BASIC ECONOMETRICS. 3RD ED.
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| TITLE : BASIC ECONOMETRICS. 3RD ED. |
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Part 1 Single-Equation Regression Models
1 The Nature of Regression Analysis 15
1.1 Historical Origin of the Term "Regression" 15
1.2 The Modern Interpretation of Regression 16
Examples 16
1.3 Statistical vs. Deterministic Relationships 19
1.4 Regressions. Causation 20
1.5 Regression vs. Correlation 21
1.6 Terminology and Notation 22
1.7 The Nature and Sources of Data for Econometric
Analysis 23
Types of Data 23
The Sources of Data 24
The Accuracy of Data 26
1.8 Summary and Conclusions 27
Exercises 28
Appendix 1A 29
1A.1 Sources of Economic Data 29
1A.2 Sources of Financial Data 31
2 Two-Variable Regression Analysis: Some Basic Ideas 32
2.1 A Hypothetical Example 32
2.2 The Concept of Population Regression Function (PRF) 36
2.3 The Meaning of the Term "Linear" 36
Linearity in the Variables 37
Linearity in the Parameters 37
2.4 Stochastic Specification of PRF 38
2.5 The Significance of the Stochastic Disturbance Term 39
2.6 The Sample Regression Function (SRF) 41
2.7 Summary and Conclusions 45
Exercises 45
3 Two-Variable Regression Model: The Problem of Estimation 52
3.1 The Method of Ordinary Least Squares . 52
3.2 The Classical Linear Regression Model: The Assumptions
Underlying the Method of Least Squares 59
How Realistic Are These Assumptions? 68
3.3 Precision or Standard Errors of Least-Squares Estimates 69
3.4 Properties of Least-Squares Estimators: The Gauss-Markov
Theorem 72
3.5 The Coefficient of Determination r2: A Measure of "Goodness of
Fit" 74
3.6 A Numerical Example 80
3.7 Illustrative Examples 83
Coffee Consumption in the United States, 1970-1980 83
Keynesian Consumption Function for the United States, 1980-1991
84
3.8 Computer Output for the Coffee Demand Function 85
3.9 A Note on Monte Carlo Experiments 85
3.10 Summary and Conclusions 86
Exercises 87
Questions 87
Problems 89
Appendix 3A 94
3A.1 Derivation of Least-Squares Estimates 94
3A.2 Linearity and Unbiasedness Properties of Least-Squares
Estimators 94
3A.3 Variances and Standard Errors of Least-Squares Estimators 95
3A.4 Covariance between/3s and/32 96
3A.5 The Least-Squares Estimator Of (J2 96
3A.6 Minimum-Variance Property of Least-Squares Estimators 97
3A.7 SAS Output of the Coffee Demand Function (3.7.1)
4 The Normality Assumption: Classical Normal Linear Regression Model
(CNLRM)
4.1 The Probability Distribution of Disturbances us 101
4.2 The Normality Assumption 102
4.3 Properties of OLS Estimators under the Normality Assumption 104
4.4 The Method of Maximum Likelihood (ML) 107
4.5 Probability Distributions Related to the Normal Distribution:
The t, Chi-square (x2), and F Distributions 107
4.6 Summary and Conclusions 109
Appendix 4A 110
Maximum Likelihood Estimation of Two-Variable Regression Model 110
Maximum Likelihood Estimation of the Consumption-Income Example 113
Appendix 4A Exercises 113
5 Two-Variable Regression: Interval Estimation and Hypothesis
Testing
115
5.1 Statistical Prerequisites 115
5.2 Interval Estimation: Some Basic Ideas 116
5.3 Confidence Intervals for Regression Coefficients,
and ~2
Confidence Interval for ~2 117
Confidence Interval for ,BI 119
Confidence Interval for ,Bi and ~2 Simultaneously 120
5.4 Confidence Interval fortr2 120
5.5 Hypothesis Testing: General Comments 121
5.6 Hypothesis Testing: The Confidence-Interval Approach 122
Two-Sided or Two-Tail Test 122
One-Sided or One-Tail Test 124
5.7 Hypothesis Testing: The Test-of-Significance Approach 124
Testing the Significance of Regression Coefficients:
The t-Test 124
Testing the Significance Of xJ2: The x2 Test 128
