Linear regression: a mathematical introduction
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| Format: | Book |
| Language: | English |
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Los Angeles ; London ; New Delhi ; Singapore ; Washington DC ; Melbourne
SAGE
[2019]
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| Series: | Quantitative applications in the social sciences
177 |
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| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Physical Description: | xxiv, 242 Seiten Diagramme |
| ISBN: | 9781544336572 |
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| adam_text | LINEAR REGRESSION
/ GUJARATI, DAMODAR N.YYEAUTHOR
: 2019
TABLE OF CONTENTS / INHALTSVERZEICHNIS
THE LINEAR REGRESSION MODEL (LRM)
THE CLASSICAL LINEAR REGRESSION MODEL (CLRM)
THE CLASSICAL NORMAL LINEAR REGRESSION MODEL: THE METHOD OF MAXIMUM
LIKELIHOOD (ML)
LINEAR REGRESSION MODEL : DISTRIBUTION THEORY AND HYPOTHESIS TESTING
EXTENSIONS OF THE CLASSICAL LINEAR REGRESSION MODEL : GENERALIZED LEAST
SQUARES (GLS)
EXTENSIONS OF THE CLASSICAL LINEAR REGRESSION MODEL : THE CASE OF
STOCHASTIC OR ENDOGENOUS REGRESSORS
SELECTED TOPICS IN LINEAR REGRESSION
DIESES SCHRIFTSTUECK WURDE MASCHINELL ERZEUGT.
CONTENTS
List of Figures xv
Series Editor’s Introduction xvii
Preface xix
About the Author xxi
Acknowledgments xxiii
Chapter 1: The Linear Regression Model (LRM) 1
1.1 Introduction 1
1.2 Meaning of “Linear” in Linear Regression 3
L3 Estimation of the LRM: An Algebraic Approach 5
1.4 Goodness of Fit of a Regression Model:
The Coefficient of Determination (i?2) 12
1.5 i?2 for Regression Through the Origin 15
1.6 An Example: The Determination of the
Hourly Wages in the United States 15
1.7 Summary 16
Exercises 17
Appendix 1 A: Derivation of the Normal Equations 20
Chapter 2: The Classical Linear Regression Model (CLRM) 23
2.1 Assumptions of the CLRM 23
2.2 The Sampling or Probability Distributions
of the OLS Estimators 27
2.3 Properties of OLS Estimators:
The Gauss—Markov Theorem 32
2.4 Estimating Linear Functions of the OLS Parameters 34
2.5 Large-Sample Properties of OLS Estimators 35
2.5.1 Consistency of OLS Estimators 35
2.5.2 Consistency of the OLS Estimator of the
Error Variance
36
2.5.3 Independence of the OLS Estimators
and the Residual Term, e 37
2.5.4 Large-Sample Distribution of b: Asymptotic
Normality of the OLS Estimators 37
2.5.5 Asymptotic Normality of S 2 38
2.6 Summary 38
Exercises 38
Chapter 3: The Classical Normal Linear Regression
Model: The Method of Maximum Likelihood (ML) 43
3.1 Introduction 43
3.2 The Mechanics of ML 44
3.3 The Likelihood Function of the ¿-Variable
Regression Model 48
3.4 Properties of the ML Method 49
3.4.1 Consistency of the ML Estimators 49
3.4.2 Asymptotic Unbiasedness of the ML Estimators 50
3.4.3 Invariance of the ML Estimators 51
3.4.4 Asymptotic Normal Distribution of the
ML Estimators 51
3.4.5 Asymptotic Efficiency of the ML Estimators 52
3.4.6 Sufficiency of the ML Estimators 52
3.5 Summary 53
Exercises 54
Appendix 3 A: Asymptotic Efficiency of the
ML Estimators of the LRM 55
Chapter 4: Linear Regression Model: Distribution
Theory and Hypothesis Testing 59
4.1 Introduction 59
4.2 Types of Hypotheses 59
4.3 Procedure for Hypothesis Testing 60
4.3.1 Null and Alternative Hypotheses 60
4.3.2 Test Statistic 60
4.3.3 Decision Rule 61
4.4 The Determination of Hourly Wages
in the United States 63
4.4.1 Interpretation of the Estimated Regression 63
4.5 Testing Hypotheses About an Individual
Regression Coefficient 65
4.5.1 One-Sided or Two-Sided Alternative Hypotheses 61
4.5.2 A Note on Rejecting or Accepting a
Null Hypothesis 61
4.5.3 Confidence Internal Approach to
Hypothesis Testing 67
4.6 Testing the Hypothesis That All the
Regressors Collectively Have No Influence on the
Regressand 69
4.6.1 Relationship Between the t Test and the F Test 71
4.6.2 Relationship Between R2 and F
Tests of Significance 71
4.7 Testing the Incremental Contribution of a
Regressor 72
4.8 Confidence Interval for the Error Variance r2 74
4.8.1 A Caution on the Confidence Interval
Approach to Hypothesis Testing 16
4.9 Large-Sample Tests of Hypotheses 76
4.9.1 The Likelihood Ratio Test 11
4.9.2 The Wald Test 78
4.9.3 The Lagrange Multiplier or the Score Test 81
4.9.4 Relationships Among the Three Tests 82
