Econometrics in Theory and Practice

Panchanan Das

Econometrics in Theoryand PracticeAnalysis of Cross Section, Time Seriesand Panel Data with Stata 15.1

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Panchanan DasDepartment of EconomicsUniversity of CalcuttaKolkata, India

ISBN 978-981-32-9018-1 ISBN 978-981-32-9019-8 (eBook)https://doi.org/10.1007/978-981-32-9019-8

© Springer Nature Singapore Pte Ltd. 2019This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, expressed or implied, with respect to the material containedherein or for any errors or omissions that may have been made. The publisher remains neutral with regardto jurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd.The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721,Singapore

Dedicated to my father late Bibhuti BhusanDas

Preface

This book is an outcome of my experience in learning and teaching econometricssince more than three decades. Good quality books of econometrics are available,but there is a dearth of user-friendly books with a proper combination of theory andapplication with statistical software. The books particularly by Maddala,Wooldridge, Greene, Enders, Maddala and Kim, Hsiao and Baltagi are very muchinvaluable. The book by Gujarati is also a good one in its ability to elaborateeconometric theories for graduate students. However, many scholars and studentsand researchers, today, use statistical software to do empirical analysis. I also haveused both EViews and Stata in my teaching and research works and personallyfound that Stata is as powerful or flexible compared to EViews. Furthermore, Statais used extensively to process large data sets. This book is a proper combination ofeconometric theory and application with Stata 15.1.

The basic purpose of this text is to introduce econometric analysis of crosssection, time series and panel data with the application of statistical software. Thisbook may serve as a basic text for those who wish to learn and apply econometricanalysis in empirical research. The level of presentation is kept as simple as pos-sible to make it useful for undergraduate as well as graduate students. It containsseveral examples with real data and Stata programmes and interpretation of theresults.

This book is intended primarily for graduate and post-graduate students inuniversities in India and abroad and researchers in the social sciences, business,management, operations research, engineering or applied mathematics. In this book,we view econometrics as a subject dealing with a set of data analytic techniques thatare used in empirical research extensively. The aim is to provide students with theskills required to undertake independent applied research using modern econo-metric methods. It covers the statistical tools needed to understand empirical eco-nomic research and to plan and execute independent research projects. It attempts toprovide a balance between theory and applied research. Various concepts andtechniques of econometric analysis are supported by carefully developed examples

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with the use of statistical software package, Stata 15.1. Hopefully, this book willsuccessfully bridge the gap between learning econometrics and learning how to useStata.

It is an attempt to incorporate econometric theories in a student-friendly mannerto understand properly the techniques needed for empirical research. It demandsboth students and professional analysts because of its balanced discussion of thetheories with software applications. However, this book should not be claimed as asubstitute for the well-established texts that are being used in academia; rather it canserve as a supplementary text in both undergraduate- and post-graduate-leveleconometric courses. The discussion in this book is based on the assumption thatthe reader is somewhat familiar with the Stata software and other statistical pro-gramming. The Stata help manuals from the Stata Corporation offer detailedexplanation and syntax for all the commands used in this book. The data used forillustration are taken mainly from official sources like CSO, NSSO and ILO.

The topics covered in this book are divided into four parts. Part I is the discussionon introductory econometric methods covering the syllabus of graduate courses inthe University of Calcutta, Delhi University and other leading universities in Indiaand abroad. This part of the book provides an introduction to basic econometricmethods for data analysis that economists and other social scientists use to estimatethe economic and social relationships, and to test hypotheses about them, usingreal-world data. There are 5 chapters in this part covering the data managementissues, details of linear regression models and the related problems due to the vio-lation of the classical assumptions. Chapter 1 provides some basic steps used ineconometrics and statistical software, Stata 15.1, for useful application of econo-metric theories. Chapter 2 discusses linear regression model and its application withcross section data. Chapter 3 deals with this problem of statistical inference of a linearregression model. Chapter 4 relaxes the homoscedasticity and non-autocorrelationassumptions of the random error of a linear regression model and shows how theparameters of the linear model are correctly estimated. Chapter 5 discusses thedetection of multicollinearity and alternatives for handling the problem.