5.8 Hypothesis Testing: Some Practical Aspects 129
The Meaning of "Accepting" or "Rejecting" a Hypothesis 129
The "Zero" Null Hypothesis and the "2-t" Rule of Thumb 129
Forming the Null and Alternative Hypotheses 130
Choosing cr, the Level of Significance 131
The Exact Level of Significance: The p Value 132
Statistical Significance versus Practical Significance 133
The Choice between Confidence-Interval and
Test-of-Significance Approaches to Hypothesis Testing 134
5.9 Regression Analysis and Analysis of Variance 134
5.10 Application of Regression Analysis: The Problem of
rediction 137
Mean Prediction 137
Individual Prediction 138
5.11 Reporting the Results of Regression Analysis 140
5.12 Evaluating the Results of Regression Analysis 140
Normality Test 141
Other Tests of Model Adequacy 144
5.13 Summary and Conclusions 144
Exercises 145
Questions 145
Problems 147
Appendix 5A 152
5A.1 Derivation of Equation (5.3.2) 152
5A.2 Derivation of Equation (5.9.1) 152
5A.3 Derivations of Equations (5.10.2) and (5.10.6) 153
Variance of Mean Prediction 153
Variance of Individual Prediction 153
6 Extensions of the Two-Variablex Linear Regression Model 155
6.1 Regression through the Origin ' 155
r2 for Regression-through-Origin Model 159
An Illustrative Example: The Characteristic Line of Portfolio
Theory 159
6.2 Scaling and Units of Measurement 161
A Numerical Example: The Relationship between GPDI and GNP,
United States, 1974-1983 163
A Word about Interpretation 164
6.3 Functional Forms of Regression Models 165
6.4 How to Measure Elasticity: The Log-Linear Model 165
An Illustrative Example: The Coffee Demand Function Revisited
167
6.5 Semilog Models: Log-Lin and Lin-Log Models 169
How to Measure the Growth Rate: The Log-Lin Model 169
The Lin-Log Model 172
6.6 Reciprocal Models 173
An Illustrative Example: The Phillips Curve for the United
Kingdom, 1950-1966 176
6.7 Summary of Functional Forms 176
6.8 A Note on the Nature of the Stochastic Error Term: Additive
versus Multiplicative Stochastic Error Term 178
6.9 Summary and Conclusions 179
Exercises 180
Questions 180
Problems 183
Appendix 6A 186
6A.1 Derivation of Least-Squares Estimators for Regression
through the Origin 186
6A.2 SAS Output of the Characteristic Line (6.1.12) 189
6A.3 SAS Output of the United Kingdom Phillips Curve
Regression (6.6.2) 190
7 Multiple Regression Analysis: The Problem of Estimation
7.1 The Three-Variable Model: Notation and Assumptions 192
7.2 Interpretation of Multiple Regression Equation 194
7.3 The Meaning of Partial Regression Coefficients 195
7.4 OLS and ML Estimation of the Partial Regression
Coefficients 197
OLS Estimators 197
Variances and Standard Errors of OLS Estimators 198
Properties of OLS Estimators 199
Maximum Likelihood Estimators 201
7.5 The Multiple Coefficient of Determination R2
and the Multiple Coeffficient of Correlation R 201
7.6 Example 7.1: The Expectations-Augmented Phillips
Curve for the United States, 1970-1982 203
7.7 Simple Regression in the Context of Multiple
Regression: Introduction to Specification Bias 204
7.8 R2 and the Adjusted R2 207
Comparing Two R2 Values 209
Example 7.2: Coffee Demand Function Revisited 210
The "Game" of Maximizing R2 211
7.9 Partial Correlation Coefficients 211
Explanation of Simple and Partial Correlation
Coefficients 211
Interpretation of Simple and Partial Correlation
Coefficients 213
7.10 Example 7.3: The Cobb-Douglas Production Function:
More on Functional Form 214
7.11 Polynomial Regression Models 217
Example 7.4: Estimating the Total Cost Function 218
Empirical Results 220
7.12 Summary and Conclusions 221
Exercises 221
Questions 221
Problems 224
Appendix 7A 231
7A.1 Derivation of OLS Estimators Given in Equations (7.4.3)
