Introduction to Time Series Analysis and Forecasting
Description
Bring the latest statistical tools to bear on predicting future variables and outcomes
A huge range of fields rely on forecasts of how certain variables and causal factors will affect future outcomes, from product sales to inflation rates to demographic changes. Time series analysis is the branch of applied statistics which generates forecasts, and its sophisticated use of time oriented data can vastly impact the quality of crucial predictions. The latest computing and statistical methodologies are constantly being sought to refine these predictions and increase the confidence with which important actors can rely on future outcomes.
Time Series Analysis and Forecasting presents a comprehensive overview of the methodologies required to produce these forecasts with the aid of time-oriented data sets. The potential applications for these techniques are nearly limitless, and this foundational volume has now been updated to reflect the most advanced tools. The result, more than ever, is an essential introduction to a core area of statistical analysis.
Readers of the third edition of Time Series Analysis and Forecasting will also find:
- Updates incorporating JMP, SAS, and R software, with new examples throughout
- Over 300 exercises and 50 programming algorithms that balance theory and practice
- Supplementary materials in the e-book including solutions to many problems, data sets, and brand-new explanatory videos covering the key concepts and examples from each chapter.
Time Series Analysis and Forecasting is ideal for graduate and advanced undergraduate courses in the areas of data science and analytics and forecasting and time series analysis. It is also an outstanding reference for practicing data scientists.
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Persons
Douglas C. Montgomery, PhD, is Regents Professor of Industrial Engineering and ASU Foundation Professor of Engineering at Arizona State University, USA. He holds a PhD in Engineering from Virginia Tech and has researched and published extensively on industrial statistics and experimental design.
Cheryl Jennings, PhD, is Associate Teaching Professor at Arizona State University. She has decades of industrial experience in manufacturing and financial services, and has taught undergraduate and graduate courses on modeling and analysis, performance management, process control, and related subjects.
Murat Kulahci, PhD, is Professor of Industrial Statistics at the Technical University of Denmark and Professor at the Luleå University of Technology, Sweden. He holds a PhD in Industrial Engineering from the University of Wisconsin, Madison. He has published widely on time series analysis, experimental design, process monitoring and related subjects.
Content
Preface xi
About the Companion Website xv
1 Introduction to Time Series Analysis and Forecasting 1
1.1 The Nature and Uses of Forecasts 1
1.2 Some Examples of Time Series 9
1.3 The Forecasting Process 16
1.4 Data for Forecasting 19
1.4.1 The Data Warehouse 19
1.4.2 Data Wrangling and Cleaning 21
1.4.3 Imputation 22
1.5 Resources for Forecasting 23
Exercises 24
2 Statistics Background for Time Series Analysis and Forecasting 27
2.1 Introduction 27
2.2 Graphical Displays 28
2.2.1 Time Series Plots 28
2.2.2 Plotting Smoothed Data 32
2.3 Numerical Description of Time Series Data 37
2.3.1 Stationary Time Series 37
2.3.2 Autocovariance and Autocorrelation Functions 39
2.3.3 The Variogram 45
2.4 Use of Data Transformations and Adjustments 49
2.4.1 Transformations 49
2.4.2 Trend and Seasonal Adjustments 51
2.5 General Approach to Time Series Modeling and Forecasting 65
2.6 Evaluating and Monitoring Forecasting Model Performance 69
2.6.1 Forecasting Model Evaluation 69
2.6.2 Choosing Between Competing Models 78
2.6.3 Monitoring a Forecasting Model 81
2.7 R Commands for Chapter 2 89
Exercises 101
3 Regression Analysis and Forecasting 115
3.1 Introduction 115
3.2 Least Squares Estimation in Linear Regression Models 118
