Benchmarking, Temporal Distribution, And Reconciliation Methods for Time Series

by Dagum, Estela Bee; Cholette, Pierre A.

• ISBN13: 9780387311029
• ISBN10: 0387311025
• Format: Paperback
• Copyright: 2006-05-30
• Publisher: Springer Verlag

Summary

In modern economies, time series play a crucial role at all levels of activity. They are used by decision makers to plan for a better future, by governments to promote prosperity, by central banks to control inflation, by unions to bargain for higher wages, by hospital, school boards, manufacturers, builders, transportation companies, and by consumers in general.A common misconception is that time series data originate from the direct and straightforward compilations of survey data, censuses, and administrative records. On the contrary, before publication time series are subject to statistical adjustments intended to facilitate analysis, increase efficiency, reduce bias, replace missing values, correct errors, and satisfy cross-sectional additivity constraints. Some of the most common adjustments are benchmarking, interpolation, temporal distribution, calendarization, and reconciliation.This book discusses the statistical methods most often applied for such adjustments, ranging from ad hoc procedures to regression-based models. The latter are emphasized, because of their clarity, ease of application, and superior results. Each topic is illustrated with many real case examples. In order to facilitate understanding of their properties and limitations of the methods discussed, a real data example, the Canada Total Retail Trade Series, is followed throughout the book.This book brings together the scattered literature on these topics and presents them using a consistent notation and a unifying view. The book will promote better procedures by large producers of time series, e.g. statistical agencies and central banks. Furthermore, knowing what adjustments are made to the data and what technique is used and how they affect the trend, the business cycles and seasonality of the series, will enable users to perform better modeling, prediction, analysis and planning.This book will prove useful to graduate students and final year undergraduate students of time series and econometrics, as well as researchers and practitioners in government institutions and business.