Part II discusses some advanced topics used frequently in empirical researchwith cross section data. This part contains 3 chapters to include some specificproblems of regression analysis. Chapter 6 explains how qualitative explanatoryvariables can be incorporated into a linear model. Chapter 7 provides econometricmodels with limited dependent variables and problems of truncated distribution,sample selection bias and multinomial logit. Special emphasis is given to multi-variate analysis, particularly principal component analysis and factor analysis,because of their popularity in empirical research with cross section data. Chapter 8captures these issues.

Part III deals with time series econometric analysis. Time series data have somespecial features, and they should be taken care of very much cautiously. Time serieseconometrics was developed in modern approach since the early 1980s with thepublications of Engle and Granger, and it becomes very much popular in empiricalresearch with the development of user-friendly software. This book covers inten-sively both the univariate and multivariate time series econometric models and their

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applications with software programming in 6 chapters. This part starts with thediscussion on data generating process of time series data in Chap. 9. Chapter 10deals with different features of the data generating process (DGP) of a time series ina univariate framework. The presence of unit roots in macroeconomic time serieshas received a major area of theoretical and applied research since the early 1980s.Chapter 11 presents some issues regarding unit root tests and explores some of theimplications for macroeconomic theory and policy. Chapter 12 explores the basicconceptual issues involved in estimating the relationship between two or morenonstationary time series with unit roots. Chapter 13 examines the behaviour ofvolatility in terms of conditional heteroscedasticity model. Forecasting is importantin economics, commerce and various disciplines of social science and pure science.Chapter 14 aims to provide an overview of forecasting based on time seriesanalysis.

Part IV takes care of panel data analysis in 4 chapters. Panel data have severaladvantages over the cross section and time series data. Panel data econometricsgains popularity because of the availability of panel data in the public domaintoday. Different aspects of fixed effects and random effects are discussed here.I have extended panel data analysis by taking dynamic panel data models which arethe most suitable for macroeconomic research. Chapter 15 discusses different typesof panel data model in a static framework. Chapter 16 deals with testing ofhypotheses to examine panel data in a static framework. Panel data with long timeperiod have been used predominately in applied macroeconomic research likepurchasing power parity, growth convergence, business cycle synchronisation andso on. Chapter 17 provides some theoretical issues and their application in testingfor unit roots in panel data. Dynamic model in panel data framework is very muchpopular in empirical research. Chapter 18 focuses on some issues of dynamic paneldata model.

All chapters in this book provide applications of econometric models by usingStata. Simple presentation of some difficult topics in a rigorous manner is the majorstrength of this book. While the Bayesian econometrics, nonparametric and semi-parametric, are popular methods today to capture the behaviour of the data in amore complex real situation, I do not attempt to cover these topics because of mycomparative disadvantage in these areas and to keep the technical difficulty at alower possible level. Despite these limitations, the topics covered in this book arebasics and necessary for econometrics training of every student in economics andother disciplines. I hope the students of econometrics will share my enthusiasm andoptimism in the importance of different econometric methods they will learnthrough reading this book. Hopefully, it will enhance their interest in empiricalresearch in economics and other fields of social science.

Kolkata, India Panchanan DasMay 2019

Preface ix

Acknowledgements

My interest in econometrics was initiated by my teachers at different level sincemore than three decades back. I acknowledge the contribution of Amiya KumarBagchi, my teacher and Ph.D. supervisor, in the field of empirical research thatencourages me to learn econometrics at least indirectly. Among others I shouldmention Dipankor Coondoo of Indian Statistical Institute, Kolkata, who helped meto understand clearly different issues of the subject. Sankar Kumar Bhoumik, mysenior colleague and friend, helped a lot to learn the subject by providing access toteaching at post-graduate level at the Department of Economics, University ofCalcutta, even much before my joining the Department as a permanent faculty.I also gratefully acknowledge my teacher, Manoj Kumar Sanyal, who in fact is acontinuous source of encouragement in learning and thinking. I think, in some way,they have prepared the background for this book being written.

A number of friends and colleagues have commented on earlier drafts of thebook, or helped in other ways. I am grateful to Maniklal Adhikary, AninditaSengupta, Pradip Kumar Biswas and others for their assistance and encouragement.Discussions with Oleg Golichenko and Kirdina Svetlana of Higher SchoolEconomics, Moscow, were helpful in clarifying some of my ideas.

Any remaining errors and omissions are, of course, my responsibility, and I shallbe glad to have them brought to my attention.

