcutt procedure , 602 coefficient of determination 70 71, analysis of variance coefficient of multiple determination 131, 146, , 177 adjusted autocorrelation 242, logit/probit models compared , measurements 368 time-series models cofactors 44, 45 cointegration bootstrap resampling methods 608 cointegrating regression concept error correction models and 571 MEH testing REH testing summary assessment tests vector autoregressions and collinearity perfect see also multicollinearity column ranks 49 column vectors 40 common deterministic trends 564 common factor restriction 263 common stochastic trends 564 complement 13 condition number (CN) , 305 conditional distributions conditional omitted variable (COV) estimator conditional probabilities 14 confidence coefficients confidence ellipses 138, 139 confidence intervals bootstrap resampling methods estimation multiple regression simple regression 78 80, 81, 87 88, tests of hypotheses and 31 32

3 I NDEX [ 623 ] confidence regions 135, 138 consistency properties constant elasticity of substitution (CES) constrained least squares method 291 continuous random variables 17 correlation errors in variables multicollinearity 289 see also autocorrelation; partial correlations; serial correlation correlation coefficients 70 correlograms 483 COV see conditional omitted variable estimator covariance analysis of multicollinearity 281 covariance matrix 52 covariance stationary 266, 484 Cowles Foundation approach 386, 391 Cox test 437 Cramer Rao lower bound 121, 125, Cramer s rule cross-equation constraints cross-sectionally augmented Dickey-Fuller (CADF) test 562 cross-validation cumulative distribution function 17 cumulative sums (CUSUM) 411 cumulative sums of squares (CUSUMSQ) 411 CUSUM see cumulative sums CUSUMSQ see cumulative sum of squares data selection search 415, 417 data sets multiple regression time-series analysis 507 data transformation Davidson and MacKinnon test deflation procedures degrees of freedom demand and supply models 5 adaptive expectations model instrumental variable method OLS estimation rational expectations model as simultaneous equations , two-stage least squares method , density gradient model dependent variables 59 dummy 329 determinants properties third order deterministic relationships deterministic seasonal trends 564 deterministic trends 564 DF-GLS test DFBETAS 413 DFFITS estimation diagnostic checking ARCH effects 404 BLUS residuals 407 bounded influence estimation DFFITS exercises least squares residuals omitted variables predicted residuals 405, , 428 recursive residuals studentized residuals 407, 428 Dickey-Fuller test , 269, 554, 556, 557 augmented , 572 cross-sectionally augmented 562 difference-stationary processes (DSP) 267, , differencing tests direct regression 71 discrete random variables 17 disproportionate sampling distributed lag models of expectations estimation finite lags infinite lags polynomial lags rational lags disturbances 5, 63 see also errors double-log specifications 95 96, 99 double-summation operators 16 DSP see difference-stationary processes dummy dependent variables 329 dummy variables autocorrelation cross-equation constraints dependent variables 329 disproportionate sampling exercises heteroskedasticity 327 intercept term changes

4 [ 624 ] I NDEX dummy variables (cont.) linear discriminant function 329, linear probability model , 333, logit models 329, omission bias 319 probit models 329, recursive residuals 409 slope coefficient changes stability tests studentized residuals 407 Durbin Watson test , , , 404, 495 panel data strategies when significant Durbin s alternative test Durbin s h -test 258, , 495 Durbin s t-test dynamic panel data models serial correlation 591 state dependency 591 dynamic simultaneous equations model 552 econometric models 4 5, 6 8 Econometric Society 3 econometrics aims 6, 8 inference aspect 6 meaning 3 4 methodologies 6 8 schematic descriptions 6 8 specification aspect 6 economic models and theories 4 5, 6 8 confirming 8 mathematical formulation 4 testing 8 9 efficiency properties 23, eigenvalues eigenvectors encompassing test endogenous expectations 510 endogenous variables 357 identification error correction models , 563 cointegrating regression 566 cointegration and 571 errors assumptions autocorrelation exogeneity heteroskedastic see heteroskedasticity hypotheses testing moving average 261 multiple regression 128 nonnormality observation 5 predictions 85 87, , regression analysis 63 65, 128 simple regression 63 65, testing type I/II see also standard errors errors in variables classical model , 471 correlated errors exercises Hausman s test instrumental variable methods meaning 452 multiple equations panel data proxy variables 452, reverse regression single equation model see single equation methods ESS see explained sum of squares estimation, levels vs first differences exact identification exhaustive events 13 exogeneity error testing Granger causality and Hausman s test , strict 387 strong 387, 389 superexogeneity 387, 389 tests 391 weak 387, 389 exogenous expectations 510 exogenous variables 357 identification expectations models adaptive see adaptive expectations model development distributed lag models see distributed lag models exercises overview 509 rational see rational expectations model variables 520, expected values explained sum of squares (ESS) 70, explained variables 59

