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Author(s): Valérie Mignon
This textbook teaches the basics of econometrics and focuses on the acquisition of methods and skills that are essential for any student to succeed in their studies, as well as for any practitioner interested in applying econometric techniques. Employing a pedagogical and easy-to-follow style, the book puts into practice the various concepts presented, such as statistics, tests, and methods, among others. Numerous examples and empirical applications using existing econometric and statistical software are given after each theoretical presentation.
The book addresses students at the undergraduate and graduate levels in economics and management, as well as students of engineering and business schools. It will further appeal to professionals and practitioners of econometrics, such as economists and researchers in companies and institutions, who will find practical solutions to the different problems they are confronted with.
Contents
About the Author
1 Introductory Developments
1.1 What Is Econometrics? Some Introductory Examples
1.1.1 Answers to Many Questions
1.1.2 The Example of Consumption and Income
1.1.3 The Answers to the Other Questions Asked
1.2 Model and Variable
1.2.1 The Concept of Model
1.2.2 Different Types of Data
1.2.3 Explained Variable/Explanatory Variable
1.2.4 Error Term
1.3 Statistics Reminders
1.3.1 Mean
1.3.2 Variance, Standard Deviation, and Covariance
1.3.3 Linear Correlation Coefficient
1.3.4 Empirical Application
1.4 A Brief Introduction to the Concept of Stationarity
1.4.1 Stationarity in the Mean
1.4.2 Stationarity in the Variance
1.4.3 Empirical Application: A Study of the Nikkei Index
1.5 Databases and Software
1.5.1 Databases
1.5.2 Econometric Software
Conclusion
The Gist of the Chapter
Further Reading
2 The Simple Regression Model
2.1 General
2.1.1 The Linearity Assumption
Linearity in the Variables
Linearity in the Parameters
Linear Model
2.1.2 Specification of the Simple Regression Model and Properties of the Error Term
The Nullity of the Mean Error
The Absence of Autocorrelation in Errors
The Homoskedasticity of Errors
The Normality of Errors
2.1.3 Summary: Specification of the Simple Regression Model
2.2 The Ordinary Least Squares (OLS) Method
2.2.1 Objective and Reminder of Hypotheses
2.2.2 The OLS Principle
2.2.3 The OLS Estimators
Searching for Estimators
Example: The Phillips Curve and the Natural Unemployment Rate
A Cross-Sectional Example: The Consumption-Income Relationship
Summary and Properties
2.2.4 Properties of OLS Estimators
Linear Estimators
Unbiased Estimators
Consistent and Minimum Variance Estimators
2.2.5 OLS Estimator of the Variance of the Error Term
Finding the Estimator of the Error Variance
Estimation of the Variances of the OLS Estimators
2.2.6 Empirical Application
2.3 Tests on the Regression Parameters
2.3.1 Determining the Distributions Followed by the OLS Estimators
2.3.2 Tests on the Regression Coefficients
Test on α
Test on β
Test on σ2
2.3.3 Empirical Application
2.4 Analysis of Variance and Coefficient of Determination
2.4.1 Analysis of Variance (ANOVA)
2.4.2 Coefficient of Determination
2.4.3 Analysis of Variance and Significance Test of the Coefficient β
2.4.4 Empirical Application
2.5 Prediction
2.6 Some Extensions of the Simple Regression Model
2.6.1 Log-Linear Model
2.6.2 Semi-Log Model
2.6.3 Reciprocal Model
2.6.4 Log-Inverse or Log-Reciprocal Model
Conclusion
The Gist of the Chapter
Further Reading
Appendix 2.1: Demonstrations
Appendix 2.1.1: Demonstration of the Linearity of the OLS Estimators
Appendix 2.1.2: Demonstration of the Unbiasedness Property of the OLS Estimators
Appendix 2.1.3: Demonstration of the Consistency and Minimum Variance Property of the OLS Estimators
Appendix 2.1.4: Calculation of the Estimator of the Variance of the Error Term
Appendix 2.1.5: Calculation of the Standard Deviation of the Forecast Error and Prediction Interval
Appendix 2.2: Normal Distribution and Normality Test
Appendix 2.3: The Maximum Likelihood Method
3 The Multiple Regression Model
3.1 Writing the Model in Matrix Form
3.2 The OLS Estimators
3.2.1 Assumptions of the Multiple Regression Model
Hypothesis 1: The Matrix X Is Nonrandom
Hypothesis 2: The Matrix X Is of Full Rank
Hypothesis 3: The Expectation of the Error Term Is Zero
Hypothesis 4: Homoskedasticity and the Absence of Autocorrelation of Errors
Hypothesis 5: Normality of Errors
3.2.2 Estimation of Coefficients