and (7.4.5) 231
7A.2 Equality between al of (7.3.5) and p2 of (7.4.7) 232
7A.3 Derivation of Equation (7.4.19) 232
7A.4 Maximum Likelihood Estimation of the Multiple Regression
Model 233
7A.5 TheProofthat E(b,2) = ~2 + p3b32 (Equation 7.7.4) 234
7A.6 SAS Output of the Expectations-Augmented Phillips Curve
(7.6.2) 236
7A.7 SAS Output of the Cobb-Douglas Production
Function (7.10.4) 237
8 Multiple Regression Analysis: The Problem of Inference 238
8.1 The Normality Assumption Once Again 238
8.2 Example 8.1: U.S. Personal Consumption and Personal
Disposal Income Relation, 1956-1970 239
8.3 Hypothesis Testing in Multiple Regression: General
Comments 242
8.4 Hypothesis Testing about Individual Partial Regression
Coeffficients 242
8.5 Testing the Overall Significance of the Sample
Regression 244
The Analysis of Variance Approach to Testing the Overall
Significance of an Observed Multiple Regression: The F Test
245
An Important Relationship between R2 and F 248
The "Incremental," or "Marginal," Contribution of an
Explanatory Variable 250
8.6 Testing the Equality of Two Regression Coefficients 254
Example 8.2: The Cubic Cost Function Revisited 255
8.7 Restricted Least Squares: Testing Linear Equality
restrictions 256
The t Test Approach ,. 256
The F Test Approach: Restricted Least Squares 257
Example 8.3: The Cobb-Douglas Production Function
for Taiwanese Agricultural Sector, 1958-1972 259
General F Testing 260
8.8 Comparing Two Regressions: Testing for Structural
Stability of Regression Models 262
8.9 Testing the Functional Form of Regression: Choosing
between Linear and Log-Linear Regression Models 265
Example 8.5: The Demand for Roses 266
8.10 Prediction with Multiple Regression 267
8.11 The Troika of Hypothesis Tests: The Likelihood Ratio
(LR), Wald (W), and Lagrange Multiplier (LM) Tests 268
8.12 Summary and Conclusions 269
The Road Ahead 269
Exercises 270
Questions 270
Problems 273
Appendix 8A 280
Likelihood Ratio (LR) Test 280
9 The Matrix Approach to Linear Regression Model 282
9.1 The k-Variable Linear Regression Model 282
9.2 Assumptions of the Classical Linear Regression Model
in Matrix Notation 284
9.3 OLS Estimation 287
An Illustration ^ 289
Variance-Covariance Matrix of , 290
Properties of OLS Vector ~291
9.4 The Coefficient of Determination R2 in Matrix Notation 292
9.5 The Correlation Matrix 292
9.6 Hypothesis Testing about Individual Regression
Coefficients in Matrix Notation 293
9.7 Testing the Overall Significance of Regression: Analysis
of Variance in Matrix Notation 294
9.8 Testing Linear Restrictions: General F Testing Using
Matrix Notation 295
9.9 Prediction Using Multiple Regression: Matrix Formulation 296
Mean Prediction 296
Individual Prediction 296
Variance of Mean Prediction 297
Variance of Individual Prediction 298
9.10 Summary of the Matrix Approach: An Illustrative Example 298
9.11 Summary and Conclusions 303
Exercises 304
Appendix 9A 309
9A. I Derivation of k Normal or Simultaneous Equations 309
9A.2 Matrix Derivation of Normal Equations 310
9A.3 Variance-Covariance Matrix of ~310
9A.4 Blue Property of OLS Estimators 311
Part 2 Relaxing the Assumptions of the Classical Model
10 Multicollinearity and Micronumerosity 319
10.1 The Nature of Multicollinearity 320