5 I NDEX [ 625 ] explanatory variables 59 dummy linearly dependent extraneous estimates F -distribution 19, 20 F -ratios posterior odds criteria regression selection criteria F -test analysis of variance 84 85, , equality of variances 176 multiple regression 135, 139, 140 nonnested hypotheses relationship with t-test stability , 326 Fieller s method 100, 101, 265 finite lags first difference equations , 300 first-order autoregressive errors 246 first-order partial correlations 146 Fisher test 32, fixed effects models vs random effects FM OLS method 569 forecasts bootstrap resampling methods 605 Box Jenkins models , 551 see also predictions frequency frequency domain method 482 full information methods functions of parameters 158 Gauss Markoff theorem Gauss Newton method generalized instrumental variable estimation (GIVE) generalized least squares (GLS) , estimators panel data 592 generalized moments method (GMM) geometric lag , 526, 533 GIVE see generalized instrumental variable estimation Glejser s test , 223, 227 GLS see generalized least squares GMAT scores GMM see generalized moments method Goldfeld Quandt test 215, 216, goodness of fit measurement , 368 tests time series analysis Granger causality 387, 389, Granger representation theorem 571 GRE scores grid-search procedures , 493 grouping methods hat matrix 405, 407 Hausman s test , errors in variables exogeneity , omitted variable interpretation Hendry s approach to model selection heteroskedasticity assumptions 214 autoregressive conditional bootstrap resampling methods 608 consequences deflators density gradient model detection dummy variables 327 example exercises linear vs log-linear form OLS estimators and Ramsey s test 403 solutions tests higher-order serial correlation Hildreth Lu procedure Hocking s criterion 420, , homogenous equations 49 homoskedasticity 211 hypotheses nested/nonnested , null 28, 29 31, 81 unit roots hypotheses testing 22, bootstrap resampling methods combining tests 32 confidence intervals and errors model selection compared 440 multiple regression , 177 power of test 30

6 [ 626 ] I NDEX hypotheses testing (cont.) simple regression type I/II errors hypotheses-testing search 415 idempotent matrices 55 identification endogenous/exogenous variables necessary and sufficient conditions , order condition 360, 363, 367 rank condition , 397 simultaneous linear equations model through reduced form Working s concept identity matrices 43 Im Pesaran Shin (IPS) test impulse response functions 553 independent events 14 endogenous 357, independent variables 59 indirect least squares method 357, 358, 365 inference Bayesian econometric work 6 small-sample statistical 21 22, 76 83, infinite lag distribution 533 instrumental variable methods errors in variables grouping panel data problems 392 rational expectations 535, 536, simultaneous equations model , 392 intercept term changes interpretive searches 415, 416 intersection of events 13 interval hypothesis 28 intervals estimation 22, one-sided separate vs joint 139 two-sided 27 inverse predictions invertibility condition 487 IPS see Im Pesaran Shin test irrelevant variables inclusion 160, matrix notation iterated weighted least squares method IV see instrumental variable methods J-test jackknife resampling methods Jarque Bera test 441 Johansen procedure 569, 572 joint distributions joint effects , joint estimation method jointly determined variables 357 Koyck lag , 526, 533 KPSS test lag distribution 526 lag operator 487 lagged dependant variables, serial correlation and Lagrangian multiplier (LM) test , , , 495 serial correlation see also Rao score test large-sample theory generalized instrumental variable estimation generalized method of moments maximum likelihood method latent roots latent variables 333 latent vectors least squares with dummy variables (LSDV) model , , see also generalized least squares leastsquaresmethod best linear unbiased estimators constrained 291 generalized see generalized least squares heteroskedasticity and indirect 357, 358, 365 inverse predictions matrix notation multiple regression sampling distribution simple regression 68 71, 76 simultaneous equations model see also ordinary least squares; two-stage least squares (2SLS)

