5.1 Introduction to Time Series Analysis

Time Series Models

A time series model is a function, algorithm, or method for finding, approximating, or predicting the values of a given time series. The basic idea behind time series models is that previous values should provide some indication as to how future values behave. In other words, there is some kind of function that takes the previous values of the time series as input and produces the next value as output:

However, in all but the most ideal situations, a function that predicts the next value of the time series with perfect accuracy does not exist. Random variation and other factors not accounted for in the model will produce error, which is defined as the extent to which predictions differ from actual observations. Thus, we should always incorporate an error term (often denoted by the Greek letter $\epsilon$, called “epsilon”) into the model. Moreover, instead of expecting the model to produce the next term exactly, the model generates predicted values. Often, the predicted values of a times series are denoted by $\left({\hat{x}}_n\right)$ to distinguish them from the actual values $\left(x_n\right)$. Thus,

In Time Series Forecasting Methods and Forecast Evaluation Methods, we will delve into the details of building time series models and accounting for error.

Time Series Forecasting

Typically, the goal of time series analysis is to make predictions, or extrapolations, about future values of the time series, a process known as forecasting. As a general rule, the accuracy of a forecast decreases as predictions are made further into the future. When future predictions become no more accurate than tossing a coin or rolling a die, then forecasting at the point or beyond becomes ineffective. In practice, time series models are updated regularly to accommodate new data.

Depending on the situation and the nature of the data, there are many different ways to forecast future data. The simplest of all methods, which is known as the naïve or flat forecasting method, is to use the most recent value as the best guess for the next value. For example, since the S&P 500 had a value of 3,839.5 at the end of 2022, it is reasonable to assume that the value will be relatively close to 3,839.5 at the end of 2023. Note, this would correspond to the time series model, ${\hat{x}}_{n+1}=x_n$. The naïve method has only limited use in practice.

Instead of using only the last observed value to predict the next one, a better approach might be to take into consideration a number of values, $x_n,x_{n-1},x_{n-2},\dots ,$ to find the estimate ${\hat{x}}_{n+1}$. One might average the last $T$ values together (for some predefined value of $T$). This is known as a simple moving average, which will be defined explicitly in Components of Time Series Analysis. For now, let’s illustrate the idea intuitively. Suppose we used the average of the most recent $T=3$ terms to estimate the next term. The time series model would be:

Based on the data in Table 5.1, the prediction for the S&P index value at the end of 2023 would work out as follows.

Another simple method of forecasting is to fit the data to a linear regression model and then use the regression model to predict future values. (See Linear Regression for details on linear regression.) Linear regression does a better job at capturing the overall direction of the data compared to using only the last data point. On the other hand, linear regression will not be able to model more subtle structures in the data such as cyclic patterns. Moreover, there is a hidden assumption that the data rises or falls more or less uniformly throughout the period we would like to predict, which is often not a valid assumption to make.

Example 5.1

Problem

Find a linear regression model for the data from Table 5.1 and use it to forecast the value of the S&P at the end of the years 2024 and 2025.

Solution

Using standard statistical software, the linear regression is found to be $\hat{y}=304.437t-\mathrm{611,288}$, where $t$ is the year. Alternatively, in Excel, the regression line can be added to the line graph using the “Trendline” feature, with option “Linear.” A graph of the regression line is shown in Figure 5.3.

Figure 5.3 Line Chart of the S&P 500 Index with the Linear Regression (data source: adapted from S&P 500 [SPX] Historical Data)

The forecast for the end of year 2024 is $\hat{y}=304.437\left(2024\right)-\mathrm{611,288}=\mathrm{4,892.5}$.

Similarly, the prediction for 2025 is $\hat{y}=304.437\left(2025\right)-\mathrm{611,288}=\mathrm{5,196.9}$.

In this chapter, we will develop more sophisticated tools that can detect patterns that averages or linear models simply cannot find, including autoregressive models, moving averages, and the autoregressive integrated moving average (ARIMA) model. However, creating a fancy model for time series data is only part of the process. An essential part of the process is testing and evaluating your model. Thus, we will also explore measures of error and uncertainty that can determine how good the model is and how accurate its forecasts might be.

Working with and Visualizing Time Series Data in Excel

You may recall from What Are Data and Data Science? that spreadsheet applications are very well suited for basic analysis and visualization of time series data. Simply create two columns, one for the time periods and the other for the data. Highlight both columns (together with their headings) and insert a line chart using the “Recommended Charts” feature in Excel¹, resulting in a chart such as the one shown in Figure 5.4.

Figure 5.4 Using Excel to Create a Line Chart. Instead of using “Recommended Charts,” you could also choose line chart from the panel of choices located on the same tab. (Used with permission from Microsoft)

Working with and Visualizing Time Series Data in Python

In Python, the pandas library is incredibly useful for reading in CSV files and managing them as data structures called DataFrames. It also includes a simple command for visualizing the time series, plot(). Here is how to create a line chart of the daily values of the S & P 500 Index from 2014 through 2024 from the dataset SP500.csv.

Python Code

      import pandas as pd
      import matplotlib.pyplot as plt
      import matplotlib.ticker as ticker
      
      # Read the CSV file into a Pandas DataFrame
      df = pd.read_csv('SP500.csv')
      
      # Create a plot
      fig, ax = plt.subplots()
      df.plot(ax=ax)
      
      # Use FuncFormatter to add commas to the y-axis
      ax.yaxis.set_major_formatter(ticker.FuncFormatter(lambda x, pos: '{:,}'.format(int(x))))
      
      # Display the plot
      plt.show()
      

The resulting output will look like this:

Notice that the simple plot feature did not label the $x$ or $y$ axes. Moreover, the x-axis does not show actual dates, but instead the number of time steps. For simple visualizations that you do not intend to share with a wider audience, this may be ok, but if you want to create better-looking and more informative graphs, use a library such as matplotlib.pyplot. In the next set of Python code, you will see the same dataset visualized using matplotlib. (Note that there is a line of code that uses the to_datatime() command. Dates and times are notoriously inconsistent from dataset to dataset, and software packages may not interpret them correctly without a little help.)

Python Code

      # Import libraries
      import pandas as pd ## for dataset management
      import matplotlib.pyplot as plt ## for visualizations
      from matplotlib.ticker import FuncFormatter ## formatting Y-axis values
      
      # Read the CSV file into a Pandas DataFrame
      df = pd.read_csv('SP500.csv')
      
      # Convert the 'DATE' column to datetime format
      df['Date'] = pd.to_datetime(df['Date'], errors='coerce')
      
      # Function to format the Y-axis values
      def y_format(value, tick_number):
        return f'{value:,.0f}'
      
      # Plotting the time series data using Matplotlib
      plt.figure(figsize=(10, 6))
      plt.plot(df['Date'], df['SP500'], marker='', linestyle='-', color='b')
      plt.title('S&P 500 Time Series')
      plt.xlabel('Date')
      plt.ylabel('S&P 500 Value')
      plt.grid(True)
      
      # Apply the formatter to the Y-axis
      plt.gca().yaxis.set_major_formatter(FuncFormatter(y_format))
      
      plt.show()
      

The resulting output will look like this: