Forecasting Aggregated Vector ARMA Processes

335 Pages·1987·8.43 MB·English

by  Prof.Dr. Helmut Lütkepohl (auth.)

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continuation on page 325
Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and W. Krelle
284
Helmut Lutkepohl
Forecasting Aggregated
Vector ARMA Processes
Springer-Verlag
Berlin Heidelberg New York London Paris Tokyo
Editorial Board
H.A1bach M.Beckmann (Managing Editor)
P. Ohrymes G. Fandel J. Green W. Hildenbrand W. Krelle (Managing Editor)
H. P. Ki..inzi K. Ritter RSato U. Schittko P. Schonfeld R. Setten
Managing Editors
Prof. Dr. M. Beckmann
Brown University
Providence, RI 02912, USA
w.
Prof. Dr. KreUe
InsliM fUr Gesellschahs-und Wirtschahswissenschahen
der Universitlit Bonn
.A.d.e..n.a uerallee 24-42, D-5300 Bonn, FRG
Prof. Dr. Helmut li..itkepohl
Universitiit Hamburg
lostitut fUr Slatistik und Okonometrie
Von-Melle-Park 5, 0·2000 Hamburg 13, FRG
ISBN·13: 978·3·540·17208·6 e-ISBN·13: 978-3·642-61584-9
001: 10.10071978-3-642-61584-9
This WOI'k is subject to copyright. All righla are.---:l, whether the whole or part of the material
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§ ~ 01 the German Copyright u w whttru copies are made lor other than private \1&8, _ leu is
pIIy_boIe to -Vllrwertungsg&eellschaft Wort", Munich.
C Sprioger-Ver1ag Bertin Heidelberg 1967
2142/3140-543210
To
Sab,[ne.
PREFACE
This study is concerned with forecasting time series variables and the
impact of the level of aggregation on the efficiency of the forecasts.
Since temporally and contemporaneously disaggregated data at various
levels have become available for many countries, regions, and variables
during the last decades the question which data and procedures to use
for prediction has become increasingly important in recent years. This
study aims at pointing out some of the problems involved and at pro
viding some suggestions how to proceed in particular situations.
Many of the results have been circulated as working papers, some
have been published as journal articles, and some have been presented
at conferences and in seminars. I express my gratitude to all those who
have commented on parts of this study. They are too numerous to be
listed here and many of them are anonymous referees and are therefore
unknown to me. Some early results related to the present study are
contained in my monograph "Prognose aggregierter Zeitreihen" (Lutkepohl
(1986a)) which was essentially completed in 1983. The present study
contains major extensions of that research and also summarizes the
earlier results to the extent they are of interest in the context of
this study.
This monograph is based on research I have carried out while I
have been employed by the University of Bielefeld (1979-B1), the Uni
versity of Osnabruck (1981-84), the University of California, San Diego
(1984/85), and the University of Hamburg (since 1985). I gratefully
acknowledge the technical support by these institutions and many useful
discussions and comments by my colleagues at these institutions.
Sabine Dornbusch has performed the computations reported in this
study. I take this opportunity to thank her for outstanding research
assistance during the last few years. Last not least I am most grateful
to Mrs. Marlene Hakelberg for typing the final manuscript with patience
and skill.