I am grateful to the Department of Economics, University of Calcutta, forproviding an adequate infrastructure where I spent time during my learning andteaching of economics. Special thanks are due to the Head of the Department ofEconomics and the authority of the University of Calcutta.

I am extremely grateful to my wife, Krishna, who took over many of my roles inthe household during the preparation of the manuscripts.

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Finally, thanks to the editorial team of Springer for help with indexing andproof-reading. I am grateful to Sagarika Ghosh of Springer for encouragement for thisproject.

Kolkata, India Panchanan DasMay 2019

xii Acknowledgements

Contents

Part I Introductory Econometrics

1 Introduction to Econometrics and Statistical Software . . . . . . . . . . 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Economic Model and Econometric Model . . . . . . . . . . . . . . . 61.3 Population Regression Function and Sample Regression

Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Parametric and Nonparametric or Semiparametric Model . . . . . 101.5 Steps in Formulating an Econometric Model . . . . . . . . . . . . . . 11

1.5.1 Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5.3 Testing of Hypothesis . . . . . . . . . . . . . . . . . . . . . . . 141.5.4 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.6.1 Cross Section Data . . . . . . . . . . . . . . . . . . . . . . . . . 151.6.2 Time Series Data . . . . . . . . . . . . . . . . . . . . . . . . . . 161.6.3 Pooled Cross Section . . . . . . . . . . . . . . . . . . . . . . . 161.6.4 Panel Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.7 Use of Econometric Software: Stata 15.1 . . . . . . . . . . . . . . . . 171.7.1 Data Management . . . . . . . . . . . . . . . . . . . . . . . . . . 181.7.2 Generating Variables . . . . . . . . . . . . . . . . . . . . . . . . 211.7.3 Describing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.7.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.7.5 Logical Operators in Stata . . . . . . . . . . . . . . . . . . . . 231.7.6 Functions Used in Stata . . . . . . . . . . . . . . . . . . . . . 24

1.8 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.8.1 Matrix and Vector: Basic Operations . . . . . . . . . . . . 241.8.2 Partitioned Matrices . . . . . . . . . . . . . . . . . . . . . . . . 281.8.3 Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 281.8.4 Inverse Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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1.8.5 Positive Definite Matrix . . . . . . . . . . . . . . . . . . . . . 311.8.6 Trace of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 311.8.7 Orthogonal Vectors and Matrices . . . . . . . . . . . . . . . 321.8.8 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . 32

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2 Linear Regression Model: Properties and Estimation . . . . . . . . . . . 372.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 The Simple Linear Regression Model . . . . . . . . . . . . . . . . . . . 382.3 Multiple Linear Regression Model . . . . . . . . . . . . . . . . . . . . . 422.4 Assumptions of Linear Regression Model . . . . . . . . . . . . . . . . 46

2.4.1 Non-stochastic Regressors . . . . . . . . . . . . . . . . . . . . 462.4.2 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.3 Zero Unconditional Mean . . . . . . . . . . . . . . . . . . . . 472.4.4 Exogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.4.5 Homoscedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 482.4.6 Non-autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . 482.4.7 Full Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.4.8 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . 50

2.5 Methods of Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.5.1 The Method of Moments (MM) . . . . . . . . . . . . . . . . 512.5.2 The Method of Ordinary Least Squares (OLS) . . . . . 512.5.3 Maximum Likelihood Method . . . . . . . . . . . . . . . . . 59

2.6 Properties of the OLS Estimation . . . . . . . . . . . . . . . . . . . . . . 632.6.1 Algebraic Properties . . . . . . . . . . . . . . . . . . . . . . . . 632.6.2 Statistical Properties . . . . . . . . . . . . . . . . . . . . . . . . 66

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3 Linear Regression Model: Goodness of Fit and Testingof Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.2 Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.2.1 The R2 as a Measure of Goodness of Fit . . . . . . . . . 763.2.2 The Adjusted R2 as a Measure of Goodness

of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.3 Testing of Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.3.1 Sampling Distributions of the OLS Estimators . . . . . 823.3.2 Testing of Hypothesis for a Single Parameter . . . . . . 833.3.3 Use of P-Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.3.4 Interval Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 893.3.5 Testing of Hypotheses for More Than One