7 I NDEX [ 627 ] least squares residuals 446 diagnostic testing problems least variance ratio (LVR) method 379 Levin Lin (LL) tests 561 Leybourne and McCabe test likelihood function 117 see also maximum likelihood method likelihood ratio (LR) 15, 119, , , 426 heteroskedasticity test 177, limited-information maximum likelihood (LIML) function limited-information methods limiting distribution 24 LIML see limited information maximum likelihood function linear discriminant function 329, linear equation solutions, matrix algebra linear estimators 113 linear functions, matrix algebra linear probability models , 333 disproportionate sampling explanatory variable changes goodness of fit logit/probit models compared , linear regression model see simple regression model linearvslog-linear functional form test linearly independent vectors 46 Liu critique 386 LM see Lagrangian multiplier test log-likelihood function 249 log-odds ratio 334 logit models 329, disproportionate sampling explanatory variable changes goodness of fit probit/linear probability models compared , long-run effects, differencing and LR see likelihood ratio LSDV see least squares with dummy variables model Lucas critique 386 LVR see least variance ratio method MA see moving average Maddala Wu (MW) test Mallow s criterion 420, , marginal distributions market efficiency hypothesis (MEH) testing matrix algebra 11, addition/subtraction 41 determinants exercises inverse irrelevant variables least squares method linear equation solutions multiple regression model multiplication 41, nonsymmetric matrices 308, omitted variables orthogonal matrices 48 postmultiplication 43 premultiplication 43 quadratic functions rank 49 reversal laws 43, 48 significance tests simultaneous equations model 49 52, stability tests traces transpose 41, 43 maximum likelihood (ML) method , equation solutions heteroskedasticity , 230 iterative technique large sample tests panel data 591 mean encompassing test 439 mean-squared error of prediction measurement errors regression analysis 64 ridge regression single equation methods median regression 105 MEH see market efficiency hypothesis method of moments minimum-variance unbiased estimator (MVUE) 23 misspecification dynamics 254, models , 163 tests ML see maximum likelihood method mode regression 105

8 [ 628 ] I NDEX model selection 7, 401, Bayes theorem cross-validation techniques data selection search 415, 417 exercises Hendry s approach hypotheses testing compared 440 hypotheses-testing search 415 interpretive search 415, 416 misspecification tests post-data model construction 415, posterior odds proxy variable search 415, regressors simplification search 415, 416 models assumptions 4 5 comparisons 9 definition 4 misspecified , 163 simple models 4 5 variables 5 see also specific models e.g. simple regression model etc moments method Monte Carlo methods 601, efficiency 603 response surface 603 moving average (MA) errors 261 estimation models multicointegration 571 multicollinearity 147, 167, additional data characteristic roots characteristic vectors condition number , 305 dropping variables examples exercises extraneous estimates first differences 300 measurement problems measures , 305 overview perfect 150, 280, 304 principal component regression , 310 ratios 300 ridge regression , 310 solutions when problematical , 289 multiple correlation coefficients 131 multiple correlations multiple equations errors multiple regression model 60, adjusted R analysis of variance , assumptions auxiliary , 162 confidence intervals data sets degrees of freedom errors 128 estimations examples exercises general case formulas hypotheses testing , 177 inclusion of irrelevant variable interpretation of coefficients least squares method matrix notation multiple correlation 147 omission of relevant variable partial correlation 145, predictions regression coefficients relationships 99 sample calculation stability tests statistical inference three explanatory variables model two explanatory variables model multiple time-series analysis 481 multiplication matrix algebra 41, probability 14 multivariate normal distributions multivariate time-series analysis 481 mutual exclusiveness 13 MVUE see minimum-variance unbiased estimator negative definite matrices negative semidefinite matrices nested hypotheses Newton-Raphson iteration method , 194, 198 Neyman Pearson approach NLLS see nonlinear least squares

