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Springer Texts in Business and Economics

John D. Levendis

Time Series Econometrics Learning Through Replication

Springer Texts in Business and Economics

Springer Texts in Business and Economics (STBE) delivers high-quality instructional content for undergraduates and graduates in all areas of Business/ Management Science and Economics. The series is comprised of self-contained books with a broad and comprehensive coverage that are suitable for class as well as for individual self-study. All texts are authored by established experts in their fields and offer a solid methodological background, often accompanied by problems and exercises.

More information about this series at http://www.springer.com/series/10099

John D. Levendis

Time Series Econometrics Learning Through Replication

123

John D. Levendis Department of Economics Loyola University New Orleans New Orleans, LA, USA

ISSN 2192-4333 ISSN 2192-4341 (electronic) Springer Texts in Business and Economics ISBN 978-3-319-98281-6 ISBN 978-3-319-98282-3 (eBook) https://doi.org/10.1007/978-3-319-98282-3 Library of Congress Control Number: 2018956137 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of 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 transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the 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 this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Catherine and Jack

Preface

What makes this book unique? It follows a simple ethos: it is easier to learn by doing. Or, “Econometrics is better taught by example than abstraction” (Angrist and Pischke 2017, p. 2). The aim of this book is to explain how to use the basic, yet essential, tools of time-series econometrics. The approach is to be as simple as possible so that real learning can take place. We won’t try to be encyclopedic, nor will we break new methodological ground. The goal is to develop a practical understanding of the basic tools you will need to get started in this exciting field. We progress methodically, building as much as possible from concrete examples rather than from abstract first principles. First we learn by doing. Then, with a bit of experience under our belts, we’ll begin developing a deeper theoretical understanding of the real processes. After all, when students learn calculus, they learn the rules of calculus first and practice taking derivatives. Only after they’ve gained familiarity with calculus do students learn the Real Analysis theory behind the formulas. In my opinion, students should take applied econometrics before econometric theory. Otherwise, the importance of the theory falls on deaf ears, as the students have no context with which to understand the theorems. Other books seem to begin at the end, with theory, and then they throw in some examples. We, on the other hand, begin where you are likely to be and lead you forward, building slowly from examples to theory. In the first section, we begin with simple univariate models on well-behaved (stationary) data. We devote a lot of attention on the properties of autoregressive and moving-average models. We then investigate deterministic and stochastic seasonality. Then we explore the practice of unit root testing and the influence of structural breaks. The first section ends with models of non-stationary variance. In the second section, we extend the simpler concepts to the more complex cases of multi-equation multi-variate VAR and VECM models. By the time you finish working through this book, you will not only have studied some of the major techniques of time series, you will actually have worked through many simulations. In fact, if you work along with the text, you will have replicated some of the most influential papers in the field. You won’t just know about some results, you’ll have derived them yourself.

vii

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Preface

No textbook can cover everything. In this text we will not deal with fractional integration, seasonal cointegration, or anything in the frequency domain. Opting for a less-is-more approach, we must leave these and other more complicated topics to other textbooks. Nobody works alone. Many people helped me complete this project. They deserve thanks. Several prominent econometricians dug up—or tried to dig up—data from their classic papers. Thanks, specifically, to Richard T. Baillie, David Dickey, Jordi Gali, Charles Nelson, Dan Thornton, and Jean-Michel Zakoian. Justin Callais provided tireless research assistance, verifying Stata code for the entire text. Donald Lacombe, Wei Sun, Peter Wrubleski, and Jennifer Moreale reviewed early drafts of various chapters and offered valuable suggestions. Mehmet F. Dicle found some coding and data errors and offered useful advice. Matt Lutey helped with some of the replications. The text was inflicted upon a group of innocent undergraduate students at Loyola University. These bright men and women patiently pointed out mistakes and typos, as well as passages that required clarification. For that, I am grateful and wish to thank Justin Callais, Rebecca Driever, Patrick Driscoll, William Herrick, Christian Mays, Nate Straight, David Thomas, Tom Whelan, and Peter Wrobleski. This project could not have been completed without the financial support of Loyola University, the Marquette Fellowship Grant committee, and especially Fr. Kevin Wildes. Thanks to Lorraine Klimowich from Springer for believing in the project and encouraging me to finish it. Finally, and most importantly, I’d like to thank my family. My son Jack: you are my reason for being; I hope to make you proud. My wife Catherine: you are a constant source of support and encouragement. You are amazing. I love you. New Orleans, LA, USA

John D. Levendis

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 What Makes Time-Series Econometrics Unique? . . . . . . . . . . . . . . . . . 1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Statistical Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Specifying Time in Stata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Installing New Stata Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 5 7 8 9

