TOWARDS A METHODOLOGY OF ECONOMETRIC MODELLING

TOWARDS A METHODOLOGY OF ECONOMETRIC MODELLING

Index

acceptance region, 286-7

and confidence region 304-5

actual DGP 20 661

adjusted R*, 382

admissible estimator 236

almost sure convergence 188

alternative hypothesis 287

approximate MLE’s 533-6

a priori restrictions 377

exclusion (zero-one), 616-19

linear, 396-401, 422-7

linear homogeneous 615-19

non-linear, 427-32

ARIMA (p.d.q) process, 156

stability conditions, 161

ARMA (p.q) process, 159-61

ARCH test 550

AR(1) process 150-5

error 506-7

estimation 279-8]

stability condition, 1524

ARi(m) process 155-8, 506-7

asymptotic expansions 203-8

asymptotic independence 140-1, 501

asymptotic moments 192

asymptotic power function, 327

asymptotic properties of estimators, 244-7

consistency 244

efficiency 247

normality 246

unbiasedness 247

asymptotic properties of tests 326-8

consistency 327

locally UMP 328

UMP, 327-8

unbiasedness 327

asymptotic stationarity, 153 asymptotic test procedures, 328- 35 asymptotic uncorrelatedness 141 autocorrelation, 134

autocorrelation, errors 501-3, 505-11 tests for, 513 21

autocovariance, 134 autoproduct moment, 134 auxillary regressions, 446-7, 460-1, 467

470

Bassman test, 652 Bayes’ formula, 121 Bayesian approach, 220 Bernoulli distribution, 62-3, 166 Bernoulhi’s theorem, 165 best linear unbiased estimator (BLUE),

239, 255-6, 450- | best linear unbiased scalar (BLUS) residuals, 407

beta distribution 401, 479 bias, 235

binomial distribution, 63-4, 166 bivariate distributions, 79-93 binomial, 84

exponential, 92, 124 logistic, 91, 125 normal, 83 88, 93, 120, 122 Pareto, 84, 124

Borel field, 41, 52 Borel function, 95 Box-Cox transformation, 455-7 Box~Jenkins approach (see ARMA, ARIMA)

Breusch—Pagan test, 469-70

Brownian motion process, 149 50

690 Index

CAN estimators, 271

canonical correlations, 314

Carleman’s condition, 74

Cauchy distribution, 70-1, 105

causality (see Granger non-causality)

central limit theorem

De Moivre-Laplace, 64, 165

Liapounov, 174

Lindeberg—Feller, 174

Lindeberg-Levy, 173

characteristic function, 73-4

Chebyshev’s inequality, 73

chi-square distribution, 98-9, 108, 111

non-central, 108, 111

Chow test, 487-8

collinearity, exact, 432-4

‘near’, 434-40

common factor restrictions, 507-11

condition numbers, 436

conditional distributions, 89 94

exponential, 92

logistic, 91

normal 93

Pareto, 92

conditional expectation, 121 7

wit a o-field, 125-7

wrt an observed value, 121-5

properties, 122, 125, 126-7

conditional moments, i22 5

mean, 122.-3

variance, 122-3

conditional probability, 43-4

confidence region, 303-6

confluence analysis, 12

consistency, weak, 244-6

strong, 246

constant, in linear regression, 370-1, 410

constrained MLE’s, 423-4

continuous rv’s, 56

convergence, mathematical, 185-8

of a function, 185-6

of a sequence, 185

pointwise, 187

uniform, 187

convergence, modes of, 188-92

almost sure, 188, 167

in distribution, 189, 167

in probability, 189 166

in rth mean, 188

convergence of moments, 192-4

correlation coefficient, 119

covariance, 119

matrix, 312-3

Cramer-Rao, lower bound, 237

regularity conditions, 237

Cramer-Wold lemma, 191

cross-correlation, 135 cross-covariance, 135 cross-section data, 342-3 cumulants, 74

cumulative distribution function (see distribution function)

cumulative frequency, 25 CUSUM test, 477 CUSUMSQ test, 477 data, economic, 342-6 and the probability model, 346-9 degrees of freedom, 108, 111-13 o-method, 201

demand function, 10-11

de Moivre-Laplace CLT, 64, 165 density function, definition, 57 conditional, 90

joint, 82 marginal 86 properties, 59 diagnostic checking, 557 difference equation, 155-6, 543-5

differencing, 161, 479-81, 528

differentiation of vectors and matrices,

603-4

distribution function, 55-60 conditional, 89-92 Joint, 78-85 marginal, 85S~7 distribution of the sample, 216 disturbance (see error term, non-systematic component)