3.3 Statistical Inference in Linear Regression 127
3.3.1 Test for Significance of Regression 128
3.3.2 Tests on Individual Regression Coefficients and Groups of Coefficients 131
3.3.3 Confidence Intervals on Individual Regression Coefficients 137
3.3.4 Confidence Intervals on the Mean Response 138
3.4 Prediction of New Observations 141
3.5 Model Adequacy Checking 143
3.5.1 Residual Plots 143
3.5.2 Scaled Residuals and PRESS 146
3.5.3 Measures of Leverage and Influence 151
3.6 Variable Selection Methods in Regression 153
3.7 Generalized and Weighted Least Squares 160
3.7.1 Generalized Least Squares 160
3.7.2 Weighted Least Squares 163
3.7.3 Discounted Least Squares 168
3.8 Regression Models for General Time Series Data 184
3.8.1 Detecting Autocorrelation: The Durbin-Watson Test 186
3.8.2 Estimating the Parameters in Time Series Regression Models 191
3.9 Econometric Models 213
3.10 R Commands for Chapter 3 216
Exercises 226
4 Exponential Smoothing Methods 243
4.1 Introduction 243
4.2 First-Order Exponential Smoothing 249
4.2.1 The Initial Value, ¿0 251
4.2.2 The Value of ¿ 251
4.3 Modeling Time Series Data 254
4.4 Second-Order Exponential Smoothing 257
4.5 Higher-Order Exponential Smoothing 269
4.6 Forecasting 270
4.6.1 Constant Process 270
4.6.2 Linear Trend Process 272
4.6.3 Estimation of se2 284
4.6.4 Adaptive Updating of the Discount Factor 285
4.6.5 Model Assessment 287
4.7 Exponential Smoothing for Seasonal Data 288
4.7.1 Additive Seasonal Model 288
4.7.2 Multiplicative Seasonal Model 292
4.8 Exponential Smoothing of Biosurveillance Data 299
4.9 Exponential Smoothers and ARIMA Models 308
4.10 R Commands for Chapter 4 309
Exercises 321
5 Autoregressive Integrated Moving Average (ARIMA) Models 339
5.1 Introduction 339
5.2 Linear Models for Stationary Time Series 340
5.2.1 Stationarity 341
5.2.2 Stationary Time Series 341
5.3 Finite Order Moving Average Processes 345
5.3.1 The First-Order Moving Average Process, MA(1) 347
5.3.2 The Second-Order Moving Average Process, MA(2) 349
5.4 Finite Order Autoregressive Processes 350
5.4.1 First-Order Autoregressive Process, AR(1) 350
5.4.2 Second-Order Autoregressive Process, AR(2) 354
5.4.3 General Autoregressive Process, AR(p) 359
5.4.4 Partial Autocorrelation Function, PACF 360
5.5 Mixed Autoregressive-Moving Average Processes 367
5.5.1 Stationarity of ARMA(p, q) Process 368
5.5.2 Invertibility of ARMA(p, q) Process 368
5.5.3 ACF and PACF of ARMA(p, q) Process 369
5.6 Nonstationary Processes 376
5.6.1 Some Examples of ARIMA(p, d, q) Processes 377
5.7 Time Series Model Building 380
5.7.1 Model Identification 380
5.7.2 Parameter Estimation 381
5.7.3 Diagnostic Checking 382
5.7.4 Examples of Building ARIMA Models 383
5.8 Forecasting Arima Processes 392
5.9 Seasonal Processes 401
5.10 Arima Modeling of Biosurveillance Data 407
5.11 Final Comments 413
5.12 R Commands for Chapter 5 415
Exercises 427
6 Transfer Functions and Intervention Models 447
6.1 Introduction 447
6.2 Transfer Function Models 448
6.3 Transfer Function-Noise Models 456
6.4 Cross-Correlation Function 456
6.5 Model Specification 458
6.6 Forecasting with Transfer Function-Noise Models 476
6.7 Intervention Analysis 481
6.8 R Commands for Chapter 6 494
Exercises 508
7 Other Time Series Analysis and Forecasting Methods 517
7.1 Multivariate Time Series Models and Forecasting 517
7.1.1 Multivariate Stationary Process 518
7.1.2 Vector ARIMA Models 519
7.1.3 Vector AR (VAR) Models 520
7.2 State Space Models 526
7.3 Arch and Garch Models 531
7.4 Direct Forecasting of Percentiles 536
7.5 Combining Forecasts to Improve Prediction Performance 542
7.6 Aggregation and Disaggregation of Forecasts 547
7.7 Neural Networks and Forecasting 551
7.8 Spectral Analysis 559
7.9 Bayesian Methods in Forecasting 565
7.10 Some Comments on Practical Implementation and Use of Statistical Forecasting Procedures 572
7.11 R Commands for Chapter 7 576
Exercises 581
Appendix A Statistical Tables 595
Appendix B Data Sets for Exercises 615
Appendix C Introduction to R 683
Bibliography 689
Index 697
CHAPTER 1
