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5.7 Forecasting Models—Trend, Seasonal, and Random Variations

When both trend and seasonal components are present in a time series, the forecasting model selected must address these. The decomposition method, which uses seasonal indices, is a very common approach. Multiple regression models are also common, with dummy variables used to adjust for seasonal variations in an additive time-series model.

The Decomposition Method

The process of isolating linear trend and seasonal factors to develop more accurate forecasts is called decomposition. The first step is to compute seasonal indices for each season as we have done with the Turner Industries data. Then, the data are deseasonalized by dividing each number by its seasonal index, as shown in Table 5.11. Figure 5.5 provides a graph of the original data, as well as a graph of the deseasonalized data. Notice how smooth the deseasonalized data are.

Table 5.11 Deseasonalized Data for Turner Industries

SALES ($1,000,000s)	SEASONAL INDEX	DESEASONALIZED SALES ($1,000,000s)
108	0.85	127.059
125	0.96	130.208
150	1.13	132.743
141	1.06	133.019
116	0.85	136.471
134	0.96	139.583
159	1.13	140.708
152	1.06	143.396
123	0.85	144.706
142	0.96	147.917
168	1.13	148.673
165	1.06	155.660

Table 5.11 Full Alternative Text

A trend line is then found using the deseasonalized data. Using computer software with these data, we have³

\begin{array}{l} b_{1} & = & 2.34 \\ b_{0} & = & 124.78 \end{array}

$\begin{array}{l} b_{1} & = & 2.34 \\ b_{0} & = & 124.78 \end{array}$

The trend equation is

\hat{Y} = 124.78 + 2.34 X

$\hat{Y} = 124.78 + 2.34 X$

where

X = time

$X = time$

This equation is used to develop the forecast based on trend, and the result is multiplied by the appropriate seasonal index to make a seasonal adjustment. For the Turner Industries data, the forecast for the first quarter of year 4 (time period $X = 13$ $X = 13$ and seasonal index $I_{1} = 0.85$ $I_{1} = 0.85$ ) would be found as follows:

A scatterplot showing sales data compared to deseasonalized sales data. — Figure 5.5 Scatterplot of Turner Industries Original Sales Data and Deseasonalized Data

Figure 5.5 Full Alternative Text

\begin{array}{l} \hat{Y} & = & 124.78 + 2.34 X \\ = & 124.78 + 2.34 (13) \\ = & 155.2 (forecast before adjustment for seasonality) \end{array}

$\begin{array}{l} \hat{Y} & = & 124.78 + 2.34 X \\ = & 124.78 + 2.34 (13) \\ = & 155.2 (forecast before adjustment for seasonality) \end{array}$

We multiply this by the seasonal index for quarter 1, and we get

\hat{Y} \times I_{1} = 155.2 \times 0.85 = 131.92

$\hat{Y} \times I_{1} = 155.2 \times 0.85 = 131.92$

Using the same procedure, we find the forecasts for quarters 2, 3, and 4 of the next year. Table 5.12 shows the unadjusted forecast based on the trend line for each of the four quarters of year 4. The last column of this table gives the final forecasts that were found by multiplying each trend forecast by the appropriate seasonal index. A scatter diagram showing the original data and the deseasonalized data, as well as the unadjusted forecast and the final forecast, for these next four quarters is provided in Figure 5.6.

Table 5.12 Turner Industry Forecasts for Four Quarters of Year 4

YEAR	QUARTER	TIME PERIOD (X)	TREND FORECAST Y = 124 + 2.34X	SEASONAL INDEX	FINAL (ADJUSTED) FORECAST
4	1	13	155.20	0.85	131.92
	2	14	157.54	0.96	151.24
	3	15	159.88	1.13	180.66
	4	16	162.22	1.06	171.95

The scatterplot from the previous figure is replicated here, with the sales data plotted as circles and the deseasonalized sales data now shown as triangles.

