Forecasting with Univariate Box Jenkins Models Concepts and Cases

ALAN PANKRATZ

CONTENTS
PART I. BASIC CONCEPTS
1 Overview
Planning and forecasting 3
What this book is about 4
Time-series data 6
Single-series (univariate) analysis 8
When may UBJ models be used? 9
The Box-Jenkins modeling procedure 16
UBJ models compared with other models
Summary 20
Questions and problems 21
Introduction to Box-Jenkins analysis of a single data series 24
2.1 Differencing 24
2.2 Deviations from the mean 29
2.3 Two analytical tools: the estimated autocorrelation function (acf) and estimated partial autocorrelation function (pacf) 29
Summary 43
Questions and problems 44
3 Underlying statistical principles 45
3.1 Process, realization, and model 45
3.2 Two common processes 47
33 Statistical inference at the identification stage 67
Summary 74
Appendix 3 A Expected value rules and definitions 76
Questions and problems 77
4 An introduction to the practice of ARIMA modeling
4.1 What is a good model? 80
4.2 Two examples of UBJ-ARIMA modeling 83
Summary 91
Questions and problems 92
5 Notation and the interpretation of ARIMA models
5.1 Three processes and ARIMA( p , d, q ) notation 95
5.2 Backshift notation 96
5.3 Interpreting ARIMA models I: optimal extrapolation
of past values of a single series 103
5.4 Interpreting ARIMA models 11: rationalizing them
from their context 105
5.5 Interpreting ARIMA models 111: ARIMA(0, d, q )
models as exponentially weighted moving
averages 109
Summary 115
Questions and problems 117
6 Identification: stationary models
6.1 Theoretical acfs and pacfs for five common
processes 121
6.2 Stationarity 130
63 Invertibility 133
6.4 Deriving theoretical acf s for the MA( 1) process 136
6.5 Deriving theoretical acf s for the AR( 1) process 142
Summary 148
and invertibility 150
and forecast performance 152
Questions and problems 152
Appendix 6A: The formal conditions for stationarity
Appendix 6B Invertibility, uniqueness,
7 Identification: nonstationary models
7.1 Nonstationary mean 155
7.2 Nonstationary variance 175
7.3 Differencing and deterministic trends 186
Summary 189
Appendix 7A: Integration 190
8 Estimation
8.1 Principles of estimation 192
8.2 Nonlinear least-squares estimation 197
8.3 Estimation-stage results: have we found a good model? 200
Summary 208
Appendix 8A Marquardt’s compromise 209
8A.1 Overview 210
8A.2 Application to an MA(1) 212
Appendix 8B Backcasting 220
8B. 1 Conditional least squares 220
8B.2 Unconditional least squares 22 1
9 Diagnostic checking
9.1 Are the random shocks independent? 224
9.2 Other diagnostic checks 230
9.3 Reformulating a model 233
Summary 237
Questions and problems 238
10 Forecasting
10.1 The algebra of ARIMA forecasts 241
10.2 The dispersion of ARIMA forecasts 252
10.3 Forecasting from data in logarithmic form 256
10.4 The optimality of ARIMA forecasts 258
Summary 259
Appendix 1OA The complementarity of ARIMA
models and econometric models 261
Questions and problems 263
11 Seasonal and other periodic models
11.1 Periodic data 268
11.2 Theoretical acf s and pacf s for seasonal processes 270
11.3 Seasonal differencing 274
11.4 Seasonal-nonseasonal multiplicative models 280
11.5 An example of a seasonal-nonseasonal multiplicative
model 282
11.6 Nonmultiplicative models 288
Summary 292
Questions and problems 293
PART 11. THE ART OF ARIMA MODELING
12 pnrcticaldes
Case Studies: Introduction
GROUP A STATIONARY, NONSEASONAL MODELS
Case 1
Case 2 Saving rate
Case3 Coalproduction
Case 4 Housing permits
Change in business inventories
GROUP B NONSTATIONARY,
NONSEASONAL MODELS
Case 5 Rail freight
Case 6 AT&T stock price
Case 7 Real-estate loans
Case 8 Parts availability
GROUP C NONSTATIONARY, SEASONAL MODELS
Case 9 Air-carrier freight
Case 10 Profit margin
Case 11 Boston armed robberies
Case 12 Machine-tool shipments
Case 13 Cigarconsumption
Case 14 College enrollment
Case15 Exports
zdistribution table
Chi-squared distribution table
References
Index

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