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211 INDEX Cointegration and vector autoregressions, 592-597 Cointegration tests, 598-600 Condition number (in multicoUinearity), Conditional probability, 14 Confidence intervals defined, 27-28, 79-80 joint and separate intervals, 138-139 relationship with testing of hypotheses, 32-33 Consistency, 24 Correlations relationships with /-ratios, 145-147 simple, partial, and multiple, 147-148 Correlogram, 528 Cramers rule, 50 Cross-validation, 504 Davidson and MacKinnons 7-test, 515-516 Deletion of variables " pitfalls in using /-ratios, 168 use of F-ratios, 167 Determinants, 44-47 Detrending, 258-259 DFFITS criterion for outliers, 487-488 Diagnostic checking, 476 Dickey-Fuller test, 259, 583 Disproportionate sampling, 330-331 Distributed lags, 408-412 Dropping variables in multicoUinearity, 289-291 use of F-ratios, 167-168, 500-503 DSP model, 259 Dummy variables for changes in intercepts, 307-312 for changes in slopes, 313-315 for cross-equation constraints, 315-317 dummy dependent variables, 322-338 under heteroskedasticity and autocorrelation, 321-322 linear discriminant function, 325-326 linear probability model, 323-325 logit model, 327-329 measures of R-, 333-334 for predictions and standard errors of prediction, 319-320 probit model, 327-329 for testing stability, 318-319 Econometrics aims and methodology, 4-7 definition, 1-2 tests of economic theories, 7 types of models, 2-4 Encompassing test, 517 Endogenous variables, 357 Error correction models, 420, 597 Errors in variables classical model, 449-450 correlated errors, 470 general model, 451-458 grouping methods, 463 Hausmans test for errors in variables, 508 instrumental variable methods, 461-462 Krasker-Pratt criterion, 467 multiple equations, 469 proxy variables, 464-466 reverse regression, 459-461 Estimation interval estimation, 27-28 method of least squares, 69-72, 130-132 method of maximum likelihood (ML), 118 method of moments, 66 point estimation, 22-26 Estimator properties asymptotic unbiasedness, 25 asymptotic variance, 25 BLUE property of least squares, 114-115 consistency, 24 efficiency, 24 unbiasedness, 23 Exogeneity defined, 390-391 relationship with causality, 394 test for exogeneity, 395 Exogenous variables, 357 Expectations adaptive expectations, 408-410 expectations and adjustment lags, 415 model-consistent expectations, 433 naive models, 406-408 rational expectadons, 431-433 sufficient expectations, 433 Extraneous estimators, 293
F-distribution, 21 F-ratios implied by different criteria for model selection, 500-503 relationship to higher R, 167-168 Instrumental variables, 112, 367-369, 461-462 Inverse of a matrix, 48 Inverse prediction, 101-102 Irrelevant variables, 164-165 Generalized least squares, 212-213, 227, 238 Glejsers test, 204 Goldfeld-Quandt test, 206 Goodness of fit, 131, 332-334, 369, 540- 541, 550-552 Granger casuality, 393 Grangers test, 393 Grid-search method, 239 Grouping methods, 463 Hausmans test, 506-509 Hendrys approach to model construction, 494-495 Heteroskedasticity Anscombes test, 204 Breusch and Pagan test, 207-209 consequences, 209-211 deflators, 215-217 estimation of correct variance, 211 Glejsers test, 204 Goldfeld and Quandt test, 206 likeUhood ratio test, 206 maximum likelihood method, 213 Ramseys RESET test, 204 weighted least squares, 212-213 Whites test, 204 Hockings 5p criterion, 498-499 Hypothesis testing general theory, 27-32 relationship with confidence intervals, 32-33 Idempotent matrices, 55 Identification necessary and sufficient conditions, 363-364 through reduced form, 358-360 Workings concept. 385 Inference classical, 22-26 muhiple regression, 134-135 simple regression, 76-78 Joint, marginal, and conditional distributions, 18 Joint and separate confidence intervals, 7-test, 515-516 Koyck model estimation in autoregressive form, 411 estimation in distribution lag form, 412 Lagrangian multiplier (LM) test, 119-124, 177, 251 Levels vs. first differences, 232-234, 259-263 Likelihood ratio (LR) test, 119-124, 206 Limited information maximum likelihood (LIML), 381 Limiting distribution, 25 Linear and quadratic forms, 52 Linear probability (LP) model. 323-325 Linear vs. log-linear forms, 96-97, 220- Liu critique, 389 Logit model, 327-329 Lucas critique, 389 Mallows Cp criterion, 498 Matrix addition, 42 Matrix multiplication, 43 Methods of estimation generalized least squares, 212-213, 227, 238 indirect least squares, 359-360 instrumental variables, 112, 367-369, 461-462 least squares, 69-72, 130-132 limited information maximum likelihood (LIML), 381 maximum likelihood, 118 moments, 66 Model selection, 492-496 Moving average process, 531
index MulticoUinearity condition number, 275 definition, 270 dropping variables, 289-291 extraneous estimates, 293 principal component regression, 284-285 problems in measuring, 276 ridge regression, 281-283 Theils measure, 275 Multiple regression analysis of variance, 157 degrees of freedom and R-, 165-168 inclusion of irrelevant variables, 164-165 interpretation, 143-144 omission of relevant variables, 161-162 partial correlation and multiple correlation, 147-148 prediction, 154-155 statistical inference, 134-135 test for linear functions of parameters, tests for stability, 170-177 tests of hypotheses, 156-159 Multivariate normal distribution, 54 Nonnested hypothesis tests, 514-518 Normal distribution, 19 Normalization in simultaneous equations, 377 Omitted variables autocorrelation, 253 bias in least squares estimators, 161-163 interpretation of Hausmans test, 510-512 test for, 477 Orthognal matrices, 49 Oudiers bounded influence estimation. 488 DFFITS criterion, 487-488 illustrated, 89-95 Partial adjustment models, 419-423 PE test, 223 Piecewise regression, 185 Polynomial lags, 423-427 Posterior odds for model selection, 503-505 Principal component regression, 284-285 Probability, 12-14 Probability distributions, 17-21 Probit model, 327-329 Proxy variables, 464-466 PSW test, 513-514 e-statistics, 540-541 Qualitative variables {see Dummy variables) Ramseys RESET test, 478 Raos score test, 119-124 Rational expectations defined,431-433 estimation of a demand and supply model, 436-442 serial correlation problem, 443 tests for rationality, 434-435, 599 Rational lags, 429 Recursive systems, 387 Reduced form, 358-360 Regression analysis of variance, 84, 156-157 assumptions and specification, 65 causation in regression, 75-76 censored regression model, 339-342 cointegrating regression, 590-591 interpretation, 143-145 inverse prediction, 101-102 irrelevant variables, 164-165 method of least squares, 69-71 method of moments, 66 outliers, 89-95 prediction, 85, 154-155, 319-320 statistical inference, 76-78, 134-135 tests of hypotheses, 80 truncated regression model, 342-343 Regression fallacy, 105 Regression with no constant term, 83 Residuals BLUS residuals, 483 predicted residuals, 481 problems with least squares residuals, 479-480
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