9 I NDEX [ 629 ] nonhomogenous equations 49 noniid errors nonlinear least squares (NLLS) 193, 195 nonlinear regressions nonnested hypotheses , problems 440 nonnormality, errors nonoccurrence 13 nonsignificant variable omission nonsingular matrices 46 nonstationary time series detrending nonsymmetric matrices 308, normal distributions 18 19, 54 55, , 345 normal equations multiple regression 129, 130 simple regression 66, 69, 83 see also leastsquaresmethod normal population samples 26 normality tests 441 normalization condition 292, normit models 334 null hypothesis 28, 29 31, 81 null matrices 41 observed significance levels OLS see ordinary least squares omission bias 319 omitted variables autocorrelations as error source 64 estimators 298 Hausman test matrix notation nonsignificant rational expectations model regression analysis 64, relevant tests order condition 360, 363, 367 ordinary least squares (OLS) effect of AR(1) errors heteroskedasticity and multicollinearity 298 recursive models residuals 411 simultaneous equation models variance under heteroskedasticity 221 orthogonal matrices 48 orthogonal regression 75 out-of-sample predictions 86, 428 outliers bounded influence estimation detection 407, 411 DFFITS regression analysis studentized residuals 407 overidentification P-value panel data analysis correlated effects dynamic models errors in variables fixed effects models , LSDV model , , meaning random coefficient model random effects model serial correlation simultaneity SUR model 597 unit root tests variance components model 586 Pareto distribution 441 partial adjustment model adaptive expectations partial autocorrelations 505 partial correlations 145 coefficient computations coefficient relationships general case 146 partial effects 129, PE test 231 percentile methods 605 percentile-t method 606 perfect collinearity perfect multicollinearity 150, 280, 304 personal pignic probability 13 pignic probability 13 Plosser Schwert White (PSW) differencing test point estimation 22 point hypothesis 28 polynomial lags alternative distributed models choosing degree 529, example problems 529 population regression function 65

10 [ 630 ] I NDEX positive definite matrices positive semidefinite matrices post-data model construction 415, posterior distribution posterior odds 15, posterior probability power of test 30, 557 PP tests 557 practically significant results 29 predetermined variables 357 predicted residual sum of squares (PRESS) 406, 428 predicted residuals 405, , 428 predictions errors 85 87, , expected values inverse predictions least squares method minimizing the mean-squared error multiple regression model out-of-sample predictions 86, 428 simple regression model 85 88, stability tests , 191 unbiasedness test 176 variables 61 within-sample prediction 86, 428 see also forecasts PRESS see predicted residual sum of squares pretesting problems 31 principal component regression 310 example multicollinearity , 310 normalization condition 292 prior adjustment prior distribution prior information 21, 297 prior odds 15, prior probability probability addition rules 13 classical view 12 conditional 14 frequency view multiplication rules 14 product operators 15, 16 subjective view 13 summation probability density function probability distributions meaning 17 normal random variables and probit models 329, disproportionate sampling explanatory variable changes goodness of fit logit/linear probability models compared , product operators 15, 16 production functions management bias stability proxy variable search 415, proxy variables coefficient errors in variables 452, PSW see Plosser Schwert White Q statistic 495 quadratic functions, matrix algebra quasi-difference transformation 248 quasi-first differences 246 Ramsey s test , 263, 403 random coefficient model random effects model vs fixed effects models random processes, time series models random variables covariance matrix 52 probability distributions and random walk , 563 regression spurious trends time series models 486 trendless 268 with zero drift 268 rank condition , 397 ranks, matrices 49 Rao score test 121, see also Cramer Rao lower bound ratio to moving average method 500 rational expectations model demand and supply model estimation estimation hypotheses (REH) testing serial correlation problem two-stage least squares method weak version hypotheses 536 rational lags