INTRODUCTION TO TIME SERIES ANALYSIS AND FORECASTING
It is difficult to make predictions, especially about the future
NEILS BOHR, Danish physicist
1.1 THE NATURE AND USES OF FORECASTS
A forecast is a prediction of some future event or events. As suggested by Neils Bohr, making good predictions is not always easy. Famously "bad" forecasts include the following from the book Bad Predictions:
- "The population is constant in size and will remain so right up to the end of mankind." L'Encyclopedie, 1756.
- "1930 will be a splendid employment year." U.S. Department of Labor, New Year's Forecast in 1929, just before the market crash on October 29.
- "Computers are multiplying at a rapid rate. By the turn of the century there will be 220,000 in the U.S." Wall Street Journal, 1966.
Forecasting is an important problem that spans many fields including business and industry, government, economics, environmental sciences, public health, medicine, social science, politics, and finance. Forecasting problems are often classified as short-term, medium-term, and long-term. Short-term forecasting problems involve predicting events only a few time periods (days, weeks, and months) into the future. Medium-term forecasts extend from 1 to 2 years into the future, and long-term forecasting problems can extend beyond that by many years. Short- and medium-term forecasts are required for activities that range from operations management to budgeting and selecting new research and development projects. Long-term forecasts impact issues such as strategic planning. Short- and medium-term forecasting is typically based on identifying, modeling, and extrapolating the patterns found in historical data. Because these historical data usually exhibit inertia and do not change dramatically very quickly, statistical methods are very useful for short- and medium-term forecasting. This book is about the use of these statistical methods.
Most forecasting problems involve the use of time series data. A time series is a time-oriented or chronological sequence of observations on a variable of interest. For example, Figure 1.1 shows the market yield on US Treasury Securities at 10-year constant maturity from April 1953 through December 2006 (data in Appendix B, Table B.1). This graph is called a time series plot. The rate variable is collected at equally spaced time periods, as is typical in most time series and forecasting applications. Many business applications of forecasting utilize daily, weekly, monthly, quarterly, or annual data, but any reporting interval may be used. Furthermore, the data may be instantaneous, such as the viscosity of a chemical product at the point in time where it is measured; it may be cumulative, such as the total sales of a product during the month; or it may be a statistic that in some way reflects the activity of the variable during the time period, such as the daily closing price of a specific stock on the New York Stock Exchange.
FIGURE 1.1 Time series plot of the market yield on US Treasury Securities at 10-year constant maturity.
Source: US Treasury.
Because time series data exhibits the inertial effects mentioned previously, they usually do not satisfy the usual assumptions made in most statistical methods. That is, time series data are usually not independent. This means that special statistical methods that takes this into account are required for most forecasting and time series analysis problems. Those methods are the focus of this work.
The reason that forecasting is so important is that prediction of future events is a critical input into many types of planning and decision-making processes, with application to areas such as the following:
- Operations Management. Business organizations routinely use forecasts of product sales or demand for services in order to schedule production, control inventories, manage the supply chain, determine staffing requirements, and plan capacity. Forecasts may also be used to determine the mix of products or services to be offered and the locations at which products are to be produced.