dummy variables, 369, 536-7 Durbin’s hr test, 541-2 Durbin—Watson test, 515-18 dynamic linear regression model, 526-70 dynamic multipliers, 601-2

econometrics, definition of, 3, 676 Edgeworth expansion, 206-7 efficiency, relative, 234-5

full, 237-8

efficient score test (see Lagrange multiplier

test)

eigenvalue, 433, 436 eigenvector, 433, 436 elliptical family of distributions, 458 empirical distribution function, 228-9 empirical econometric model, 21, 23, 670 encompassing, 568, 670

endogeneity (non-exogeneity), 629 endogenous variables, 608 Engel’s law, 6

equilibrium, long-run, 558-9

ergodicity, 143 500

Index

error, autocorrelation, 505-7

error bounds, Berry—Esseen, 202-3

error-correction model, 554

errors-in-variables, 12

error term, 349-50, 374-5 (see also non-

systematic component)

estimable model 23, 668

estimate, 231

estimation, methods, 252-84

estimation, properties of estimators,

231-49

estimators,

CAN 271

FIML, 625

GLS, 463, 503, 587-8

IV, 637-44

k-class, 632

LIML, 629, 631, 633

OLS, 449-52

3SLS, 639-40

2SLS, 635-7

estimator generation equation, 624-6

events, 38

elementary, 38

impossible, 39

mutually exclusive, 44

sure, 39

exclusion restrictions (see a priori

restrictions}

exogeneity, weak, 273, 376-7, 421-2

strong, 505, 629-30

tests for, 653

expectation, 68-9

conditional, 121-7

properties of, 70-1, 116-20

experimental design, 366-7

exponential distribution, 76, 92, 124

exponential family of distributions, 68, 299

F distribution, central, 104, 108, 113, 319

non-central 113 319 320, 324

F test 398-402 425-6, $53

power of 401

F-type misspecification test 446

homoskedasticity 467, 547, 555

linearity 460- 1, 547-8, 555

parameter time-invariance, 477

sample independence 511 541

structural invariance 482-6 556

FIML, 625

final form, 601

finite sample properties 232-44

efficiency, 234-8

linearity, 238

sufficiency, 242-4

unbiasedness, 232-4

691

Fisher-Neyman factorisation, 242 Fisher’s F distribution (see F distribution)

Fisher paradigm, 7-9

forecasting (see prediction) frequency curves, 27 frequency polygon, 24 functions of r.v.’s, distribution of, 96-107 addition, 100-2

min, 105-6

quotient, 102-5 Gamma function, 99 Gaussian distribution (see normal distribution)

Gauss linear model, 6-8, 348, 353, 357-68 Gauss~Markov theorem, 239, 449

generalised F-test, 590-3 GIVE PC (computer package), xviii

GIVE (estimator), 638

GLS, 463-6, 503, 587 goodness of fit (see R*)

Gram-Charlier series A, 205

Granger non-causality, 505, 509, 529

test for 544

hermite polynomials, 204

heteroskedasticity, 463-71 versus parameter time-dependence, 473

histogram, 23

homogeneous non-stationary process, 161,

479-81, 527-8

homoskedasticity, 126, 378, 463-71 misspecification tests for, 464-71, 547,

648-9

hypothesis testing, 285-303

alternative, 287

composite, 287

null, 286-7

simple, 287

idempotent matrix, 319, 381, 411

identically distributed r.v.’s, 94, 216-17 identification 614-9

exact, 618 order condition, 615 overidentification, 618 rank condition, 617-8 underidentification, 618

impact multipliers, 601-2

incidental parameters, 136, 346-7, 499 independence, 44, 87, 93

independent r.v.’s, 87 linear, 118

necessary and sufficient condition,

117-18

versus orthogonality, 118

692 Index

independent r.v.’s (continued)

versus uncorrelatedness, 118

independent sample, 217, 378

misspecification tests for, 511-21

indirect MLE, 621-2, 627

information matrix, 239

asymptotic, 247

sample, 239

single observation, 247

information matrix test procedure, 467

initial conditions, 151-3, 156, 528, 531

innovation process, 147

instrumental variables, 637-44

estimator, 638

instrumentalism, 665

integral, Riemann-Stieltjes, 69

integrability, 193

square, 203

uniform, 193

interim multipliers, 601

intersept (see constant)

interval estimation (see confidence region)