November 1986 Helmut Lutkepohl
CONTENTS
CHAPTER 1. PROLOGUE
1.1 Objective of the Study ••••••••••••.••••••••••••••••• 1
1.2 Survey of the Study ••••.•••••••••••••••.•••••••.•••• 3
CHAPTER 2. VECTOR STOCHASTIC PROCESSES ••••••••••••••••.••.•••• 6
2.1 Discrete-Time, Stationary Vector Stochastic Processes 6
2.1.1 General Assumptions ••.••••.••••••••••..••••••• 6
2.1.2 The Wold or Moving Average Representation ••••• 8
2.1.3 Autoregressive Representation ••••••••••••••••• 10
2.1.4 Spectral Representation ••••••••••••••••••••••• 12
2.2 Nonstationary Processes ••••••••••••••••••••••••••••• 14
2.3 Vector Autoregressive Moving Average Processes •••••• 16
2.3.1 Stationary Processes •••••••••••••••.•.•••..••• 16
2.3.2 Nonstationary Processes ••••••••••••.•••••••••• 20
2.4 Estimation. • • • • • • • • • • • . • • • • • • • • • • • • • • • • • • • • . . • • • • • • • 21
2.4.1 Maximum Likelihood Estimation of Stationary
Gaussian Vector ARMA Processes of Known Order.. 21
2.4.2 Estimation of Vector Autoregressive Processes
of Known Order •••••••••••••.••••••••••••••.••• 23
2.4.3 Multivariate Least Squares Estimation of AR
Processes With Unknown Order •••••••••.•••••••. 32
2.4.4 Nonstationary Processes •••.•••••••••••.••••.•• 36
2.5 Model Specification •••.•••••••••.•••••.••.••.••••••• 36
2.5.1 AR Order Determination •••••••••••••••••••••••• 36
2.5.2 Subset Autoregressions ••••.••••••••••..••.•••• 41
2.5.3 The Box-Jenkins Approach •••••••••••••••••••••• 47
2.6 Summary ••••••••••••••••.•••••••••••••••••••••••••.•• 47
CHAPTER 3. FORECASTING VECTOR STOCHASTIC PROCESSES ••••••••.••• 49
3.1 Forecasting Known Processes ••••••••••••••••••••••••. 50
3.1.1 Predictors Based on the Moving Average
Representation •••••••••••••••••••••••••••••••• 50
3.1.2 Predictors Based on the Autoregressive
Representation ••••••••••••••••••••••••••••.••• 54
3.1.3 Forecasting Known Vector ARMA Processes ••.•••• 57
3.2 Forecasting Vector ARMA Processes with Estimated
Coefficients •••••.••••.•.••••••••••••••••.•••••••••• 58
3.2.1 The General Case •.•••••••••••••.••••••••••••.• 58
3.2.2 Finite Order AR Processes ••••••••••••••••••••• 65
VIII
3.3 Forecasting Autoregressive Processes of Unknown Order 73
3.3.1 The Asymptotic MSE Matrix •••..••••••.••••••••• 74
3.3.2 Proof of Proposition 3.2 ..••••.•.••••..•••.•.• 77
3.4 Forecasting Nonstationary Processes ••••..••••••••.•• 85
3.4.1 Known Processes .•••.•••••.••.••••.•••••••.•.•• 86
3.4.2 Estimated Coefficients •.••••••.••.••••••••••.• 92
3.4.3 Unknown Order ...••••.•....•••••.•••.••••••.••• 94
3.5 Comparing Forecasts .•.•..••••••••••.••••••••••••.•.• 94
3.6 Summary. • . • • • • • • • • . . . . . . • . • . • • . • • . . • • • . • • • • •• • • • . • . • 95
CHAPTER 4. FORECASTING CONTEMPORANEOUSLY AGGREGATED KNOWN
PROCESSES ••.••..•.••.......•..•••.•..••••.••....••. 97
4.1 Linear Transformations of vector Stochastic Processes 98
4.2 Forecasting Linearly Transformed Stationary vector
Stochastic Processes •.••••••••••••.•.•.•.••.•••••••• 100
4.2.1 The Predictors •.••.•••••••••••••.•••••••..•.•• 100
4.2.2 Comparison of the Predictors ••...•••••.••••.•• 104
4.2.3 Equality of the Predictors •••••••.•••••.••••.• 105
4.2.4 Granger-Causality •••••.•••••••••••••.•••••.••. 108
4.3 Forecasting Linearly Transformed Nonstationary
Processes •••••.•.•.••.•••.••..••.••.••.•.••.•••.••.. 110
4.4 Linearly Transformed vector ARMA Processes ••.•••.••. 113
4.4.1 Finite Order MA Processes ••••.•••.•••.••••.... 113
4.4.2 ARMA Processes ..•.••.•.•••••••.••••••.•••••••• 114
4.5 Summary and Comments •.••.•••.••••.••••.•.•.•.••••.•• 117
CHAPTER 5. FORECASTING CONTEMPORANEOUSLY AGGREGATED ESTIMATED
PROCESSES ••••.••.•..••••.••••••••••••••••••.•••.••. 119
5.1 Summary of Assumptions and Predictors •••••.••••••••. 120
5.2 Estimated Coefficients •••.••••.••..•••.••.•..•..•••• 121