10.2 Estimation in the Presence of Perfect Multicollinearity 323
10.3 Estimation in the Presence of "High" but "Imperfect"
Multicollinearity 325
10.4 Multicollinearity: Much Ado about Nothing?
Theoretical Consequences of Multicollinearity 325
10.5 Practical Consequences of Multicollinearity 327
Large Variances and Covariances of OLS Estimators 328
Wider Confidence Intervals 329
"Insignificant" t Ratios 330
A High R2 but Few Significant t Ratios 330
Sensitivity of OLS Estimators and Their Standard Errors to
Small Changes in Data 331
Consequences of Micronumerosity 332
10.6 An Illustrative Example: Consumption Expenditure in Relation
to Income and Wealth 332
10.7 Detection of Multicollinearity 335
10.8 Remedial Measures 339
10.9 Is Multicollinearity Necessarily Bad? Maybe Not If the
Objective Is Prediction Only 344
10.10 Summary and Conclusions 345
Exercises 346
Questions 346
Problems 351
11 Heteroscedasticity 355
11.1 The Nature of Heteroscedasticity 355
11.2 OLS Estimation in the Presence of Heteroscedasticity 359
11.3 The Method of Generalized Least Squares (GLS) 362
Difference between OLS and GLS 364
11.4 Consequences of Using OLS in the Presence of
Heteroscedasticity 365
OLS Estimation Allowing for Heteroscedasticity 365
OLS Estimation Disregarding Heteroscedasticity 366
11.5 Detection of Heteroscedasticity 367
Informal Methods 368
Formal Methods 369
11.6 Remedial Measures 381
When < Is Known: The Method of Weighted Least Squares 381
When d Is Not Known 382
11.7 A Concluding Example 387
11.8 Summary and Conclusions 389
Exercises 390
Questions 390
Problems 392
Appendix 11A 398
11 A.1 Proof of Equation (11.2.2) 398
11 A.2 The Method of Weighted Least Squares 399
12 Autocorrelation 400
12.1 The Nature of the Problem 400
12.2 OLS Estimation in the Presence of Autocorrelation 406
12.3 The BLUE Estimator in the Presence of Autocorrelation 409
12.4 Consequences of Using OLS in the Presence of Autocorrelation
410
OLS Estimation Allowing for Autocorrelation 410
OLS Estimation Disregarding Autocorrelation 411
12.5 Detecting Autocorrelation 415
Graphical Method 415
The Runs Test 419
Durbin-Watson d Test 420
Additional Tests of Autocorrelation 425
12.6 Remedial Measures 426
When the Structure of Autocorrelation Is Known 427
When p Is Not Known 428
12.7 An Illustrative Example: The Relationship between
Help-Wanted Index and the Unemployment Rate, United States:
Comparison of the Methods 433
12.8 Autoregressive Conditional Heteroscedasticity (ARCH) Model 436
What to Do If ARCH Is Present? 438
A Word on the Durbin-Watson d Statistic and the ARCH Effect 438
12.9 Summary and Conclusions 439
Exercises 440
Questions 440
Problems 446
Appendix 12A 449
12A.1 TSP Output of United States Wages (Y)
Productivity (X) Regression, 1960-1991 449
13 Econometric Modeling I: Traditional Econometric Methodology 452
13.1 The Traditional View of Econometric Modeling: Average Economic
Regression (AER) 452
13.2 Types of Specification Errors 455
13.3 Consequences of Specification Errors 456
Omitting irrelevant Variable (Overfitting a Model) 456
Inclusion of an Irrelevant Variable (Overfitting a Model) % 458
13.4 Tests of Specification Errors 459
Detecting the Presence of Unnecessary Variables 460
