Table Of ContentForecasting with
Dynamic Regression Models
Forecasting with
Dynamic Regression Models
ALAN PANKRATZ
DePauw University
Greencastle, Indiana
Wiley-Interscience Publication
JOHN WILEY & SONS INC.
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Library of Congress Cataloging in Publication Data:
Pankratz, Alan, 1944-
Forecasting with dynamic regression models / Alan Pankratz.
p. cm. — (Wiley series in probability and mathematical
statistics. Applied probability and statistics, ISSN 0271-6356)
"A Wiley-Interscience publication."
Includes bibliographical references and index.
ISBN 0-471-61528-5
1. Time-series analysis. 2. Regression analysis. 3. Prediction
theory. I. Title. II. Series.
QA280.P368 1991
519.5'5-dc20 91-12484
CIP
10 9 8 7 6 5
To Aaron Mark Dietrich and
Sherith Hope Laura
Contents
Preface
Chapter 1 Introduction and Overview
1.1 Related Time Series, 1
1.2 Overview: Dynamic Regression Models, 7
1.3 Box and Jenkins' Modeling Strategy, 15
1.4 Correlation, 17
1.5 Layout of the Book, 21
Questions and Problems, 22
Chapter 2 A Primer on ARIMA Models
2.1 Introduction, 24
2.2 Stationary Variance and Mean, 27
2.3 Autocorrelation, 34
2.4 Five Stationary ARIMA Processes, 39
2.5 ARIMA Modeling in Practice, 49
2.6 Backshift Notation, 52
2.7 Seasonal Models, 54
2.8 Combined Nonseasonal and Seasonal Processes, 57
2.9 Forecasting, 59
2.10 Extended Autocorrelation Function, 62
2.11 Interpreting ARIMA Model Forecasts, 64
Questions and Problems, 69
viii CONTENTS
Case 1 Federal Government Receipts (ARIMA) 72
Chapter 3 A Primer on Regression Models 82
3.1 Two Types of Data, 82
3.2 The Population Regression Function (PRF) with One Input, 82
3.3 The Sample Regression Function (SRF) with One Input, 86
3.4 Properties of the Least-Squares Estimators, 88
3.5 Goodness of Fit (R2), 89
3.6 Statistical Inference, 92
3.7 Multiple Regression, 93
3.8 Selected Issues in Regression, 96
3.9 Application to Time Series Data, 103
Questions and Problems, 113
Case 2 Federal Government Receipts (Dynamic Regression) 115
Case 3 Kilowatt-Hours Used 131
Chapter 4 Rational Distributed Lag Models 147
4.1 Linear Distributed Lag Transfer Functions, 148
4.2 A Special Case: The Koyck Model, 150
4.3 Rational Distributed Lags, 156
4.4 The Complete Rational Form DR Model and Some
Special Cases, 163
Questions and Problems, 165
Chapter 5 Building Dynamic Regression Models: Model Identification 167
5.1 Overview, 167
5.2 Preliminary Modeling Steps, 168
5.3 The Linear Transfer Function (LTF) Identification Method, 173
5.4 Rules for Identifying Rational Distributed Lag Transfer
Functions, 184
Questions and Problems, 193
Appendix 5A The Corner Table, 194
Appendix 5B Transfer Function Identification Using Prewhitening
and Cross Correlations, 197
CONTENTS ix
Chapter 6 Building Dynamic Regression Models: Model Checking,
Reformulation, and Evaluation 202
6.1 Diagnostic Checking and Model Reformulation, 202
6.2 Evaluating Estimation Stage Results, 209
Questions and Problems, 215
Case 4 Housing Starts and Sales 217
Case 5 Industrial Production, Stock Prices, and Vendor Performance 232
Chapter 7 Intervention Analysis 253
7.1 Introduction, 253
7.2 Pulse Interventions, 254
7.3 Step Interventions, 259
7.4 Building Intervention Models, 264
7.5 Multiple and Compound Interventions, 272
Questions and Problems, 276
Case 6 Year-End Loading 279
Chapter 8 Intervention and Outlier Detection and Treatment 290
8.1 The Rationale for Intervention and Outlier Detection, 291
8.2 Models for Intervention and Outlier Detection, 292
8.3 Likelihood Ratio Criteria, 299
8.4 An Iterative Detection Procedure, 313
8.5 Application, 315
8.6 Detected Events Near the End of a Series, 319
Questions and Problems, 320
Appendix 8A BASIC Program to Detect AO, LS, and IO
Events, 321
Appendix 8B Specifying IO Events in the SCA System, 322
Chapter 9 Estimation and Forecasting 324
9.1 DR Model Estimation, 324
9.2 Forecasting, 328
Questions and Problems, 340
Appendix 9A A BASIC Routine for Computing the
Nonbiasing Factor in (9.2.24), 340
X CONTENTS
Chapter 10 Dynamic Regression Models in a Vector ARMA
Framework 342
10.1 Vector ARMA Processes, 342
10.2 The Vector AR (IT Weight) Form, 345
10.3 DR Models in VAR Form, 346
10.4 Feedback Check, 349
10.5 Check for Contemporaneous Relationship and
Dead Time, 354
Questions and Problems, 356
Appendix 357
Table A Student's / Distribution, 357
Table B x2 Critical Points, 359
Table C F Critical Points, 360
Data Appendix 362
References 376
Index 381
Preface
Single-equation regression models are one of the most widely used statistical
forecasting tools. Over the last two decades many ideas relevant to regression
forecasting have arisen in the time series literature, starting with the first
edition of Box and Jenkins' text, Time Series Analysis: Forecasting and
Control, Holden-Day (1976). Those who apply regression methods to time
series data without knowledge of these ideas may miss out on a better
understanding of their data, better models, and better forecasts.