5.2.1 Comparison of YAOt (h) and YA t(h) •••••••.••••.•.•. 123
5.2.2 Compar~. son of yAOt( h) and YAUt (h) ••.••••••.••••••. 127
5.3 Unknown Orders and Estimated Coefficients ••••••.•.•• 129
5.4 Nonstationary Processes .•••••••••••••.•••••••••••••• 132
5.5 Small Sample Results ....•.••••.•.•••••.•••...••••.•• 133
5.5.1 Design of the Monte Carlo Experiment ..••••.••• 133
5.5.2 Simulation Results for AR Process I •.•.•••.••. 134
5.5.3 Simulation Results for AR Process II .•.•••.••. 139
5.5.4 Simulation Results for MA Process I ••••••••••• 142
5.5.5 Simulation Results for MA Process II •.•••••••• 145
5.5.6 Simulation Results for MA Process III .•••••.•• 147
5.6 An Empirical Example ••••••••..••••••••••••.••••••••. 149
5.7 Conclusions. • . . • • . • . . . . • • • • • . . . • . • . • . • • • • • • • . • • • • . •• 158
IX
CHAPTER 6. FORECASTING TEMPORALLY AND CONTEMPORANEOUSLY
AGGREGATED KNOWN PROCESSES ...•....•.••.........•••• 161
6.1 Macro Processes ..•...•....•.•••••..••.•..•...••....• 163
6.2 Six Predictors •..••.••••..•••••••.•.•..•.•..•.•..••. 166
6.3 Comparison of Predictors .•.••..•.•.••...•....••.•••. 173
6.4 Nonstationary Processes •..•..•.•...•••..••..•.•....• 179
6.4.1 Differencing to Obtain Stationarity ...•..•.... 179
6.4.2 Forecasting Aggregated Nonstationary Processes 181
6.5 Temporally and Contemporaneously Aggregated vector
ARMA Processes ••......•••.•.•......••....•..•.•....• 182
6.6 Conclusions and Comments ...•..•...•......••.....••.. 186
CHAPTER 7. TEMPORAL AGGREGATION OF STOCK VARIABLES -
SYSTEMATICALLY MISSING OBSERVATIONS ••...••...••...• 187
7. 1 Forecasting Known Processes With Systematically
Missing Observations ••...•••..••.......•.•.•••..•.•• 189
7.2 Processes With Estimated Coefficients .•..•.•••••..•• 192
7.3 Processes With Unknown Orders and Estimated
Coefficients •..•..•.••.•...•..•...•.•.•.••...••..... 198
7.4 Nonstationary Time Series With Systematically Missing
Observations •..•....•.....•.........•.•......•.....• 200
7.5 Monte Carlo Results ..•.•.••.•.••........••.•....•..• 203
7.5.1 Univariate AR Processes •.....•.•.••.•.•••.•..• 203
7.5.2 Bivariate AR Process ........•.••....•.•....•.• 206
7.5.3 MA(m) Processes ••.•.••...•..•.•....•....•.••.• 208
7.5.4 Univariate MA(l) Process ••........•.•..••...•. 211
7.5.5 Summary of Small Sample Results .••.••••....•.• 215
7.6 Empirical Examples ••..•...•...••....••...•......•.•• 215
7.6.1 Consumption Expenditures ••....•.•............• 216
7.6.2 Investment ..........••..••.•....•.••....•....• 219
7.7 Concluding Remarks •••••..••.••••••••••...••••••••••• 220
7.A Appendix: Proof of Relation (7.2.18) •••.•....•....•. 222
CHAPTER 8. TEMPORAL AGGREGATION OF FLOW VARIABLES •..•••••••••• 224
8.1 Forecasting With Known Processes •...•..•.••....•.••. 225
8.2 Forecasts Based on Processes With Estimated
Coefficients •.•.•••••••.••.•.....•••.•.••..•..•..•.. 230
8.3 Forecasting With Autoregressive Processes of
Unknown Order •.•..••..•.•....•...••....•••...•.•.••. 236
8.4 Temporally Aggregated Nonstationary Processes ......• 238
8.5 Small Sample Comparison .•....•.....•.•.•...••.•..•.. 240
8.5.1 A Univariate AR Process .•.••••••.•••••••••.••. 240
8.5.2 A Univariate MA(2) Process .••.••.•••.••••••••. 242
8.5.3 A Univariate 1~(3) Process •.•..••.•.•.••...••. 244
8.5.4 A Bivariate MA Process .•....•...••.....••••••• 246
8.5.5 A System With a Stock and a Flow Variable .•••. 248
x
8.6 Examples. . • • • . • • • • • . • . • . . • • • • • • • • . • • • • . • • • • • • • . . • • •• 250
8.6.1 Consumption •......•..••......•.•.•.....•...•.. 250
8.6.2 Investment •...••...•..•••.••.•..•.•••.••...•.. 253
8.7 Summary and Conclusions •.•.•••••••••.•....•.•...•.•• 255
8.A Appendix: Proof of Relation (8.2.23) .••••.•..••.•.•. 256
CHAPTER 9. JOINT TEMPORAL AND CONTEMPORANEOUS AGGREGATION ..•.. 258