Parameter: t Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.3.6 Testing Significance of the Regression: F Test . . . . . 913.3.7 Testing for Linearity . . . . . . . . . . . . . . . . . . . . . . . . 93

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3.3.8 Tests for Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 953.3.9 Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . 963.3.10 The Likelihood-Ratio, Wald and Lagrange

Multiplier Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.4 Linear Regression Model by Using Stata 15.1 . . . . . . . . . . . . 101

3.4.1 OLS Estimation in Stata . . . . . . . . . . . . . . . . . . . . . 1013.4.2 Maximum Likelihood Estimation (MLE) in Stata . . . 104

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4 Linear Regression Model: Relaxing the Classical Assumptions . . . . 1094.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.2 Heteroscedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.2.1 Problems with Heteroscedastic Data . . . . . . . . . . . . . 1104.2.2 Heteroscedasticity Robust Variance . . . . . . . . . . . . . 1124.2.3 Testing for Heteroscedasticity . . . . . . . . . . . . . . . . . 1154.2.4 Problem of Estimation . . . . . . . . . . . . . . . . . . . . . . 1164.2.5 Illustration of Heteroscedastic Linear Regression

by Using Stata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.3 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.3.1 Linear Regression Model with AutocorrelatedError . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.3.2 Testing for Autocorrelation: Durbin–Watson Test . . . 1284.3.3 Consequences of Autocorrelation . . . . . . . . . . . . . . . 1304.3.4 Correcting for Autocorrelation . . . . . . . . . . . . . . . . . 1314.3.5 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 132

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5 Analysis of Collinear Data: Multicollinearity . . . . . . . . . . . . . . . . . 1375.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.2 Multiple Correlation and Partial Correlation . . . . . . . . . . . . . . 1385.3 Problems in the Presence of Multicollinearity . . . . . . . . . . . . . 1405.4 Detecting Multicollinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.4.1 Determinant of (X′X) . . . . . . . . . . . . . . . . . . . . . . . . 1435.4.2 Determinant of Correlation Matrix . . . . . . . . . . . . . . 1435.4.3 Inspection of Correlation Matrix . . . . . . . . . . . . . . . 1435.4.4 Measure Based on Partial Regression . . . . . . . . . . . . 1435.4.5 Theil’s Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445.4.6 Variance Inflation Factor (VIF) . . . . . . . . . . . . . . . . 1445.4.7 Eigenvalues and Condition Numbers . . . . . . . . . . . . 146

5.5 Dealing with Multicollinearity . . . . . . . . . . . . . . . . . . . . . . . . 1475.6 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . . . . . . . . 149References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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Part II Advanced Analysis of Cross Section Data

6 Linear Regression Model: Qualitative Variables as Predictors . . . . 1556.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.2 Regression Model with Intercept Dummy . . . . . . . . . . . . . . . . 157

6.2.1 Dichotomous Factor . . . . . . . . . . . . . . . . . . . . . . . . 1576.2.2 Polytomous Factors . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.3 Regression Model with Interaction Dummy . . . . . . . . . . . . . . 1606.4 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

7 Limited Dependent Variable Model . . . . . . . . . . . . . . . . . . . . . . . . 1677.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1677.2 Linear Probability Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1687.3 Binary Response Models: Logit and Probit . . . . . . . . . . . . . . . 170

7.3.1 The Logit Model . . . . . . . . . . . . . . . . . . . . . . . . . . 1737.3.2 The Probit Model . . . . . . . . . . . . . . . . . . . . . . . . . . 1747.3.3 Difference Between Logit and Probit Models . . . . . . 174

7.4 Maximum Likelihood Estimation of Logit and ProbitModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1757.4.1 Interpretation of the Estimated Coefficients . . . . . . . . 1767.4.2 Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . 1787.4.3 Testing of Hypotheses . . . . . . . . . . . . . . . . . . . . . . . 1797.4.4 Illustration of Binary Response Model by Using

Stata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1807.5 Regression Model with Truncated Distribution . . . . . . . . . . . . 185

7.5.1 Illustration of Truncated Regression byUsing Stata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.6 Problem of Censoring: Tobit Model . . . . . . . . . . . . . . . . . . . . 1917.6.1 Illustration of Tobit Model by Using Stata . . . . . . . . 193

7.7 Models with Sample Selection Bias . . . . . . . . . . . . . . . . . . . . 1957.7.1 Illustration of Sample Selection Model by Using

Stata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1997.8 Multinomial Logit Regression . . . . . . . . . . . . . . . . . . . . . . . . 201

7.8.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 203References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

8 Multivariate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2078.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2078.2 Displaying Multivariate Data . . . . . . . . . . . . . . . . . . . . . . . . . 208

8.2.1 Multivariate Observations . . . . . . . . . . . . . . . . . . . . 2088.2.2 Sample Mean Vector . . . . . . . . . . . . . . . . . . . . . . . 2118.2.3 Population Mean Vector . . . . . . . . . . . . . . . . . . . . . 2118.2.4 Covariance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 212

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8.2.5 Correlation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 2138.2.6 Linear Combination of Variables . . . . . . . . . . . . . . . 215

8.3 Multivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . . . 2188.4 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . 219

8.4.1 Calculation of Principal Components . . . . . . . . . . . . 2208.4.2 Properties of Principal Components . . . . . . . . . . . . . 2238.4.3 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 223

8.5 Factor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2258.5.1 Orthogonal Factor Model . . . . . . . . . . . . . . . . . . . . 2268.5.2 Estimation of Loadings and Communalities . . . . . . . 2288.5.3 Factor Loadings Are not Unique . . . . . . . . . . . . . . . 2328.5.4 Factor Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2328.5.5 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 233

8.6 Multivariate Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2368.6.1 Structure of the Regression Model . . . . . . . . . . . . . . 2368.6.2 Properties of Least Squares Estimators of B . . . . . . . 2388.6.3 Model Corrected for Means . . . . . . . . . . . . . . . . . . . 2398.6.4 Canonical Correlations . . . . . . . . . . . . . . . . . . . . . . 239

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

Part III Analysis of Time Series Data

9 Time Series: Data Generating Process . . . . . . . . . . . . . . . . . . . . . . 2479.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2479.2 Data Generating Process (DGP) . . . . . . . . . . . . . . . . . . . . . . . 248

9.2.1 Stationary Process . . . . . . . . . . . . . . . . . . . . . . . . . . 2509.2.2 Nonstationary Process . . . . . . . . . . . . . . . . . . . . . . . 252

9.3 Methods of Time Series Analysis . . . . . . . . . . . . . . . . . . . . . . 2539.4 Seasonality and Seasonal Adjustment . . . . . . . . . . . . . . . . . . . 2549.5 Creating a Time Variable by Using Stata . . . . . . . . . . . . . . . . 255References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

10 Stationary Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26110.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26210.2 Univariate Time Series Model . . . . . . . . . . . . . . . . . . . . . . . . 26210.3 Autoregressive Process (AR) . . . . . . . . . . . . . . . . . . . . . . . . . 264

10.3.1 The First-Order Autoregressive Process . . . . . . . . . . 26510.3.2 The Second-Order Autoregressive Process . . . . . . . . 26910.3.3 The Autoregressive Process of Order p . . . . . . . . . . 27510.3.4 General Linear Processes . . . . . . . . . . . . . . . . . . . . . 276

10.4 The Moving Average (MA) Process . . . . . . . . . . . . . . . . . . . . 27810.4.1 The First-Order Moving Average Process . . . . . . . . . 27810.4.2 The Second-Order Moving Average Process . . . . . . . 279

Contents xvii

10.4.3 The Moving Average Process of Order q . . . . . . . . . 28010.4.4 Invertibility in Moving Average Process . . . . . . . . . 281

10.5 Autoregressive Moving Average (ARMA) Process . . . . . . . . . 28110.6 Autocorrelation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

10.6.1 Autocorrelation Function for AR(1) . . . . . . . . . . . . . 28510.6.2 Autocorrelation Function for AR(2) . . . . . . . . . . . . . 28710.6.3 Autocorrelation Function for AR(p) . . . . . . . . . . . . . 29010.6.4 Autocorrelation Function for MA(1) . . . . . . . . . . . . 29110.6.5 Autocorrelation Function for MA(2) . . . . . . . . . . . . 29210.6.6 Autocorrelation Function for MA(q) . . . . . . . . . . . . 29310.6.7 Autocorrelation Function for ARMA Process . . . . . . 293