Table of Contents

1 Introduction	1	(14)
1.1 Benchmarking	2	(3)
1.2 Interpolation and Temporal Distribution	5	(2)
1.3 Signal Extraction and Benchmarking	7	(1)
1.4 A Unified View	8	(1)
1.5 Calendarization	8	(2)
1.6 Data Requirements for Benchmarking and Calendarization	10	(1)
1.7 Reconciliation or Balancing Systems of Time Series	11	(1)
1.8 Book Outline	12	(3)
2 The Components of Time Series	15	(36)
2.1 Introduction	15	(1)
2.2 Time Series Decomposition Models	16	(4)
2.3 The Secular or Long-Term Trend	20	(5)
2.3.1 Deterministic Trend Models	21	(3)
2.3.2 Stochastic Trends	24	(1)
2.4 The Business Cycle	25	(5)
2.4.1 Deterministic and Stochastic Models for the Business Cycle	26	(1)
2.4.2 Limitations of Same-Month Comparisons	27	(3)
2.5 Seasonality	30	(5)
2.5.1 The Causes and Costs of Seasonality	30	(3)
2.5.2 Models for Seasonality	33	(2)
2.6 The Moving-Holiday Component	35	(4)
2.7 The Trading-Day Component	39	(6)
2.7.1 Causes and Costs of Daily Patterns of Activity	39	(3)
2.7.2 Models for Trading-Day Variations	42	(1)
2.7.3 Sunday Opening of Stores	43	(2)
2.8 The Irregular Component	45	(6)
2.8.1 Redistribution Outliers and Strikes	46	(1)
2.8.2 Models for the Irregular Component and Outliers	47	(4)
3 The Cholette-Dagum Regression-Based Benchmarking Method - The Additive Model	51	(34)
3.1 Introduction	51	(6)
3.2 Simple Benchmarking Methods	57	(3)
3.3 The Additive Benchmarking Model	60	(5)
3.4 A Conservative Specification of Deterministic Trends	65	(3)
3.5 Flow, Stock and Index Series	68	(1)
3.6 Matrix Representation of the Model	69	(6)
3.6.1 Temporal Sum Operators	70	(1)
3.6.2 Solution of the Model	71	(4)
3.7 Other Properties of the Regression-Based Benchmarking Method	75	(5)
3.8 Proportional Benchmarking with the Regression-Based Model	80	(2)
3.9 A Real Data Example: the Canadian Total Retail Trade Series	82	(3)
4 Covariance Matrices for Benchmarking and Reconciliation Methods	85	(28)
4.1 Introduction	85	(2)
4.2 Minimization of an Objective Function	87	(6)
4.3 Weak versus Strong Movement Preservation	93	(10)
4.4 Weak versus Strong Proportional Movement Preservation	103	(1)
4.5 Minimizing the Size of the Corrections	104	(1)
4.6 Other ARMA Error Models and Movement Preservation	105	(5)
4.7 Guidelines on the Selection of Sub-Annual Error Models	110	(1)
4.8 The Covariance Matrix of the Benchmarks	111	(2)
5 The Cholette-Dagum Regression-Based Benchmarking Method - The Multiplicative Model	113	(22)
5.1 Introduction	113	(1)
5.2 The Multiplicative Benchmarking Model	114	(3)
5.3 Matrix Representation	117	(1)
5.4 Non-Linear Estimation of the Multiplicative Model	118	(4)
5.5 Other Properties of the Regression-Based Multiplicative Benchmarking Model	122	(6)
5.6 A Real Data Example: The Canadian Total Retail Trade Series	128	(7)
6 The Denton Method and its Variants	135	(24)
6.1 Introduction	135	(1)
6.2 The Original and Modified Additive First Difference Variants of the Denton Method	136	(10)
6.2.1 Preserving Continuity with Previous Benchmarked Values	141	(2)
6.2.2 Approximation of the Original and Modified Denton Variants by the Additive Regression-Based Model	143	(2)
6.2.3 Preferred Variant of Movement Preservation	145	(1)
6.3 The Proportional First Difference Variants of the Denton Method	146	(7)
6.3.1 Approximation of the Original and Modified Proportional Variants by the Additive Regression-Based Model	149	(1)
6.3.2 Preferred Variant of Proportional Movement Preservation	150	(3)
6.4 Other Variants of the Denton Method	153	(6)
6.4.1 The Additive Second Difference Variants	154	(3)
6.4.2 The Proportional Second Different Variants	157	(2)
7 Temporal Distribution, Interpolation and Extrapolation	159	(34)
7.1 Introduction	159	(3)
7.2 Ad Hoc Interpolation and Distribution Methods	162	(3)
7.3 Interpolation and Temporal Distribution Based on Regression Methods	165	(9)
7.4 The Chow-Lin Regression-Based Method and Dynamic Extensions	174	(4)
7.5 ARIMA Interpolation, Temporal Distribution and Extrapolation	178	(9)
7.5.1 Trend Stationary Models	179	(1)
7.5.2 Difference Stationary Models	180	(5)
7.5.3 The Stram and Wei Approach	185	(2)
7.6 Combining Sub-Annual and Annual Forecasts	187	(6)
8 Signal Extraction and Benchmarking	193	(16)