Figure 5.6 Full Alternative Text

Steps to Develop a Forecast Using the Decomposition Method

Compute seasonal indices using CMAs.
Deseasonalize the data by dividing each number by its seasonal index.
Find the equation of a trend line using the deseasonalized data.
Forecast for future periods using the trend line.
Multiply the trend line forecast by the appropriate seasonal index.

Most forecasting software, including Excel QM and QM for Windows, includes the decomposition method as one of the available techniques. This will automatically compute the CMAs, deseasonalize the data, develop the trend line, make the forecast using the trend equation, and adjust the final forecast for seasonality.

Software for Decomposition

Program 5.6A shows the QM for Windows input screen that appears after selecting the Forecasting module, selecting New-Time Series, and specifying 12 past observations. The multiplicative decomposition method is selected, and you can see the inputs that are used for this problem. When this is solved, the output screen in Program 5.6B appears. Additional output is available by selecting details or graph from the Windows drop-down menu (not shown in Program 5.6B) that appears after the problem is solved. The forecasts are slightly different from the ones shown in Table 5.12 due to round-off error.

Excel QM can be used in Excel 2016 for this problem also. From the QM Ribbon, select the menus By Chapter and select Chapter 5-Forecasting-Decomposition. Specify there are 12 past observations and 4 seasons, and use a centered moving average.

Using Regression with Trend and Seasonal Components

Multiple regression may be used to forecast with both trend and seasonal components present in a time series. One independent variable is time, and other independent variables are dummy variables to indicate the season. If we forecast quarterly data, there are four categories (quarters), so we would use three dummy variables. The basic model is an additive decomposition model and is expressed as follows:

A screenshot of QM for Windows instructing you on providing input for Turner Industries. A series of tip boxes guides you through the process of inputting data. — Program 5.6A QM for Windows Input Screen for Turner Industries Example

Figure 5.6A Full Alternative Text

A screenshot showing QM for Windows output for Turner Industries. Tip boxes point out the final forecast, seasonal factors, and the location of the trend line formula in the output table. — Figure 5.6B QM for Windows Output Screen for Turner Industries Example

Figure 5.6B Full Alternative Text

\hat{Y} = a + b_{1} X_{1} + b_{2} X_{2} + b_{3} X_{3} + b_{4} X_{4}

$\hat{Y} = a + b_{1} X_{1} + b_{2} X_{2} + b_{3} X_{3} + b_{4} X_{4}$

where

\begin{array}{l} X_{1} & = & time period \\ X_{2} & = & 1 i f quarter 2 \\ = & 0 otherwise \\ X_{3} & = & 1 if quarter 3 \\ = & 0 otherwise \\ X_{4} & = & 1 if quarter 4 \\ = & 0 otherwise \end{array}

$\begin{array}{l} X_{1} & = & time period \\ X_{2} & = & 1 i f quarter 2 \\ = & 0 otherwise \\ X_{3} & = & 1 if quarter 3 \\ = & 0 otherwise \\ X_{4} & = & 1 if quarter 4 \\ = & 0 otherwise \end{array}$

If $X_{2} = X_{3} = X_{4} = 0,$ $X_{2} = X_{3} = X_{4} = 0,$ then the quarter would be quarter 1. It is an arbitrary choice as to which of the quarters would not have a specific dummy variable associated with it. The forecasts will be the same, regardless of which quarter does not have a specific dummy variable.

Using Excel QM in Excel 2016 to develop the regression model, from the Excel QM ribbon, we select the menu for Chapter 5—Forecasting, and then choose the Multiple Regression method. The input screen in Program 5.7A appears, and the number of observations (12) and the number of independent variables (4) are entered. An initialized spreadsheet appears, and the values for all the variables are entered in columns B through F, as shown in Program 5.7B. Once this has been done, the regression coefficients appear. The regression equation (with rounded coefficients) is

A screenshot of the Spreadsheet Initialization window for running Multiple Regression analysis on Turner Industries.