10.7 Partial Autocorrelation Function (PACF) . . . . . . . . . . . . . . . . 29410.7.1 Partial Autocorrelation for AR Series . . . . . . . . . . . . 29610.7.2 Partial Autocorrelation for MA Series . . . . . . . . . . . 298

10.8 Sample Autocorrelation Function . . . . . . . . . . . . . . . . . . . . . . 29910.8.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 300

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

11 Nonstationarity, Unit Root and Structural Break . . . . . . . . . . . . . . 30511.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30611.2 Analysis of Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

11.2.1 Deterministic Function of Time . . . . . . . . . . . . . . . . 30711.2.2 Stochastic Function of Time . . . . . . . . . . . . . . . . . . 30811.2.3 Stochastic and Deterministic Function of Time . . . . . 310

11.3 Concept of Unit Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31211.4 Unit Root Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

11.4.1 Dickey–Fuller Unit Root Test . . . . . . . . . . . . . . . . . 31511.4.2 Augmented Dickey–Fuller (ADF) Unit Root Test . . . 31811.4.3 Phillips–Perron Unit Root Test . . . . . . . . . . . . . . . . 32611.4.4 Dickey–Fuller GLS Test . . . . . . . . . . . . . . . . . . . . . 32911.4.5 Stationarity Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 33111.4.6 Multiple Unit Roots . . . . . . . . . . . . . . . . . . . . . . . . 33411.4.7 Some Problems with Unit Root Tests . . . . . . . . . . . . 33611.4.8 Macroeconomic Implications of Unit Root . . . . . . . . 336

11.5 Testing for Structural Break . . . . . . . . . . . . . . . . . . . . . . . . . . 33711.5.1 Tests with Known Break Points . . . . . . . . . . . . . . . . 33711.5.2 Tests with Unknown Break Points . . . . . . . . . . . . . . 341

11.6 Unit Root Test with Break . . . . . . . . . . . . . . . . . . . . . . . . . . . 34911.6.1 When Break Point is Exogenous . . . . . . . . . . . . . . . 34911.6.2 When Break Point is Endogenous . . . . . . . . . . . . . . 354

11.7 Seasonal Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

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11.7.1 Unit Roots at Various Frequencies: Seasonal UnitRoot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

11.7.2 Generating Time Variable and Seasonal Dummiesin Stata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

11.8 Decomposition of a Time Series into Trend and Cycle . . . . . . 360References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

12 Cointegration, Error Correction and Vector Autoregression . . . . . 36712.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36712.2 Regression with Trending Variables . . . . . . . . . . . . . . . . . . . . 36812.3 Concept of Cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37012.4 Granger’s Representation Theorem . . . . . . . . . . . . . . . . . . . . . 37312.5 Testing for Cointegration: Engle–Granger’s Two-Step

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37412.5.1 Illustrations by Using Stata . . . . . . . . . . . . . . . . . . . 376

12.6 Vector Autoregression (VAR) . . . . . . . . . . . . . . . . . . . . . . . . 37712.6.1 Stationarity Restriction of a VAR Process . . . . . . . . 38112.6.2 Autocovariance Matrix of a VAR Process . . . . . . . . 38412.6.3 Estimation of a VAR Process . . . . . . . . . . . . . . . . . 38612.6.4 Selection of Lag Length of a VAR Model . . . . . . . . 39012.6.5 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 391

12.7 Vector Moving Average Processes . . . . . . . . . . . . . . . . . . . . . 39212.8 Impulse Response Function . . . . . . . . . . . . . . . . . . . . . . . . . . 393

12.8.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 39812.9 Variance Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39912.10 Granger Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

12.10.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 40112.11 Vector Error Correction Model . . . . . . . . . . . . . . . . . . . . . . . 403

12.11.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 40612.12 Estimation and Testing of Hypotheses of Cointegrated

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40812.12.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 413

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

13 Modelling Volatility Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41713.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41713.2 Modelling Non-constant Conditional Variance . . . . . . . . . . . . 41913.3 The ARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42113.4 The GARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42513.5 Asymmetric ARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . 42913.6 ARCH-in-Mean Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43013.7 Testing and Estimation of a GARCH Model . . . . . . . . . . . . . . 432