Tests for Ornitted Variables and Incorrect Functional Form 461
13.5 Errors of Measurement 467
Errors of Measurement in the Dependent Variable Y 468
Errors of Measurement in the Explanatory Variable X 469
An Example 470
Measurement Errors in the Dependent Variable Y Only 471
Errors of Measurement in X 472
13.6 Summary and Conclusions 472
Exercises 473
Questions 473
Problems 476
Appendix 13A 477
13A.1 The Consequences of Including an Irrelevant
Variable: The Unbiasedness Property 477
13A.2 Proof of (13.5.10) 478
14 Econometric Modeling II: Alternative Econometric Methodologies 480
14.1 Leamer's Approach to Model Selection 481
14.2 Hendry's Approach to Model Selection 485
14.3 Selected DiagnosticTests: General Comments 486
14.4 Tests of Nonnested Hypothesis 487
The Discrimination Approach 487
The Discerning Approach 488
14.5 Summary and Conclusions 494
Exercises 494
Questions 494
Problems 495
Part 3 Topics in Econometrics
15 Regression on Dummy Variables 495
15.1 The Nature of Dummy Variables 49
Example 15.1: Professor's Salary by Sex 500
15.2 Regression on One Quantitative Variable and One Qualitative
Variable with Two Classes, or Categories 502
Example 15.2: Are Inventories Sensitive to Interest Rates? 505
15.3 Regression on One Quantitative Variable and One Qualitative
Variable with More than Two Classes 505
15.4 Regression on One Quantitative Variable and Two Qualitative
Variables 507
15.5 Example 15.3: The Economics of "Moonlighting" 508
15.6 Testing for Structural Stability of Regression Models:
Basic Ideas 509
Example 15.4: Savings and Income, United Kingdom, 1946-1963 509
15.7 Comparing Two Regressions: The Dummy Variable Approach 512
15.8 Comparing Two Regressions: Further Illustration 514
Example 15.5: The Behavior of Unemployment and Unfilled
Vacancies; Great Britain, 1958-1971 514
15.9 Interaction Effects 516
15.10 The Use of Dummy Variables in Seasonal Analysis 517
Example 15.6: Profits-Sales Behavior in U.S. Manufacturing 517
15.11 Piecewise Linear Regression 519
Example 15.7: Total Cost in Relation to Output 521
15.12 The Use of Dummy Variables in Combining Time Series and
Cross-Sectional Data 522
Pooled Regression: Pooling Time Series and Cross-Sectional Data
522
Example 15.8: Investment Functions for General Motors and
Westinghouse Companies 524
15.13 Some Technical Aspects of Dummy Variable Technique 525
The Interpretation of Dummy Variables in Semilogarithmic
Regressions 525
Example 15.9: Semilogarithmic Regression with Dummy Variable525
Another Method of Avoiding the Dummy Variable Trap 526
Dummy Variables and Heteroscedasticity 527
Dummy Variables and Autocorrelation 527
15.14 Topics for Further Study 528
15.15 Summary and Conclusions 529
Exercises 530
Questions 530
Problems 535
Appendix 15A 538
15A.1 Data Matrix for Regression (15.8.2) 538
15A.2 Data Matrix for Regression (15.10.2) 539
16 Regression on Dummy Dependent Variable:
The LPM, Logit, Probit, and Tobit Models 540
16.1 Dummy Dependent Variable 540
16.2 The Linear Probability Model (LPM) 541
16.3 Problems in Estimation of LPM 542
Nonnormality of the Disturbances ui 542
Heteroscedastic Variances of the Disturbances 543
Nonfulfillment of 0 ' E(Yi | X) c 1 544
Questionable Value of R2 as a Measure of Goodness of Fit 545