This book is a companion volume to my earlier text, Forecasting with
Univariate Box-Jenkins Models: Concepts and Cases, Wiley (1983), where I
present the Box-Jenkins modeling strategy as applied to ARIMA models.
There is more to Box and Jenkins than ARIMA models. They also discuss
"combined transfer function-disturbance" models, or what I call dynamic
regression models. The purpose of the present text is to pull together recent
time series ideas in the Box-Jenkins tradition that are important for the
informed practice of single-equation regression forecasting. I pay special
attention to possible intertemporal (dynamic) patterns —distributed lag re-
sponses of the output series (dependent variable) to the input series (indepen-
dent variables), and the autocorrelation patterns of the regression disturb-
ance.
This book may be used as a main text for undergraduate and beginning
graduate courses in applied time series and forecasting in areas such as
economics, business, biology, political science, engineering, statistics, and
decision science. It may be used in advanced courses to supplement theoreti-
cal readings with applications. And it can serve as a guide to the construction
and use of regression forecasting models for practicing forecasters in business
and government.
Special features of this book include the following:
• Many step-by-step examples using real data, including cases with multiple
inputs, both stochastic and deterministic.
• Emphasis on model interpretation.
xi
xii PREFACE
• Emphasis on a model identification method that is easily applied with
multiple inputs.
■ Suggested practical rules for identifying rational form transfer functions.
• Emphasis on feedback checks as a preliminary modeling step.
• Careful development of an outlier and intervention detection method,
including a BASIC computing routine.
• A chapter linking dynamic regression models to vector ARMA models.
• Use of the extended autocorrelation function to identify ARIMA models.
While there are other books that cover some of these ideas, they tend to
cover them briefly, or at a fairly high level of abstraction, or with few
supporting applications. I present the theory at a low level, and I show how
these ideas may be applied to real data by means of six case studies, along
with other examples within the chapters. Several additional data sets are
provided in the Data Appendix. The examples and cases are drawn mainly
from economics and business, but the ideas illustrated are also applicable to
data from other disciplines.
Those who read the entire text will notice some repetition; I realize that
many readers will read only selected chapters and cases, and that most won't
already know the contents of this book before they read, so I have tried to
repeat some key ideas. I hope my readers will learn enough of the main ideas
and practices of dynamic regression forecasting to feel more comfortable and
be more insightful when doing their own data analysis, criticizing their own
procedures and results, moving ahead to more advanced literature and pract-
ices, and when moving back to less advanced methods.
I have successfully used a majority of the material in this book with
undergraduates, most of whom have had just one course in statistics. I
assume that readers have been exposed to the rudiments of probability,
hypothesis testing, and interval estimation. A background in regression meth-
ods is helpful but not required; Chapter 3 is an introduction to the fundamen-
tals of regression. I use matrix algebra in a few places, primarily Chapter
10, but a knowledge of matrix algebra is not required to understand the main
ideas in the text. For many readers the most difficult part of the text may
be the use of backshift notation. If you are one of those readers, to you I
say (1) it's only notation, (2) learning to use it is easier than learning to ride
a bicycle, and (3) it's immensely useful.
As you read this book, I hope you find parts that are interesting, informa-
tive, and helpful. When you do, you should be as grateful as I am to Gregory
Hudak, Brooks Elliott, and an anonymous reviewer who provided many
excellent comments on a draft of the manuscript. I am also grateful to Fran
Cappelletti for his perceptive comments on parts of the manuscript. I wish
I could also blame shortcomings and undetected errors on these people, but
I can't; I didn't take all of their advice, and any problems you encounter are
my responsibility. Various members of the DePauw University Computer