13.7.1 Testing for ARCH Effect . . . . . . . . . . . . . . . . . . . . . 43213.7.2 Maximum Likelihood Estimation for

GARCH (1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

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13.8 The ARCH Regression Model in Stata . . . . . . . . . . . . . . . . . . 43313.8.1 Illustration with Market Capitalisation Data . . . . . . . 434

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

14 Time Series Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43914.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43914.2 Simple Exponential Smoothing . . . . . . . . . . . . . . . . . . . . . . . 44014.3 Forecasting—Univariate Model . . . . . . . . . . . . . . . . . . . . . . . 44114.4 Forecasting with General Linear Processes . . . . . . . . . . . . . . . 44514.5 Multivariate Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44714.6 Forecasting of a VAR Model . . . . . . . . . . . . . . . . . . . . . . . . . 44714.7 Forecasting GARCH Processes . . . . . . . . . . . . . . . . . . . . . . . 44914.8 Time Series Forecasting by Using Stata . . . . . . . . . . . . . . . . . 450References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

Part IV Analysis of Panel Data

15 Panel Data Analysis: Static Models . . . . . . . . . . . . . . . . . . . . . . . . . 45715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45815.2 Structure and Types of Panel Data . . . . . . . . . . . . . . . . . . . . . 459

15.2.1 Data Description by Using Stata 15.1 . . . . . . . . . . . 46015.3 Benefits of Panel Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46515.4 Sources of Variation in Panel Data . . . . . . . . . . . . . . . . . . . . . 46515.5 Unrestricted Model with Panel Data . . . . . . . . . . . . . . . . . . . . 46715.6 Fully Restricted Model: Pooled Regression . . . . . . . . . . . . . . . 468

15.6.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 46915.7 Error Component Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47115.8 First-Differenced (FD) Estimator . . . . . . . . . . . . . . . . . . . . . . 473

15.8.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 47315.9 One-Way Error Component Fixed Effects Model . . . . . . . . . . 474

15.9.1 The “Within” Estimation . . . . . . . . . . . . . . . . . . . . . 47415.9.2 Least Squares Dummy Variable (LSDV)

Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48315.10 One-Way Error Component Random Effects Model . . . . . . . . 486

15.10.1 The GLS Estimation . . . . . . . . . . . . . . . . . . . . . . . . 49015.10.2 Maximum Likelihood Estimation . . . . . . . . . . . . . . . 49215.10.3 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 494

Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

16 Panel Data Static Model: Testing of Hypotheses . . . . . . . . . . . . . . . 49916.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49916.2 Measures of Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . 50016.3 Testing for Pooled Regression . . . . . . . . . . . . . . . . . . . . . . . . 501

xx Contents

16.4 Testing for Fixed Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50316.4.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 503

16.5 Testing for Random Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 50516.5.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 506

16.6 Fixed or Random Effect: Hausman Test . . . . . . . . . . . . . . . . . 50716.6.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 509

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

17 Panel Unit Root Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51317.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51317.2 First-Generation Panel Unit Root Tests . . . . . . . . . . . . . . . . . . 514

17.2.1 Wu (1996) Unit Root Test . . . . . . . . . . . . . . . . . . . 51517.2.2 Levin, Lin and Chu Unit Root Test . . . . . . . . . . . . . 51617.2.3 Im, Pesaran and Shin (IPS) Unit Root Test . . . . . . . 52117.2.4 Fisher-Type Unit Root Tests . . . . . . . . . . . . . . . . . . 524

17.3 Stationarity Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52617.3.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 528

17.4 Second-Generation Panel Unit Root Tests . . . . . . . . . . . . . . . . 52817.4.1 The Covariance Restrictions Approach . . . . . . . . . . . 52917.4.2 The Factor Structure Approach . . . . . . . . . . . . . . . . 531

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

18 Dynamic Panel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54118.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54218.2 Linear Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54218.3 Fixed and Random Effects Estimation . . . . . . . . . . . . . . . . . . 544

18.3.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 54718.4 Instrumental Variable Estimation . . . . . . . . . . . . . . . . . . . . . . 548

18.4.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 54918.5 Arellano–Bond GMM Estimator . . . . . . . . . . . . . . . . . . . . . . 552

18.5.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 55618.6 System GMM Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560

18.6.1 Illustration by Using Stata . . . . . . . . . . . . . . . . . . . . 562Appendix: Generalised Method of Moments . . . . . . . . . . . . . . . . . . . . 564References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