3.2.3 Properties of OLS Estimators
Linearity of the Estimator
Unbiased Estimator
Variance-Covariance Matrix of Coefficients
Minimum Variance Estimator
3.2.4 Error Variance Estimation
3.2.5 Example
Determination of OLS Estimators
Practical Calculation
3.3 Tests on the Regression Coefficients
3.3.1 Distribution of Estimators
3.3.2 Tests on a Regression Coefficient
3.3.3 Significance Tests of Several Coefficients
Test on a Particular Regression Coefficient
Test of Equality of Coefficients
Significance Test for All Coefficients
Significance Test of a Subset of Coefficients
Synthesis
3.4 Analysis of Variance (ANOVA) and Adjusted Coefficient of Determination
3.4.1 Analysis-of-Variance Equation
Case of Centered Variables
Case of Noncentered Variables
3.4.2 Coefficient of Determination
3.4.3 Adjusted Coefficient of Determination
3.4.4 Partial Correlation Coefficient
3.4.5 Example
Analysis-of-Variance Equation: Case of Centered Variables
Analysis-of-Variance Equation: Case of Noncentered Variables
Tests on the Regression Coefficients
Calculation of the Partial Correlation Coefficients
3.5 Some Examples of Cross-Sectional Applications
3.5.1 Determinants of Crime
3.5.2 Health Econometrics
3.5.3 Inequalities and Financial Openness
3.5.4 Inequality and Voting Behavior
3.6 Prediction
3.6.1 Determination of Predicted Value and Prediction Interval
3.6.2 Example
3.7 Model Comparison Criteria
3.7.1 Explanatory Power/Predictive Power of a Model
3.7.2 Coefficient of Determination and Adjusted Coefficient of Determination
3.7.3 Information Criteria
Akaike Information Criterion ( AIC)
Schwarz Information Criterion (SIC)
Hannan-Quinn Information Criterion (HQ)
3.7.4 The Mallows Criterion
3.8 Empirical Application
3.8.1 Practical Calculation of the OLS Estimators
3.8.2 Software Estimation
Conclusion
The Gist of the Chapter
Further Reading
Appendix 3.1: Elements of Matrix Algebra
General
Main Matrix Operations
Equality
Transposition
Addition and Subtraction
Matrix Multiplication and Scalar Product
Idempotent Matrix
Rank, Trace, Determinant, and Inverse Matrix
Rank of a Matrix
Trace of a Matrix
Determinant of a Matrix
Inverse Matrix
Appendix 3.2: Demonstrations
Appendix 3.2.1: Demonstration of the Minimum Variance Property of OLS Estimators
Appendix 3.2.2: Calculation of the Error Variance
Appendix 3.2.3: Significance Tests of Several Coefficients
4 Heteroskedasticity and Autocorrelation of Errors
4.1 The Generalized Least Squares (GLS) Estimators
4.1.1 Properties of OLS Estimators in the Presence of Autocorrelation and/or Heteroskedasticity
4.1.2 The Generalized Least Squares (GLS) Method
4.1.3 Estimation of the Variance of the Errors
4.2 Heteroskedasticity of Errors
4.2.1 The Sources of Heteroskedasticity
4.2.2 Estimation When There Is Heteroskedasticity
4.2.3 Detecting Heteroskedasticity
The Goldfeld and Quandt Test (1965)
The Glejser Test (1969)
The Breusch-Pagan Test (1979)
The White Test (1980)
ARCH Test
4.2.4 Estimation Procedures When There Is Heteroskedasticity
The White Estimator of the Variance-Covariance Matrix
The Newey and West Estimator of the Variance-Covariance Matrix
Hypotheses About the Form of Heteroskedasticity
Note on the Logarithmic Transformation
4.2.5 Empirical Application
The Goldfeld and Quandt Test
The Glejser Test
The Breusch-Pagan Test
The White Test
ARCH Test
Heteroskedasticity-Corrected Estimations
4.3 Autocorrelation of Errors
4.3.1 Sources of Autocorrelation
4.3.2 Estimation When There Is Autocorrelation
4.3.3 Detecting Autocorrelation
The Geary Test (1970)
The Durbin and Watson Test (1950, 1951)
The Durbin Test (1970)
The Breusch-Godfrey Test
The Box-Pierce (1970) and Ljung-Box (1978) Tests
4.3.4 Estimation Procedures in the Presence of Error Autocorrelation
Case Where the Variance of the Error Term Is Known: General Principle of GLS
Case Where the Variance of the Error Term Is Unknown: Pseudo GLS Methods
4.3.5 Prediction in the Presence of Error Autocorrelation
4.3.6 Empirical Application
Conclusion
The Gist of the Chapter
Further Reading
5 Problems with Explanatory Variables
5.1 Random Explanatory Variables and the Instrumental Variables Method
5.1.1 Instrumental Variables Estimator
5.1.2 The Hausman1978 Specification Test
5.1.3 Application Example: Measurement Error
5.2 Multicollinearity and Variable Selection
5.2.1 Presentation of the Problem
5.2.2 The Effects of Multicollinearity
5.2.3 Detecting Multicollinearity
Correlation Between Explanatory Variables
The Klein Test (1962)