16.4 LPM: A Numerical Example 546
16.5 Applications of LPM 548
Example 16.1: Cohen-Rea-Lerman study 548
Example 16.2: Predicting a Bond Rating 551
Example 16.3: Predicting Bond Defaults 552
16.6 Alternatives to LPM 552
16.7 The Logit Model 554
16.8 Estimation logit Model 556
16.9 The Logit Model: A Numerical Example 558
16.10 The Logit Model: Illustrative Examples 561
Example 16.4: "An Application of Logit Analysis to Prediction
of Merger Targets" 561
Example 16.5: Predicting a Bond Rating 562
16.11 The Probit Model 563
16.12 The Probit Model: A Numerical Example 567
Logit versus Probit 567
Comparing Logit and Probit Estimates 568
The Marginal Effect of a Unit Change in the Value of a
Regressor 569
16.13 The Probit Model: Example 16.5 569
16.14 The Tobit Model 570
16.15 Summary and Conclusions 575
Exercises 576
Questions 576
Problems 578
17 Dynamic Econometric Model: Autoregressive and Distributed-Lag
Models
584
17.1 The Role of "Ilme," or "Lag," in Economics 585
17.2 The Reasons for Lags 589
17.3 Estimation of Distributed-Lag Models 590
Ad Hoc Estimation of Distributed-Lag Models 590
17.4 The Koyck Approach to Distributed-Lag Models 592
The Median Lag 595
The Mean Lag 595
17.5 Rationalization of the Koyck Model: The Adaptive
Expectations Model 596
17.6 Another Rationalization of the Koyck Model: The Stock
Adjustment, or Partial Adjustment, Model 599
17.7 Combination of Adaptive Expectations and Partial Adjustment
Models 601
17.8 Estimation of Autoregressive Models 602
17.9 The Method of Instrumental Variables (IV) 604
17.10 Detecting Autocorrelation in Autoregressive Models: Durbinh
Test 605
17.11 A Numerical Example: The Demand for Money in India 607
17.12 Illustrative Examples 609
Example 17.7: The Fed and the Real Rate of Interest 609
Example 17.8: The Short- and Long-Run Aggregate Consumption
Functions for the United States, 1946-1972 611
17.13 The Almon Approach to Distributed-Lag Models: The Almon or
Polynomial Distributed Lag (PDL) 612
17.14 CausalityinEconomics:TheGrangerTest 620
The Granger Test 620
Empirical Results 622
17.15 Summary and Conclusions 623
Exercises 624
Questions 624
Problems % 630
Part 4 Simultaneous-Equation Models
18 Simultaneous-Equation Models 635
18.1 The Nature of Simultaneous-Equation Models 635
18.2 Examples of Simultaneous-Equation Models 636
Example 18.1: Demand-and-Supply Model 636
Example 18.2: Keynesian Model of Income Determination 638
Example 18.3: Wage-Price Models 639
Example 18.4: The IS Model of Macroeconomics 639
Example 18.5: The LM Model 640
Example 18.6: Econometric Models 641
18.3 The Simultaneous-Equation Bias: Inconsistency of OLS
Estimators 642
18.4 The Simultaneous-Equation Bias: A Numerical Example 645
18.5 Summary and Conclusions 647
Exercises 648
Questions 648
Problems 651
19 The Identification Problem 653
19.1 Notations and Definitions 653
19.2 The Identification Problem 657
Underidentification 657
Just, or Exact, Identification 660
Overidentification 663
19.3 Rules for Identification 664
The Order Condition of Identifiability 665
The Rank Condition of Identifiability 666
19.4 A Test of Simultaneity 669
Hausman Specification Test 670
Example 19.5: Pindyck-Rubinfeld Model of Public Spending 671
19.5 Tests for Exogeneity 672
