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An Introduction to Econometric Theory

An Introduction to Econometric Theory

James Davidson

ISBN: 978-1-119-48492-9

Aug 2018

272 pages

$86.99

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Description

A guide to economics, statistics and finance that explores the mathematical foundations underling econometric methods

An Introduction to Econometric Theory offers a text to help in the mastery of the mathematics that underlie econometric methods and includes a detailed study of matrix algebra and distribution theory. Designed to be an accessible resource, the text explains in clear language why things are being done, and how previous material informs a current argument. The style is deliberately informal with numbered theorems and lemmas avoided. However, very few technical results are quoted without some form of explanation, demonstration or proof.

The author — a noted expert in the field — covers a wealth of topics including: simple regression, basic matrix algebra, the general linear model, distribution theory, the normal distribution, properties of least squares, unbiasedness and efficiency, eigenvalues, statistical inference in regression, t and F tests, the partitioned regression, specification analysis, random regressor theory, introduction to asymptotics and maximum likelihood. Each of the chapters is supplied with a collection of exercises, some of which are straightforward and others more challenging. This important text:

  • Presents a guide for teaching econometric methods to undergraduate and graduate students of economics, statistics or finance
  • Offers proven classroom-tested material
  • Contains sets of exercises that accompany each chapter
  • Includes a companion website that hosts additional materials, solution manual and lecture slides 

Written for undergraduates and graduate students of economics, statistics or finance, An Introduction to Econometric Theory is an essential beginner’s guide to the underpinnings of econometrics. 

 

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Preface viii

I Fitting 1

1 Elementary Data Analysis 2

1.1 Variables and Observations 2

1.2 Summary Statistics 4

1.3 Correlation 6

1.4 Regression 9

1.5 Computing the Regression Line 12

1.6 Multiple Regression 16

1.7 Exercises 19

2 Matrix Representation 21

2.1 Systems of Equations 21

2.2 Matrix Algebra Basics 23

2.3 Rules of Matrix Algebra 26

2.4 Partitioned Matrices 27

2.5 Exercises 29

3 Solving the Matrix Equation 31

3.1 Matrix Inversion 31

3.2 Determinant and Adjoint 34

3.3 Transposes and Products 37

3.4 Cramer.s Rule 38

3.5 Partitioning and Inversion 39

3.6 A Note on Computation 41

3.7 Exercises 44

4 The Least Squares Solution 47

4.1 Linear Dependence and Rank 47

4.2 The General Linear Regression 51

4.3 De.nite Matrices 53

4.4 Matrix Calculus 56

4.5 Goodness of Fit 58

4.6 Exercises 61

II Modelling 64

5 Probability Distributions 65

5.1 A Random Experiment 65

5.2 Properties of the Normal Distribution 69

5.3 Expected Values 72

5.4 Discrete Random Variables 75

5.5 Exercises 82

6 More on Distributions 84

6.1 Random Vectors 84

6.2 The Multivariate Normal Distribution 85

6.3 Other Continuous Distributions 88

6.4 Moments 91

6.5 Conditional Distributions 93

6.6 Exercises 96

7 The Classical Regression Model 98

7.1 The Classical Assumptions 98

7.2 The Model 100

7.3 Properties of Least Squares 102

7.4 The Projection Matrices 105

7.5 The Trace 106

7.6 Exercises 109

8 The Gauss-Markov Theorem 111

8.1 A Simple Example 111

8.2 E¢ ciency in the General Model 113

8.3 Failure of the Assumptions 115

8.4 Generalized Least Squares 116

8.5 Weighted Least Squares 118

8.6 Exercises 121

III Testing 123

9 Eigenvalues and Eigenvectors 124

9.1 The Characteristic Equation 124

9.2 Complex Roots 126

9.3 Eigenvectors 127

9.4 Diagonalization 129

9.5 Other Properties 131

9.6 An Interesting Result 133

9.7 Exercises 135

10 The Gaussian Regression Model 137

10.1 Testing Hypotheses 137

10.2 Idempotent Quadratic Forms 139

10.3 Con.dence Regions 142

10.4 t Statistics 144

10.5 Tests of Linear Restrictions 147

10.6 Constrained Least Squares 149

10.7 Exercises 152

11 Partitioning and Speci.cation 155

11.1 The Partitioned Regression 155

11.2 Frisch-Waugh-Lovell Theorem 157

11.3 Misspeci.cation Analysis 158

11.4 Speci.cation Testing 162

11.5 Stability Analysis 163

11.6 Prediction Tests 165

11.7 Exercises 167

IV Extensions 170

12 Random Regressors 171

12.1 Conditional Probability 171

12.2 Conditional Expectations 173

12.3 Statistical Models Contrasted 176

12.4 The Statistical Assumptions 179

12.5 Properties of OLS 181

12.6 The Gaussian Model 184

12.7 Exercises 187

13 Introduction to Asymptotics 189

13.1 The Law of Large Numbers 190

13.2 Consistent Estimation 194

13.3 The Central Limit Theorem 197

13.4 Asymptotic Normality 201

13.5 Multiple Regression 204

13.6 Exercises 207

14 Asymptotic Estimation Theory 209

14.1 Large Sample E¢ ciency 209

14.2 Instrumental Variables 210

14.3 Maximum Likelihood 213

14.4 Gaussian ML 216

14.5 Properties of ML Estimators 217

14.6 Likelihood Inference 219

14.7 Exercises 221

V Appendices 224

A The Binomial Coe¢ cients 225

B The Exponential Function 227

C Essential Calculus 229

D The Generalized Inverse 231

Recommended Reading 234

Acknowledgements 236

Index 236

Figures

1.1 Long and short UK interest rates 6

1.2 Scatter plot of the interest rate series. 7

1.3 The regression line 10

1.4 The regression line and the data 10

1.5 The regression residual 11

1.6 Plot of y = 2 + 2x1 + x2 17

5.1 Archery target scatter 65

5.2 Frequency contours 66

5.3 Bivariate normal density 67

5.4 Normal p.d.f. 71

5.5 Binomial probabilities and the normal density function. 78

5.6 Prussian cavalry data and predictions 80

6.1 The standard Cauchy density function 90

10.1 Con.dence regions for k = 2 143

13.1 P.d.f of the sum of three uniform r.v.s, with normal curve for comparison. 199