4.9.5 Caution About the Three Tests 83
4.10 Summary 83
Exercises 84
Appendix 4A: Constrained Least Squares: OLS
Estimation Under Linear Restrictions 86
Chapter 5: Generalized Least Squares (GLS): Extensions
of the Classical Linear Regression Model 91
5.1 Introduction 91
5.2 Estimation of B With a Nonscalar
Covariance Matrix 94
5.2.1 Estimation ofLRMby Transformation 95
5.2.2 OLS Versus GLS 91
5.3 Estimated Generalized Least Squares 98
5.3.1 Properties of EGLS 98
5.4 Heteroscedasticity and Weighted Least Squares 99
5.5 White’s Heteroscedasticity-Consistent
Standard Errors 101
5.6 Autocorrelation 103
5.6.1 The Newey-West Method of Correcting
OLS Standard Errors 105
5.7 Summary 106
Exercises 107
Appendix 5A: ML Estimation of GLS 108
Chapter 6: Extensions of the Classical Linear Regression
Model: The Case of Stochastic or Endogenous Regressors 111
6.1 Introduction 111
6.2 X and u Are Distributed Independently 112
6.3 X and u Are Contemporaneously Uncorrelated 113
6.4 X and u Are Neither Independently Distributed
Nor Contemporaneously Uncorrelated 113
6.5 The Case of k Regressors 115
6.6 What Is the Solution? The Method of
Instrumental Variables (IVs) 116
6.6.1IV Regression 118
6.6.2 Estimation of the IV Regression Error Variance 119
6.6.3 Covariance Matrix of IV Estimators 119
6.7 Hypothesis Testing Under IV Estimation 121
6.8 Practical Problems in the Application of the IV Method 121
6.8.1 Test of the Endogeneity of a Regressor 122
6.8.2 How to Find Whether an Instrument
Is Weak or Strong 122
6.8.3 The Case of Multiple Instruments 122
6.8.4 Testing the Validity of Surplus Instruments 123
6.9 Regression Involving More Than One
Endogenous Regressor 123
6.10 An Illustrative Example: Earnings and Educational
Attainment of Youth in the United States 124
6.10.1 Test of the Endogeneity of a Regressor 129
6.10.2 How to Find Whether an Instrument
Is Weak or Strong 132
6.10.3 The Case of Multiple Instruments 134
6.10.4 Testing the Validity of Surplus Instruments 134
6.11 Regression Involving More Than One Endogenous Regressor 136
6.12 Summary 140
Appendix 6A: Properties of OLS When Random
X and u Are Independently Distributed 141
Appendix 6B: Properties of OLS Estimators When
Random X and u Are Contemporaneously Uncorrelated 143
Chapter 7: Selected Topics in Linear Regression 145
7.1 Introduction 145
7.2 The Nature of Multicollinearity 145
7.2.1 Estimation in the Presence of Perfect
Multicollinearity 146
7.2.2 Estimation in the Presence of Imperfect
or Near Multicollinearity 146
7.2.3 Detection of Multicollinearity 150
7.2.4 Remedial Measures 151
7.3 Model Specification Errors 153
7.3.1 Exclusion of Relevant Variables:
Underfitting a Model 154
7.3.2 Inclusion of Irrelevant Variables:
Overfitting a Model 155
7.3.3 Functional Forms of Regression Models 156
7.4 Qualitative or Dummy Regressors 161
7.5 Nonnormal Error Term 162
7.5.1 Normality Test 164
7.6 Summary 164
Exercises 165
Appendix 7A: Ridge Regression: A Solution to
Perfect Collinearity 168
Appendix 7B: Specification Errors 170
Appendix A: Basics of Matrix Algebra 175
A.l Definitions 175
A. 1.1 Matrix 175
A. 1.2 Scalar 175
A. 1.3 Column Vector 175
A. 1.4 Row Vector 176
A. 1.5 Transposition of a Matrix 176
A. 1.6 Transpose of Vectors 176
A. 1.7 Submatrix 176
A.2 Types of Matrices 177
A.2.1 Square Matrix 177
A.2.2 Diagonal Matrix 111
A.2.3 Identity or Unit Matrix III
A.2.4 Scalar Matrix 177
A.2.5 Symmetric Matrix 178
A.2.6 Null Matrix 178
A.2.7 Null Vector 178
A.2.S Equal Matrices 178
A.2.9 Upper Triangular Matrix 178
A.2.10 Lower Triangular Matrix 178
A.2.11 Idempotent Matrix 178
A.3 Matrix Operations 179
A.3.1 Matrix Addition 179
A.3.2 Matrix Subtraction 179
A.3.3 Scalar Multiplication 179
A.3.4 Matrix Multiplication 180
A.4 Matrix Transposition 182
A.5 Matrix Inversion 182
A.6 Determinants 183
A. 6.1 Evaluation of a Determinant 183
A.6.2 Properties of Determinants 184
A.7 Rank of a Matrix 184
A.7.1 Minor 185
A.7.2 Cofactor 185
A.7.3 Cofactor Matrix 186
A. 7.4 Adjoint Matrix 186
A.8 Finding the Inverse of a Square Matrix 186
A. 8.1 Properties of Inverse Matrices 187
A.9 Trace of a Square Matrix 187
A. 10 Quadratic Forms and Definite Matrices 187
A. 10.1 Some Properties of Quadratic Forms 188
A. 10.2 Mean and Variance of Quadratic Forms 188
A. 11 Eigenvalues and Eigenvectors 188
A. 12 Vector and Matrix Differentiation 191