11 I NDEX [ 631 ] rationality tests recursive models recursive residuals 255, 404, dummy variable method 409 properties 409 reduced form equations 358, 396 parameters 358 regression artificial cointegrating selection of regressors through the origin 83, 150 see also multiple regression model; simple regression model regression analysis definition deterministic relationships stochastic relationships 61, 62, terminology 61 variables 61 see also multiple regression model; simple regression model regression coefficients 63, 81 multiple regression stability tests regressive expectations 511 REH see rational expectations model, hypothesis testing relationships nonlinear simple regression 61 65, stochastic 61, 62, 102, relevant variable omission resampling methods bootstrap see bootstrap resampling methods jackknife Monte Carlo 601, RESET test , 263 residual sum of squares (RSS) analysis of variance autocorrelation distribution goodness of fit multiple regression , 150 reverse regression 72 simple regression 70, 71 see also predicted residual sum of squares; restricted residual sum of squares; unrestricted sum of squares residuals 65 autoregressive moving average (ARMA) model 494 bootstrap resampling methods 607 predicted 405, , 428 regression analysis studentized 407, 428 variance estimate 164 see also errors response surfaces 603 restricted residual sum of squares (RRSS) 119, 155, , returns to scale 164 reversal laws 43, 48 reverse regression 71 75, ridge regression Bayesian interpretation 291 constrained least squares method 291 measurement error interpretation multicollinearity , 310 row ranks 49 row vectors 40 RRSS see restricted residual sum of squares RSS see residual sum of squares sample precision 21 sample regression function 65 sampling distributions 22 least squares estimators normal population samples 26 Sargan s common factor test 263 SBC see Schwartz Bayesian information criterion scalar product of vectors 41 schematic descriptions 6 8 Schwartz Bayesian information criterion (SBC) score function 121 scoring method , 198 search procedures 98 seasonal cointegration 564 seasonality 502, 504, 511, 564 second order autoregression 246 second order partial correlations 146 second order stationary time series 484 seemingly unrelated regression (SUR) model 597 semilog forms 95, 99 semiparametric estimation SER see standard error of the regression

12 [ 632 ] I NDEX serial correlation 240 ARCH models higher order lagged dependent variable tests Lagrange multiplier test misspecified dynamics panel data rational expectation models small-sample tests 605 tests , unit root tests Shapiro-Wilk test 441 significance levels 28 29, 81 significance tests simple models 4 5 simple regression model alternative functional forms analysis of variance bivariate normal distributions confidence intervals 78 80, 81, 87 88, constant term exclusions 83 error terms 63 65, estimations exercises GRE/GMAT tests guidelines 75 hypotheses testing inverse predictions least squares method 68 71, 76 nonlinear relationships normal equations 66, 69, 83 outliers predictions regression fallacy relationship specifications 61 65, residuals analysis reverse regression statistical inference stochastic relationships 102 see also regression analysis simplification search 415, 416 simultaneous equations model causality Cowles Foundation approach 386, 391 dynamic 552 estimation exercises exogeneity , identification instrumental variables method , 392 least squares method limited-information maximum likelihood function matrix notation 49 52, necessary and sufficient conditions , OLS and reduced form two-stage least squares method , variables 357 single equation methods , both variables measured with error one explanatory variable , 471 one variable measured with error two explanatory variables single realization 482 singular matrices 46 slope coefficient changes Slutsky effect 488 SM see substitution method small-sample inference spatial correlation 240 specification testing see diagnostic checking spectral analysis 482 spurious correlations 225 spurious trends SSPR see sum of squares of predicted residuals SSSR see sum of squares of studentized residuals stability tests analysis of variance , dummy variables examples matrix notation multiple regression predictive test , 191 regression coefficients standard error of the regression (SER) 77 standard errors bootstrap method computation calculating multicollinearity simple regression stationarity as null strict 483 weak stationary time series 266, statistical background 11 57