Figure 5.7A Full Alternative Text

A screenshot shows the output screen for Turner Industries. — Program 5.7B Excel QM Multiple Regression Output Screen for Turner Industries

Figure 5.7B Full Alternative Text

In Action Forecasting at NBC Universal

NBC Universal (NBCU), a world leader in the production, distribution, and marketing of entertainment, news, and information, had revenues of over $14 billion in 2005. NBCU owns a television network and several stations in the United States, an impressive portfolio of cable networks, a major motion picture company, and very popular theme parks. Over 60 percent of NBCU’s revenues are from the sales of on-air advertising time on its television networks and stations.

Each year, the up-front market is a brief period in late May when the television networks sell a majority of their on-air advertising inventory, right after announcing their program schedules for the upcoming broadcast year. To address the challenging problem of forecasting up-front market demand, NBCU initially relied primarily on judgment models and then tried time-series forecasting models. These models proved, however, to be rather unsatisfactory due to the unique nature of the demand population. NBCU now estimates up-front demand using a novel procedure that combines the Delphi forecasting method with grassroots forecasting.

The system, which has been in place since 2004, has been used to support sales decisions each year worth over $4.5 billion. The system enables NBCU to easily analyze pricing scenarios across all of its television properties, while predicting demand with a high level of accuracy. NBCU’s sales leaders have credited the forecast system with giving the company a unique competitive advantage over its competitors.

Source: Based on S. Bollapragada, S. Gupta, B. Hurwitz, P. Miles, and R. Tyagi, “NBC-Universal Uses a Novel Qualitative Forecasting Technique to Predict Advertising Demand,” Interfaces 38, 2 (2008): 103–111, © Trevor S. Hale.

\hat{Y} = 104.1 + 2.3 X_{1} + 15.7 X_{2} + 38.7 X_{3} + 30.1 X_{4}

$\hat{Y} = 104.1 + 2.3 X_{1} + 15.7 X_{2} + 38.7 X_{3} + 30.1 X_{4}$

If this is used to forecast sales in the first quarter of the next year, we get

\hat{Y} = 104.1 + 2.3 (13) + 15.7 (0) + 38.7 (0) + 30.1 (0) = 134

$\hat{Y} = 104.1 + 2.3 (13) + 15.7 (0) + 38.7 (0) + 30.1 (0) = 134$

For quarter 2 of the next year, we get

\hat{Y} = 104.1 + 2.3 (14) + 15.7 (1) + 38.7 (0) + 30.1 (0) = 152

$\hat{Y} = 104.1 + 2.3 (14) + 15.7 (1) + 38.7 (0) + 30.1 (0) = 152$

Notice these are not the same values we obtained using the multiplicative decomposition method. We could compare the MAD or MSE for each method and choose the one that is better.

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Table of Contents for
5.7 Forecasting Models—Trend, Seasonal, and Random VariationsForecasting Models—Trend, Seasonal, and Random Variations

5.7 Forecasting Models—Trend, Seasonal, and Random Variations

The Decomposition Method

Table 5.11 Deseasonalized Data for Turner Industries

Figure 5.5 Scatterplot of Turner Industries Original Sales Data and Deseasonalized Data

Table 5.12 Turner Industry Forecasts for Four Quarters of Year 4

Figure 5.6 Scatterplot of Turner Industries’ Original Sales Data and Deseasonalized Data with Unadjusted and Adjusted Forecasts

Steps to Develop a Forecast Using the Decomposition Method

Software for Decomposition

Using Regression with Trend and Seasonal Components

Program 5.6A QM for Windows Input Screen for Turner Industries Example

Figure 5.6B QM for Windows Output Screen for Turner Industries Example

Program 5.7A Excel QM Multiple Regression Initialization Screen for Turner Industries

Program 5.7B Excel QM Multiple Regression Output Screen for Turner Industries

Table of Contents for 5.7 Forecasting Models&#8212;Trend, Seasonal, and Random VariationsForecasting Models—Trend, Seasonal, and Random Variations

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Table of Contents for
5.7 Forecasting Models—Trend, Seasonal, and Random VariationsForecasting Models—Trend, Seasonal, and Random Variations