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About the Author

Panchanan Das is a Professor of Economics, currently teaching Time Series andPanel Data Econometrics at the Department of Economics, University of Calcutta.His main research areas are Development Economics, Indian Economics, andApplied Macroeconomics. He has published several articles and book chapters ongrowth, inequality and poverty, and is a principal author of Economics I andEconomics II, graduate-level textbooks published by Oxford University Press, NewDelhi. He is also a major contributor to the West Bengal Development Report –2008, published by the Academic Foundation, New Delhi, in collaboration with thePlanning Commission, Government of India.

xxiii

List of Figures

Fig. 1.1 Income demand relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Fig. 1.2 Conditional mean function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Fig. 1.3 Sample regression function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Fig. 2.1 Spending–income relationship for households

in West Bengal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Fig. 2.2 Relation between projection and error vectors . . . . . . . . . . . . . . 66Fig. 3.1 Histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Fig. 3.2 a Two-tailed test, b one-tailed test (left tail), c one-tailed test

(right tail) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Fig. 3.3 Comparison of LR, W and LM tests. . . . . . . . . . . . . . . . . . . . . . 100Fig. 3.4 Log-likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Fig. 4.1 Distribution of Y with heteroscedastic error . . . . . . . . . . . . . . . . 111Fig. 4.2 Variability of ln(wage) with year of schooling. Source NSS

68th round (2011–2012) data on employment andunemployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Fig. 4.3 Scattered plot of residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Fig. 4.4 Pattern of residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Fig. 4.5 Pattern of corrected residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Fig. 6.1 Relation between education and income among men

and women . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Fig. 6.2 Conditional mean functions for female and male groups . . . . . . 158Fig. 7.1 Predicted probability function . . . . . . . . . . . . . . . . . . . . . . . . . . . 170Fig. 7.2 Density function for logit (green) and probit (red) models . . . . . 175Fig. 7.3 CDF for logit (blue) and probit (red) models . . . . . . . . . . . . . . . 175Fig. 9.1 Different shapes of time series . . . . . . . . . . . . . . . . . . . . . . . . . . 249Fig. 9.2 Time behaviour of BSE sensex. . . . . . . . . . . . . . . . . . . . . . . . . . 257Fig. 9.3 Time behaviour of first difference of BSE sensex . . . . . . . . . . . . 258Fig. 10.1 Stationarity region for AR(2) process . . . . . . . . . . . . . . . . . . . . . 274Fig. 10.2 Autocorrelation function of log GDP series . . . . . . . . . . . . . . . . 302Fig. 10.3 Autocorrelation function of the first difference of log

GDP series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

xxv

Fig. 10.4 Partial autocorrelation function of log GDP series . . . . . . . . . . . 303Fig. 11.1 Time path of a series without trend . . . . . . . . . . . . . . . . . . . . . . 316Fig. 11.2 Time path of a series with trend . . . . . . . . . . . . . . . . . . . . . . . . . 317Fig. 11.3 Wald test statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350Fig. 11.4 Index of industrial production . . . . . . . . . . . . . . . . . . . . . . . . . . . 356Fig. 11.5 Seasonally adjusted iip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360Fig. 12.1 Impulse response function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399Fig. 12.2 Movement of GDP and consumption expenditure. . . . . . . . . . . . 407Fig. 13.1 Time path of stock price and return . . . . . . . . . . . . . . . . . . . . . . 418Fig. 13.2 Autocorrelation function of returns and squared returns . . . . . . . 418Fig. 13.3 Time path of first-differenced series of market capitalisation. . . . 435Fig. 15.1 Line plots of GDP growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464Fig. 15.2 Line plots of GDP growth (overlay) . . . . . . . . . . . . . . . . . . . . . . 464Fig. 15.3 Relation between labour employment and labour

productivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470Fig. 15.4 Relation between labour employment and GDP growth . . . . . . . 471Fig. 15.5 Mean values of variables by country . . . . . . . . . . . . . . . . . . . . . 479Fig. 15.6 Estimated relationship between labour employment and labour

productivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

xxvi List of Figures

List of Tables

Table 7.1 Distribution of random error . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Table 15.1 Data matrix of a single variable (X) . . . . . . . . . . . . . . . . . . . . . 459

xxvii