invariance, linear transformations, 438-9

invariance of MLE’s, 266-7

inverse matrix, partitioned, 442

invertibility conditions, 161

iterative estimation procedure, 588

Jacobian transformation, 106, 257

joint central moments, 119

joint density function, 81-2

continuous, 82

discrete, 82

k-class estimator, 632

Khinchin’s WLLN, 169

King’s law, 5

Kolmogorov—Gabor polynomial, 446

Kolmogorov’s axiomatic approach, 37-43

Kolmogorov’s inequality, 171

Kolmogorov’s stochastic process

conditions, 133

Kolmogorov’s SLLN, 170-1

Kolmogorov’s WLLN, 169

Kolmogorov-Smimov test, 229, 453

Kronecker product, 573, 603-4

kurtosis, 73, 452

lag operator, 155, 161, 509

Lagrange multiplier test procedure, 330,

3334, 430-2

in misspecification testing, 446, 453, 460

466, 468~9, 519-21

Laguerre polynomials, 208

law of large numbers (see WLLN, SLLN)

leading indicator model, 554

least squares method, 6-7, 253-6, 448-450 least squares estimators, 448

GLS, 463, 503, 587-8 OLS 448-9, 638

Lehmann-Scheffe theorem, 242-3, 387, 577 level of significance of a test (see size of a test)

Liapunov’s CLT, 174

likelihood function, 258-60

likelihood ratio test procedure, 299-303,

328-9, 335, 425-6, 432

limit of a function, 186 limit of a sequence, 185 limit of moments, 192 limit, probability (see convergence in probability)

LIML estimator, 629, 631, 633 Lindeberg condition, 174-5 Lindeberg—Feller CLT 174, 177 Lindeberg—Levy CLT, 173-4 linear regression model, 369-410 linear restrictions (see a priori restrictions) linearity 370, 378, 457 63

and normality, 316 inducing transformations, 462-3

misspecification tests for, 459-61, 597,

648

locally UMP test, 335 logical empiricism, 662-3 logical positivism, 3, 662-3 logistic distribution, 91, 125 log-likelihood, 258-60 log-normal distribution 283, 457 long-run equilibrium solution, 558-9 long-run multipliers, 602

lower bound (see Cramer—Rao lower bound)

MA(p) stochastic process, 158 Malinvaud formulation, 588 Mann-Wald theorem, 197 marginal distributions, 85-9 normal distribution, 88, 317 Markov inequality, 71 Markov process, 148-9 Martingale difference process, 147, 273 5

Martingale orthogonality, 118, 171

Martingale process, 145

CLT for, 178 SLLN for, 172 WLLN for, 172

maximum likelihood, method, 257-81 mean, 25, 70-1

mean square error (MSE), 234-6, 249

measurement equations, 12

measurement information, 352, 665

Index

measurement systems, 409-11

m-dependent process, 141

median, 25, 7]

methodology, 15-21, 659 72

misspecification testing, 21, 221, 392

mixing, strong 142 179

uniform, 143, 179

MLE, 260

constrained, 423

properties, 266-82

mode, 25, 71

model selection, 523, 669-70

moments, approximate, 194

asymptotic, 192

central, 73, 119

limit of, 192

raw, 73, 118

moments, method of, 256-7

Monte Carlo, 435

mth-order Markov process, 149

multicollinearity (see collinearity)

multiplication rule of probability, 44

multiple correlation coefficient, 313-14,

318, 322-3, 382 439

multivariate linear regression model,

571-607

in relation to the SEM, 610-14

multivariate norma! distribution, 312-24

multivariate t distribution, 471

Neyman-—Pearson theorem, 296

non-centrality parameter, 108, 111-13

in the F-test, 399, 401

nonlinear model, 461-3

non-linear restrictions (see a priori

restrictions)

non-parametric inference, 218

non-parametric processes, 146-52

non-parametric tests, 453

non-random sample 218 343, 494-7

non-stochastic variables, 357

non-systematic component 350, 370, 374,

376

normal distribution 64 6

bivariate, 83 4 Xs

mean of, 70

multivariate 315 24

standard 68

variance, 71

normality, 447 57

misspecification tests for 451-5

normalising transformations 455 7

normal (Gaussian) stochastic process

135-6

time homogeneity restrictions on 139

693

nuisance parameters, 414 null hypothesis, 286 observation space, 217 ogive, 27

omitted variables bias argument, 419-21 and auxiliary regressions, 446, 458-61,

468, 471, 502, 515, 523 reformulated, 445-7 OLS, 448-51

O, o notation, 195-6

O,, 0, notation, 196-9

order condition (see identification) order of magnitude, 171, 174, 179, 194-8 orthogonal projection, 381, 411, 642 orthogonality, 118

between systematic and non-systematic components, 350, 358, 371, 381 overdifferencing, 479

overidentifying restrictions, 618 test for, 651-3

overparametrisation, 612-13 panel data, 42

parameter space, 60 parameter structural change, 481-7 parameter time-invariance, 378, 472-81

parametric family of densities (see probability model)

parametric processes, 146

Pareto distribution, 61, 339-41 partial adjustment model, 552 partial correlation coefficient, 314, 318,