The Farrar and Glauber Test (1967)
The Eigenvalue Method
Variance Inflation Factors
Empirical Application
5.2.4 Solutions to Multicollinearity
Use of Preliminary Estimates
The Ridge Regression
Other Techniques
5.2.5 Variable Selection Methods
The Method of All Possible Regressions
Backward Elimination of Explanatory Variables
Forward Selection of Explanatory Variables
The Stepwise Method
Empirical Application
5.3 Structural Changes and Indicator Variables
5.3.1 The Constrained Least Squares Method
5.3.2 The Introduction of Indicator Variables
Definition
Introductory Examples
Model Containing Only Indicator Variables
Model Containing Indicator and Usual Explanatory Variables
Interactions
Use of Indicator Variables for Deseasonalization
Empirical Application
5.3.3 Coefficient Stability Tests
Rolling Regressions and Recursive Residuals
The Chow Test (1960)
Empirical Application
Conclusion
The Gist of the Chapter
Further Reading
Appendix: Demonstration of the Formula for Constrained Least Squares Estimators
6 Distributed Lag Models
6.1 Why Introduce Lags? Some Examples
6.2 General Formulation and Definitions of DistributedLag Models
6.3 Determination of the Number of Lags and Estimation
6.3.1 Determination of the Number of Lags
6.3.2 The Question of Estimating Distributed Lag Models
6.4 Finite Distributed Lag Models: Almon Lag Models
6.5 Infinite Distributed Lag Models
6.5.1 The Koyck Approach
The Koyck Transformation
Estimation: The Instrumental Variables Method
The Partial Adjustment Model
The Adaptive Expectations Model
6.5.2 The Pascal Approach
6.6 Autoregressive Distributed Lag Models
6.6.1 Writing the ARDL Model
6.6.2 Calculation of ARDL Model Weights
6.7 Empirical Application
Conclusion
The Gist of the Chapter
Further Reading
7 An Introduction to Time Series Models
7.1 Some Definitions
7.1.1 Time Series
7.1.2 Second-Order Stationarity
7.1.3 Autocovariance Function, Autocorrelation Function, and Partial Autocorrelation Function
7.2 Stationarity: Autocorrelation Function and Unit Root Test
7.2.1 Study of the Autocorrelation Function
7.2.2 TS and DS Processes
Characteristics of TS Processes
Characteristics of DS Processes
7.2.3 The Dickey-Fuller Test
Simple Dickey-Fuller (DF) Test
Augmented Dickey-Fuller (ADF) Test
Sequential Testing Strategy
Empirical Application
7.3 ARMA Processes
7.3.1 Definitions
Autoregressive Processes
Moving-Average Processes
Autoregressive Moving-Average Processes: ARMA(p,q)
7.3.2 The Box and Jenkins Methodology
Step 1: Identification of ARMA Processes
Step 2: Estimation of ARMA Processes
Step 3: Validation of ARMA Processes
Step 4: Prediction of ARMA Processes
7.3.3 Empirical Application
Step 1: Identification
Step 2: Estimation
Step 3: Validation
7.4 Extension to the Multivariate Case: VAR Processes
7.4.1 Writing the Model
Introductory Example
General Formulation
7.4.2 Estimation of the Parameters of a VAR(p) Process and Validation
7.4.3 Forecasting VAR Processes
7.4.4 Granger Causality
7.4.5 Empirical Application
7.5 Cointegration and Error-Correction Models
7.5.1 The Problem of Spurious Regressions
7.5.2 The Concept of Cointegration
7.5.3 Error-Correction Models
7.5.4 Estimation of Error-Correction Models and Cointegration Tests: The EngleandGranger1987 Approach
Two-Step Estimation Method
Dickey-Fuller Test of No Cointegration
Example: The Relationship Between Prices and Dividends
7.5.5 Empirical Application
Conclusion
The Gist of the Chapter
Further Reading
8 Simultaneous Equations Models
8.1 The Analytical Framework
8.1.1 Introductory Example
8.1.2 General Form of Simultaneous Equations Models
8.2 The Identification Problem
8.2.1 Problem Description
8.2.2 Rank and Order Conditions for Identification
Restrictions
Conditions for Identification
8.3 Estimation Methods
8.3.1 Indirect Least Squares
8.3.2 Two-Stage Least Squares
8.3.3 Full-Information Methods
8.4 Specification Test
8.5 Empirical Application
8.5.1 Writing the Model
8.5.2 Conditions for Identification
8.5.3 Data
8.5.4 Model Estimation
OLS Estimation Equation by Equation
Two-Stage Least Squares Estimation
Three-Stage Least Squares Estimation
Full-Information Maximum Likelihood Estimation
Conclusion
The Gist of the Chapter
Further Reading
Appendix: Statistical Tables
Standard Normal Distribution
Student t Distribution: Critical Values of t
Chi-Squared Distribution: Critical Values of c
Fisher–Snedecor Distribution: Critical Values of F
Durbin–Watson Critical Values
References
Index
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