8.1 Introduction	193	(2)
8.2 ARIMA Model-Based Signal Extraction and Benchmarking: The Hillmer and Trabelsi Method	195	(4)
8.3 State Space Signal Extraction and Benchmarking: The Durbin and Quenneville Method	199	(7)
8.3.1 State-Space Model for Signal Extraction	199	(3)
8.3.2 Two-Stage Benchmarking: the Additive Model	202	(1)
8.3.3 Single-Stage Benchmarking: Additive Model	203	(3)
8.4 Non-Parametric Signal Extraction and Benchmarking: The Chen, Cholette and Dagum Method	206	(3)
9 Calendarization	209	(26)
9.1 Introduction	209	(2)
9.2 The Assignment Calendarization Procedure	211	(6)
9.3 The Fractional Calendarization Method and its Variants	217	(2)
9.4 Model-Based Calendarization Methods	219	(9)
9.4.1 Denton-Based Methods	219	(4)
9.4.2 Regression-Based Method	223	(5)
9.5 Calendarizing Multi-Weekly Data Covering 4 or 5 Weeks	228	(6)
9.6 Calendarizing Payroll Deductions	234	(1)
10 A Unified Regression-Based Framework for Signal Extraction, Benchmarking and Interpolation	235	(28)
10.1 Introduction	235	(1)
10.2 The Generalized Dynamic Stochastic Regression Model	235	(4)
10.3 Signal Extraction	239	(2)
10.4 Benchmarking With and Without Signal Extraction	241	(2)
10.5 Interpolation, Temporal Distribution and Extrapolation	243	(3)
10.6 Multiplicative Models for Signal Extraction and Benchmarking	246	(4)
10.7 A Real Case Example: the Canadian Total Retail Trade Series	250	(13)
10.7.1 Method 1: Multiplicative Benchmarking Without Signal Extraction	250	(6)
10.7.2 Method 2: Benchmarking with Signal Extraction, Interpolation and Extrapolation	256	(7)
11 Reconciliation and Balancing Systems of Time Series	263	(22)
11.1 Introduction	263	(6)
11.2 General Regression-Based Reconciliation Method	269	(3)
11.3 Choosing the Covariance Matrices	272	(5)
11.4 Data Problems	277	(3)
11.5 Strategies for Reconciliation	280	(5)
12 Reconciling One-Way Classified Systems of Time Series	285	(24)
12.1 Introduction	285	(1)
12.2 The Reconciliation Model for One-Way Classified Systems of Series	286	(4)
12.3 Implementation of the Analytical Solution	290	(3)
12.4 Redundant Constraints in the One-Way Reconciliation Model	293	(1)
12.5 An Example of One-Way Reconciliation	294	(9)
12.6 A Real Data Example: One-Way Reconciliation Model of The Seasonally Adjusted Canadian Retail Trade Series	303	(6)
13 Reconciling the Marginal Totals of Two-Way Classified Systems of Series	309	(28)
13.1 Introduction	309	(2)
13.2 The Marginal Two-Way Reconciliation Model	311	(8)
13.2.1 Deriving an Analytical Solution in Terms of the Main Partitions	313	(1)
13.2.2 Deriving an Analytical Solution for Each Series	314	(3)
13.2.3 General Analytical Solution of the Marginal Two-Way Reconciliation Model	317	(2)
13.3 Implementation of the Analytical Solution	319	(3)
13.4 Redundant Constraints	322	(2)
13.5 A Real Data Example: the Seasonally Adjusted Canadian Retail Trade Series	324	(13)
13.5.1 The Indirect Seasonal Adjustment	326	(4)
13.5.2 The Direct Seasonal Adjustment	330	(7)
14 Reconciling Two-Way Classifed Systems of Series	337	(26)
14.1 Introduction	337	(1)
14.2 The Reconciliation Model for Two-Way Classified Systems of Time Series	338	(9)
14.2.1 Deriving an Analytical Solution in Terms of the Main Partitions	341	(1)
14.2.2 Deriving an Analytical Solution for Each Series	342	(4)
14.2.3 Analytical Solution of the Two-Way Reconciliation Model	346	(1)
14.3 Particular Cases of the Two-Way Reconciliation Model	347	(3)
14.3.1 The One-Way Model as a Particular Case	347	(1)
14.3.2 The Marginal Two-Way Model as a Particular Case	347	(1)
14.3.3 The Two-Way Model Without the Grand Total	348	(2)
14.4 Input-Output Models	350	(3)
14.5 Implementation of the Two-Way Reconciliation Model	353	(5)
14.6 Redundant Constraints in the Two-Way Reconciliation Model	358	(1)
14.7 A Real Data Example of a Large Two-Way System of Series	359	(4)
Appendix A: Extended Gauss-Markov Theorem	363	(6)
Appendix B: An Alternative Solution for the Cholette-Dagum Model for Binding Benchmarks	369	(4)
Appendix C: Formulae for Some Recurring Matrix Products	373	(4)
Appendix D: Seasonal Regressors	377	(10)
Appendix E: Trading-Day Regressors	387	(6)
References	393	(10)
Index	403