323, 439-40

Pearson family of densities, 28, 452-3 Pearson paradigm, 7-8

Pillai’s trace test, 593 pivots, 295

Political ArithmeHk, 4-5

power function, 291

power of a test, 290 power set, 39 predetermined variables, 610 prediction, 221, 247-9, 306-9

in the linear regression model, 402-5

in the multivariate linear regression model, 599, 601-2, 654-5

principal components, 434 probability, definition, axiomatic, 43 classical, 34 frequency, 34 subjective, 35 probability limit (see convergence) probability model, 60-1, 214

694 Index

probability set function, 42-3

probability space, 42

quadratic forms related to the normal

distribution, 319-20

quotient of two r.v.’s, 102—5

R? (see multiple correlation coefficient)

random experiment, 37

random matrix, 135

random sample, 216-17

random variable, 48-76

continuous, 56

definition, 50

discrete, 56

functions of, 97, 99-110

minimal o-field generated by, 50

random vector, 78-93

rank condition (see identification)

Rao-Blackwell lemma, 243

realism, 663

recursive estimator, 407, 474-78

recursive system, 612-14

regression curve, 122-4

regressors, order of magnitude, 391-2

rejection region, 286

reparametrisation/restriction, 21, 352

RESET type tests, 446, 460-1, 555, 597

residuals, 405-8

BLUS, 407

recursive, 407, 474-8

residual sum of squares (RSS), 428

respecification approach, 498-502, 505-9

restrictions (see a priori restrictions)

Russian school, 36, 64

sample information, 352, 667

sample moments, 227

sample space, 38

sampling model, 215-19

score function, 260

second order stochastic process, 138-9

selection matrices, 619

sequential conditioning, 273-4, 495

serial correlation (see autocorrelation)

Shapiro-WIlk test, 452

o-field, definition, 40

generated by a r.v., 50-1

generated by a set, 40

increasing sequence of, 51

simple random sampling, 343

simultaneous equation model, 608-58

singular normal distribution, 406

size of a test, 291

skedasticity, 123-4

skewness, 26, 73

skewness-kurtosis test, 452-5 SLLN, 170-2

small sample properties (see finite sample

properties) specification testing, 392 spectral decomposition (see eigenvalues) standard deviation, 72

stationarity, strict, 137-8 Ith order, 138-9 statistic, 224 statistical GM, 344, 349-52 statistical model, 218, 339 well-defined estimated, 352, 409, 522,

668 statistical parameters of interest, 351, 371,

376, 419-22, 575, 666-7 versus theoretical parameters of interest,

351, 376, 667-9

stochastic linear regression model, 413-18

stochastic process, 131-7 realisation of, 131 stratified sampling, 343 structural form, unrestricted, 614 restricted, 616

structural parameters (see theoretical parameters of interest)

Student’s t distribution, 104, 108 multivariate, 471

sufficiency, 242 minimal, 243 supermartingale, 145 SURE formulation, 585-8, 636 systematic component, 349-51, 370-1, 375-6

t distribution (see Student’s t distribution)

t test, 364, 396-7, 392

test, definition, 294 test statistic, 289 testing, 221, 285-303

Theil’s inequality coefficient, 405

theoretical parameters of interest, 349, 351,

553, 569, 613-14, 620, 669

theoretical model, 21, 667 theory, 20, 662-6 3SLS estimator, 635-7

time-homogeneity restrictions, 137-9

time series data, 130, 342 Toeplitz matrix, 495, 505 total sum of squares, 382 2SLS estimator, 626-35 type I error, 286 type II error, 286

UMA region, 305

UMP test, 291

Index

unbiased confidence region, 306

unbiased estimator, 232

unbiased test 293

uncorrelated r.v.s, 118

underidentification, 621

uniform convergence (see convergence)

uniform distribution, 57-8

unimodal distribution, 71

univariate distributions, 62-8

variance, 72 119

properties, 73

variance—covariance matrix, 312

variance ratio tests 395-6

variance stabilising transformations, 487-8

variation free condition, 377, 422

violation of, 629

vectoring, 573 604

vector stochastic process, 135

695

Wald test procedure, 329, 332-3, 429-30

weak convergence (see convergence in

probability) weakly stationary process (see second order stationarity)

Weibull distribution, 105 white-noise process, 150, 151 White test for homoskedasticity, 465-7

Wilk’s ratio test, 593

Window, t-period, 40, 562 Wishart distribution, 321, 577, 602-3

WLLN, 168-9

Wold decomposition, 159 Yule-Walker equations 157 Zellner formulation (see SURE) Zero-one restrictions (see exclusion restrictions)