9. 1 Summary of Processes and Predictors ••..•.••..••..•.. 259
9.2 Prediction Based on Processes With Estimated
Coefficients •.••.•.••••.••.•.••.•.•••..•••••••...... 263
9.2.1 General Results .•..•.•.•.•..•...•••.•••.•.••.• 263
9.2.2 An Example •••••••..••••••••.••••.••.•••.•••... 267
9.2.3 Conclusions for Processes With Estimated
Coefficients ..•.•..•••.••.••••.•.••.•.•....... 274
9.3 Prediction Based on Estimated Processes With
Unknown Order s •••.•.....••.•••..•.••...•..••....•••. 275
9.3.1 General Comments ••..••.••.••••••••.••.•.•..•.• 275
9.3.2 Comparison of MSEs •••..••.•.••.••••••.••••••.• 278
9.3.3 Summary and Discussion of Results for Processes
With Unknown Orders •.•.•.••.••.••••••.•••.••.. 280
9.4 Monte Carlo Comparison of Predictors •••.••.•••••.••. 281
9.4.1 Simulation Results for AR Process •..•.•..••••. 282
9.4.2 Simulation Results for MA Process ••••.••.••... 289
9.4.3 Discussion of Small Sample Results ••••••••••.. 296
9.5 Forecasts of U.S. Gross Private Domestic Investment .• 299
9.5.1 First Differences of Investment Data ..••.••••. 299
9.5.2 Aggregation of Original Investment Data .•..•.. 303
9.6 Summary and Conclusions ...•..••••.•••.••••.••....••• 304
CHAPTER 1 0 . EPILOGUE. • . • • • • • . • • . . • • • . • • • • . . • • • • • . . • • • . • • . • . . • .• 309
10.1 Summary and Conclusions 309
10.2 Some Remaining Problems 311
APPENDIX. DATA USED FOR EXAMPLES ....•.•••.••••.••.••.•..•.••. 313
BIBLIOGRAPHY •.•••••••.••.••..•.•••••••.•••••••••.•••••.••••••••• 316
CHAPTER 1. PROLOGUE
1.1 OBJECTIVE OF THE STUDY
In making choices between alternative courses of action, decision mak
ers at all structural levels often need predictions of aggregated var
iables. For example, in the process of planning a government budget,
forecasts of annual tax revenues may be required. If quarterly or month
ly figures of previous revenues are available then a time series model
may be constructed for the generation process of the quarterly or month
ly data. This model can then be used to obtain predictions for the next
quarters or months and these forecasts can be aggregated to obtain an
nual forecasts of the tax revenues. Alternatively, the available month
ly or quarterly data may be aggregated to obtain an annual series of
tax revenues. Based on this series a model may be constructed to gen
erate annual forecasts.
As a second example consider the manager of a retail firm with sev
eral branches who wants to place an order of a certain product to stock
in for the next week. To minimize the storage space and at the same time
minimize the risk of running out of stock, he or she needs a precise
forecast of the demand in all branches during the next week. In other
words, the manager will be interested in a forecast of a variable that
is aggregated over time (temporally), adding up the sales of all days
during the week, and over units (contemporaneously), summing up the
sales for the individual branches. One possibility is to obtain the de
sired forecast by constructing and estimating a univariate time series
model for the total sales data of the past weeks and use this model for
generating a forecast. Alternatively, rather than using contemporane
ously and temporally aggregated weekly total sales figures, a multiple
time series model could be constructed for the disaggregate daily sales
data of all separate branches. Such a model can be used to produce daily
sales forecasts for the individual branches. The obtained forecasts can
then be aggregated to give a total sales forecast for the next week.
There are some other predictors that are relevant in the present con
text. For instance, the total daily sales, aggregated over all branches,
can be predicted using univariate time series methods. A weekly fore-