CHAPTER 1
INTRODUCTION TO TIME SERIES ANALYSIS AND FORECASTING

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:

1. 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.
2. 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.
3. 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.

4. 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.
5. 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.
6. 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.
7. 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...