- Marketing. Forecasting is important in many marketing decisions. Forecasts of sales response to advertising expenditures, new promotions, or changes in pricing polices enable businesses to evaluate their effectiveness, determine whether goals are being met, and make adjustments.
- Finance and Risk Management. Investors in financial assets are interested in forecasting the returns from their investments. These assets include but are not limited to stocks, bonds, and commodities; other investment decisions can be made relative to forecasts of interest rates, options, and currency exchange rates. Financial risk management requires forecasts of the volatility of asset returns so that the risks associated with investment portfolios can be evaluated and insured, and so that financial derivatives can be properly priced.
FIGURE 1.2 Five years of Bitcoin prices (from Yahoo Finance).
As an example of financial data, consider the Bitcoin price history for the most recent five years shown in Figure 1.2. Bitcoin is a cryptocurrency introduced in early 2009 although standard pricing did not begin until about a year later. The graph shows that there has been considerable growth in Bitcoin prices, but also considerable volatility. Investors and currency traders would be interested in modeling and forecasting the performance of this asset. However, the inherent volatility in Bitcoin price would make this a very challenging task.
- Economics. Governments, financial institutions, and policy organizations require forecasts of major economic variables, such as gross domestic product, population growth, unemployment, interest rates, inflation, job growth, production, and consumption. These forecasts are an integral part of the guidance behind monetary and fiscal policy, and budgeting plans and decisions made by governments. They are also instrumental in the strategic planning decisions made by business organizations and financial institutions.
- Industrial Process Control. Forecasts of the future values of critical quality characteristics of a production process can help determine when important controllable variables in the process should be changed, or if the process should be shut down and overhauled. Feedback and feedforward control schemes are widely used in monitoring and adjustment of industrial processes, and predictions of the process output are an integral part of these schemes.
- Demography. Forecasts of population by country and regions are made routinely, often stratified by variables such as gender, age, and race. Demographers also forecast births, deaths, and migration patterns of populations. Governments use these forecasts for planning policy and social service actions, such as spending on health care, retirement programs, and antipoverty programs. Many businesses use forecasts of populations by age groups to make strategic plans regarding developing new product lines or the types of services that will be offered.
- Public Health Applications. As an example of the use of time series analysis in the public health arena, let us consider the recent COVID-19 pandemic. This is also known as the coronavirus pandemic, caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The virus was first identified in an outbreak in the China in December 2019. Attempts to contain it there failed, allowing the virus to spread worldwide in 2020. The pandemic triggered social and economic disruption around the world, including a global recession. There were widespread supply shortages, including food shortages, resulting from supply chain disruptions. Mitigation strategies including travel restriction, business and school closures, social distancing measures, masking mandates, testing, contact tracing of infected individuals, and remote working were widespread. COVID-19 vaccines became available in late 2020 and have been widely deployed. As of mid-2023, the pandemic had caused over 700,000,000 cases and approximately 6.9 million deaths.
Figure 1.3 shows a time series plot of daily new cases from early 2020 to mid-2023. Figure 1.4 shows a plot of daily deaths from mid-2020 through mid-2023. The number of deaths declined rapidly over the last year shown in the graph due to the various mitigation strategies and the widespread availability and use of effective vaccines. Public health agencies at the national, state, and local level frequently made data such as this available. There was also interest in hospitalizations arising from the disease, as there was some concerns that hospital resources would be overwhelmed by the number of cases requiring that level of care.
FIGURE 1.3 Daily new cases of COVID-19.
FIGURE 1.4 Daily deaths from COVID-19.
A time series analysist could use this data to predict cases, deaths, and hospitalizations. It could also be possible to include the introduction of the mitigation measures including vaccines in the analysis to determine the potential effectiveness of these measures in reducing the spread of severity of the disease. A technique called intervention analysis can be used to do this. Intervention analysis is discussed in this book in Chapter...