A. 12.1 Some Differentiation Rules 191
A. 12.2 The Hessian Matrix 192
Appendix B: Essentials of Large-Sample Theory 193
B. l Some Inequalities 193
B. 1.1 Markov s Inequality 193
B.l.2 Chebyshev’s Inequality 194
B.1.3 Khinchine s Theorem (Weak Law of
Large Numbers) 195
B.l. 4 Strong Law of Large Numbers 196
B.l.5 Jensen s Inequality 196
B.l.6 Cauchy—Schwarz Inequality 196
B. 1.7 The Central Limit Theorem 197
B.2 Types of Convergence 198
B.2.1 Convergence of a Sequence of Real Numbers 198
B.2.2 Strong or Almost Sure Convergence 198
B.2.3 Convergence in Probability of a Sequence
of Random Variables 198
B. 2.4 Convergence in Mean Square 200
B.2.5 Convergence in Distribution 200
B.2.6 Relationships Among Three Modes of Convergence 200
B.3 The Order of Magnitude of a Sequence 201
B.4 The Order of Magnitude of a Stochastic Sequence 202
Appendix C: Small- and Large-Sample Properties
of Estimators 203
C.l Small-Sample Properties of Estimators 203
C.1.1 Unbiasedness 203
C.L2 Minimum Variance 203
C.l.3 Best Unbiased or Efficient Estimator 203
C.l.4 Linearity 203
C.L5 Best Linear Unbiased Estimator 204
C.L6 Minimum Mean Square Error Estimator 204
C. 1.7 Efficient Estimator 204
C.1.8 Sufficient Estimator(s) 207
C. 1.9 Uniformly Minimum Variance
Unbiased Estimator 209
C. 2 Large-Sample Properties of Estimators 209
C.2.1 Asymptotic Unbiasedness 209
C.2.2 Consistency 211
C. 2.3 Asymptotic Efficiency 215
C. 2.4 Asymptotic Normality 215
Appendix D: Some Important Probability Distributions 219
D. l The Normal Distribution and the Z Test 219
D. l.l Properties of the Normal Distribution 220
D.2 The Gamma Distribution 222
D.3 The Chi-Square (X2) Distribution and the %2 Test 223
D.3.1 Properties of the Chi-Square Distribution 224
D.4 Student’s t Distribution 225
D.5 Fisher’s F Distribution 226
D.5.1 Properties of the F Distribution 226
D.6 Relationships Among Probability Distributions 228
D.7 Uniform Distributions 228
D.7.1 Discrete Uniform Distribution 228
D.7.2 Continuous Uniform Distribution 228
D.8 Some Special Features of the Normal Distribution 229
Index
233
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| spelling | Gujarati, Damodar N. Verfasser (DE-588)129300462 aut Linear regression a mathematical introduction Damodar N. Gujarati Los Angeles ; London ; New Delhi ; Singapore ; Washington DC ; Melbourne SAGE [2019] © 2019 xxiv, 242 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Quantitative applications in the social sciences 177 Lineare Regression (DE-588)4167709-2 gnd rswk-swf Regressionsanalyse (DE-588)4129903-6 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Lineare Regression (DE-588)4167709-2 s DE-604 Regressionsanalyse (DE-588)4129903-6 s Quantitative applications in the social sciences 177 (DE-604)BV000005102 177 LoC Fremddatenuebernahme application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030584600&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030584600&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gujarati, Damodar N. Linear regression a mathematical introduction Quantitative applications in the social sciences Lineare Regression (DE-588)4167709-2 gnd Regressionsanalyse (DE-588)4129903-6 gnd |
| subject_GND | (DE-588)4167709-2 (DE-588)4129903-6 (DE-588)4151278-9 |
| title | Linear regression a mathematical introduction |
| title_auth | Linear regression a mathematical introduction |
| title_exact_search | Linear regression a mathematical introduction |
| title_full | Linear regression a mathematical introduction Damodar N. Gujarati |
| title_fullStr | Linear regression a mathematical introduction Damodar N. Gujarati |
| title_full_unstemmed | Linear regression a mathematical introduction Damodar N. Gujarati |
| title_short | Linear regression |
| title_sort | linear regression a mathematical introduction |
| title_sub | a mathematical introduction |
| topic | Lineare Regression (DE-588)4167709-2 gnd Regressionsanalyse (DE-588)4129903-6 gnd |
| topic_facet | Lineare Regression Regressionsanalyse Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030584600&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030584600&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005102 |
| work_keys_str_mv | AT gujaratidamodarn linearregressionamathematicalintroduction |
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