13 I NDEX [ 633 ] statistical inference multiple regression model simple regression model statistical relationships 61, 62 statistically significant results 29 stochastic process 482 stochastic relationships 61, 62, 102, stochastic seasonal trends 564 stochastic trends 564 strict exogeneity 387 strict stationarity 483 strong exogeneity 387, 389 structural equations 358, 396 structural parameters 358 studentization 405, 407, 411, 606 studentized residuals 407, 428 substitution method (SM) 539 subtraction, matrices 41 sum of squares of predicted residuals (SSPR) 428 sum of squares of studentized residuals (SSSR) 428 summation operators superexogeneity 387, 389 supply model see demand and supply models SUR see seemingly unrelated regression model system methods t-distribution 19, 20 t-test hypotheses testing 28 29, 80 linear functions of parameters 158 multicollinearity and 289 relationship with F-test test statistic 28 Theil s criterion 420, 421, multicollinearity third order autoregression 246 time domain method 482 time-series meaning 481 nonstationary , stationary , 563 time-series analysis ARIMA model autocorrelation function 484 autoregressive moving average process autoregressive process Box Jenkins approach coefficient of multiple determination data sets 507 exercises frequency domain method 482 models moving average process overview purely random process random walk 486 time domain method 482 tobit model estimation examples limitations total sum of squares (TSS) 70, traces of matrices transformation of data transpose of a matrix 41, 43 trend-stationary processes (TSP) 267, , trends deterministic 564 elimination 500 spurious stochastic 564 truncated normal distribution 345 truncated regression model truncated variables exercises tobit model truncated regression model TSP see trend-stationary processes TSS see total sum of squares two-stage least squares (2SLS) method , 370 estimation LIML method and rational expectations model standard errors two-step weighted least squares method SLS see two-stage least squares method type I/II errors unbiased linear estimators 113 unbiasedness properties 23, unbiasedness test in prediction 176 underidentification unionofevents 13 unit roots 551, alternative hypothesis bootstrap resampling methods 608 confirmatory analysis DF GLS test low powers of test 557

14 [ 634 ] I NDEX unit roots (cont.) null hypothesis panel data serial correlation stationarity as null structural change and tests , , 608 univariate time-series analysis 481 unpredictability error terms 64 unrestricted residual sum of squares (URSS) 119, 155, 158, , 177 URSS see unrestricted residual sum of squares variables deflation dependent 59, 329 determination 5 dropping errors in exogenous 357, expectational 520, explanatory 59, , intercorrelation linearly dependent random variables 17 18, 52 regression analysis 61 see also dummy variables; omitted variables; proxy variables; truncated variables variance analysis of see analysis of variance heteroskedasticity and multicollinearity 281 non-constant 211 ordinary least squares estimator 221 prediction errors 86 sample 26 simple regression model variance components model 586 see also random effects model variance encompassing test 439 variance-inflation factor (VIF) VARs see vector autoregression models vector autoregression models (VARs) cointegration and overparameterization problems vectors, scalar product 41 Venn diagrams 13 VIF see variance-inflation factor von Neumann ratio 255, 408 Wtestsee Wald test Wald (W) test , , , weak exogeneity 387, 389 weak stationarity weighted least squares method 216, , white noise 563 White s test , 263 wide-sense stationary 484 within-sample prediction 86, 428 Working s concept of identification WTD estimator Yule Walker equations 489

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