2

ARMA(p,q) Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 A Purely Random Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 AR(1) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Estimating an AR(1) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Impulse Responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 AR(p) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Estimating an AR(p) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Impulse Responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 MA(1) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Impulse Responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 MA(q) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Impulse Responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Non-zero ARMA Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Non-zero AR Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Non-zero MA Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Dealing with Non-zero Means . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 ARMA(p,q) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 14 15 15 20 24 25 26 28 30 32 33 33 34 37 37 38 39 40 41 42 45 46 46 ix

x

Contents

3

Model Selection in ARMA(p,q) Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 ACFs and PACFs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Theoretical ACF of an AR(1) Process . . . . . . . . . . . . . . . . . . 3.1.2 Theoretical ACF of an AR(p) Process . . . . . . . . . . . . . . . . . . 3.1.3 Theoretical ACF of an MA(1) Process . . . . . . . . . . . . . . . . . 3.1.4 Theoretical ACF of an MA(q) Process . . . . . . . . . . . . . . . . . 3.1.5 Theoretical PACFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Summary: Theoretical ACFs and PACFs . . . . . . . . . . . . . . . 3.2 Empirical ACFs and PACFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Calculating Empirical ACFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Calculating Empirical PACFs . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Putting It All Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Information Criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 48 48 52 57 59 63 64 64 68 69 72 77

4

Stationarity and Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 What Is Stationarity? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Importance of Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Restrictions on AR coefficients Which Ensure Stationarity . . . . . . 4.3.1 Restrictions on AR(1) Coefficients . . . . . . . . . . . . . . . . . . . . . 4.3.2 Restrictions on AR(2) Coefficients . . . . . . . . . . . . . . . . . . . . . 4.3.3 Restrictions on AR(p) Coefficients . . . . . . . . . . . . . . . . . . . . . 4.3.4 Characteristic and Inverse Characteristic Equations . . . 4.3.5 Restrictions on ARIMA(p,q) Coefficients . . . . . . . . . . . . . . 4.4 The Connection Between AR and MA Processes . . . . . . . . . . . . . . . . . 4.4.1 AR(1) to MA(∞) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 AR(p) to MA(∞) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Invertibility: MA(1) to AR(∞). . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 What Are Unit Roots, and Why Are They Bad? . . . . . . . . . . . . . . . . . .

81 81 82 83 83 84 92 93 94 95 95 97 97 99

5

Non-stationarity and ARIMA(p,d,q) Processes . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Differencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Example of Differencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 The Mean and Variance of the Random Walk . . . . . . . . . . 5.2.2 Taking the First Difference Makes it Stationary. . . . . . . . 5.3 The Random Walk with Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 The Mean and Variance of the Random Walk with Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Taking the First Difference Makes it Stationary. . . . . . . . 5.4 Deterministic Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Mean and Variance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 First Differencing Introduces an MA Unit Root . . . . . . . 5.5 Random Walk with Drift vs Deterministic Trend . . . . . . . . . . . . . . . . . 5.6 Differencing and Detrending Appropriately . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Mistakenly Differencing (Overdifferencing) . . . . . . . . . . . 5.6.2 Mistakenly Detrending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 102 104 104 105 106 106 107 107 107 107 108 109 113 116

Contents

xi

5.7 5.8

Replicating Granger and Newbold (1974) . . . . . . . . . . . . . . . . . . . . . . . . . 117 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6

Seasonal ARMA(p,q) Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Different Types of Seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Deterministic Seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Seasonal Differencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Additive Seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Multiplicative Seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5 MA Seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Invertibility and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 How Common are Seasonal Unit Roots? . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Using De-seasonalized Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 125 126 127 128 131 133 135 135 136 137

7

Unit Root Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Unit Root Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Dickey-Fuller Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 A Random Walk vs a Zero-Mean AR(1) Process . . . . . . 7.3.2 A Random Walk vs an AR(1) Model with a Constant . 7.3.3 A Random Walk with Drift vs a Deterministic Trend. . 7.3.4 Augmented Dickey-Fuller Tests . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 DF-GLS Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.6 Choosing the Lag Length in DF-Type Tests . . . . . . . . . . . . 7.4 Phillips-Perron Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 KPSS Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Nelson and Plosser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Testing for Seasonal Unit Roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Conclusion and Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 139 140 141 142 146 148 150 152 153 156 158 160 168 169

8

Structural Breaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Structural Breaks and Unit Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Perron (1989): Tests for a Unit Root with a Known Structural Break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Zivot and Andrews’ Test of a Break at an Unknown Date . . . . . . . . 8.3.1 Replicating Zivot and Andrews (1992) in Stata . . . . . . . . 8.3.2 The zandrews Command . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 171 173 184 185 191 194

ARCH, GARCH and Time-Varying Variance. . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Conditional vs Unconditional Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 ARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 ARCH(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...