A Note on Causality and Exogeneity 673
19.6 Summary and Conclusions 673
Exercises 674
20 Simultaneous-EquationMethods 678
20.1 Approaches to Estimation 678
20.2 Recursive Models and Ordinary Least Squares 680
20.3 Estimation of a Just Identified Equation: The Method of
Indirect Least Squares (ILS) 682
An Illustrative Example 683
Properties of ILS Estimators 686
20.4 Estimation of an Overidentified Equation: The Method of
Two-Stage Least Squares (2SLS) 686
20.5 2SLS: A Numerical Example 690
20.6 Illustrative Exam~ples 693
Example 20.1: Advertising, Concentration, and Price Margins 693
Example 20.2: Klein's Model I 694
Example 20.3: The Capital Asset Pricing Model Expressed as a
Recursive System 694
Example 20.4: Revised Form of St. Louis Model 697
20.7 Summary and Conclusions 699
Exercises 700
Questions 700
Problems 703
Appendix 20A 704
20A.1 Bias in the Indirect Least-Squares Estimators 704
20A.2 Estimation of Standard Errors of 2SLS
Estimators 705
Part 5 Time Series Econometrics
21 Time Series Econometrics I: Stationarity, Unit Roots, and
Cointegration 709
21.1 A Look at Selected U.S. Economic Time Series 710
21.2 Stationary Stochastic Process 710
21.3 Test of Stationarity Based on Correlogram 714
21.4 The Unit Root Test of Stationarity 718
Is the U.S. GDP Time Series Stationary? 720
Is the First-Differenced GDP Series Stationary? 721
21.5 Trend-Stationary (TS) and Difference-Stationary (DS)
Stochastic Process 722
21.6 Spurious Regression 724
21.7 Cointegration 725
Engle-Granger (EG) or Augmented Engle-Granger (AEG) Test 726
Cointegrating Regression Durbin-Watson (CRDW) Test 727
21.8 Cointegration and Error Correction Mechanism (ECM) 728
21.9 Summary and Conclusions 729
Exercises 730
Questions 730
Problems 731
Appendix 21A 732
21A.1 A Random Walk Model 732
22 Time Series Econometrics II: Forecasting with ARIMA and VAR Models
734
22.1 Approaches to Economic Forecasting 734
22.2 AR, MA, and ARIMA Modeling of Time Series Data 736
An Autoregressive (AR) Process 736
A Moving Average (MA) Process 737
An Autoregressive and Moving Average (ARMA) Process 737
An Autoregressive Integrated Moving Average (ARIMA) Process737
22.3 The Box-Jenkins (BJ) Methodology 738
22.4 Identification 739
22.5 Estimation of the ARIMA Model 742
22.6 Diagnostic Checking 743
22.7 Forecasting 744
22.8 Further Aspects of the BJ Methodology 745
22.9 Vector Autoregression (VAR) 746
Estimation of VAR 746
Forecasting with VAR 747
Some Problems with VAR Modeling 747
An Application of VAR: A VAR Model of the Texas Economy 750
22.10 Summary and Conclusions 752
Exercises 753
Questions 753
Problems 753
Appendixes
A A Review of Some Statistical Concepts 755
B Rudiments of Matrix Algebra 791
C A List of Statistical Computer Packages 804
D Statistical Tables 807
Table D. l Areas under the Standardized Normal Distribution 808
Table D.2 Percentage Points of the t Distribution 809
Table D.3 Upper Percentage Points of the F Distribution 810
Table D.4 Upper Percentage Points of the x2 Distribution 816
Table D.5 Durbin-Watson d Statistic: Significant Points of
dL and dU at 0.05 and 0.01 Levels of Significance 818
Table D.6 Critical Values of Runs in the Runs Test 822
Selected Bibliography 824
Indexes
Name Index 827
Subject Index 831