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Chapter 16
Causal Forecast

Measurement Need

Managers may seek to understand how many products produce (the dependent variable, or the “output”) under given demand conditions (the independent variable, or the “input”).

Solutioni

A commonly used technique in causal forecasting is linear regression. In the linear regression method, when the dependent variable (usually the vertical, or y axis on a graph) changes as a result of the change in another variable (plotted as the horizontal, or x axis), it reflects a causal relationship and is represented by a straight line drawn through closely-related data points on the graph. Linear regression helps illustrate if there is a trend to the data, and is represented by a line formula:

y = a + b x

$y = a + b x$

Where

y = dependent variable

a = intercept

b = slope of the line

x = independent variable

To calculate the line formula, both the slope of the line (designated as b above) and the intercept (designed as a” above) must be calculated. The slope of the line describes the effect of the independent variable, x, on the dependent variable, y (i.e., changes in y if x changes by one unit). If there is no relationship between the dependent and independent variables, then the slope of the line would equal 0. The intercept describes where the linear regression line intersects with the y axis. The formulas are:

\begin{array}{l} I n t e r c e p t = a = Y - b X \\ S l o p e = b = \frac{Σ x y - n X Y}{Σ x^{2} - n X^{2}} \end{array}

$\begin{array}{l} I n t e r c e p t = a = Y - b X \\ S l o p e = b = \frac{Σ x y - n X Y}{Σ x^{2} - n X^{2}} \end{array}$

Where

a = intercept

b = slope of the line

X = Σx = mean of x

Y = Σy = mean of y

n = number of periods

Once the slope of the line is determined, the strength of the relationship between the dependent and independent variables must be measured. This is known as correlation and is represented by:

r = \frac{n \sum x y - \sum x \sum y}{\sqrt{[n \sum x^{2} - {(\sum x)}^{2}] [n \sum y^{2} - {(\sum y)}^{2}]}}

$r = \frac{n \sum x y - \sum x \sum y}{\sqrt{[n \sum x^{2} - {(\sum x)}^{2}] [n \sum y^{2} - {(\sum y)}^{2}]}}$

Where

r = correlation coefficient

n = number of periods

x = independent variable

y = dependent variable

Finally, when forecasters need to calculate the percentage of variation in the dependent (y) variable that is attributed to the independent (x) variable, then the coefficient of determination is used (which measures the relationship between the dependant and independent variables). If the independent variable is changed, then what impact does that have on the dependent variable? Do the two variables “go together”? The closer the relationship, the larger the coefficient of determination, up to 1.0 (or – 1.0 for negative relationships). It is calculated by:

r = r^{2}

$r = r^{2}$

We use the following example to illustrate how these various formulas work together (see Table 16.1):

–restaurant steak house

–forecasting food sales

○how many meals will be sold each week

–forecasting inventory

○perishable food

○nonperishable food

Table 16.1: Example Data Table

A linear regression is then calculated as follows:

\begin{array}{l} X = 1, 110 / 10 = 111 \\ Y = 1, 362 / 10 = 136.20 \\ b = \frac{\sum x y - n X Y}{\sum x^{2} - n X^{2}} = \frac{(155, 365) - (10) (111) (136.20)}{(127, 400) - (10) {(111)}^{2}} \\ b = .9983 \\ a = Y - b X = 136.20 - .9983 (111) \\ a = 25.3887 \end{array}

$\begin{array}{l} X = 1, 110 / 10 = 111 \\ Y = 1, 362 / 10 = 136.20 \\ b = \frac{\sum x y - n X Y}{\sum x^{2} - n X^{2}} = \frac{(155, 365) - (10) (111) (136.20)}{(127, 400) - (10) {(111)}^{2}} \\ b = .9983 \\ a = Y - b X = 136.20 - .9983 (111) \\ a = 25.3887 \end{array}$

These results are plugged into the original line formula:

\begin{array}{l} y = a + b x \\ y = 25.3887 + .9983 (x) \end{array}

$\begin{array}{l} y = a + b x \\ y = 25.3887 + .9983 (x) \end{array}$

For x, the forecaster should select the number of meals to be served (using this example) to calculate y. Let’s select 130, as that is the approximate average number of meals served per day:

\begin{array}{l} y = 25.3887 + .9983 (130) \\ y = 155.17 \end{array}

$\begin{array}{l} y = 25.3887 + .9983 (130) \\ y = 155.17 \end{array}$

Therefore, 155 pounds of beef should be ordered.

Next, the correlation coefficient is calculated to determine the strength of the relationship (also known as “interdependence”) between x and y.

\begin{array}{l} r = \frac{n \sum x y - \sum x \sum y}{\sqrt{[n \sum x^{2} - {(\sum x)}^{2}] [n \sum y^{2} - {(\sum y)}^{2}]}} \\ r = \frac{10 (155, 365) - (1, 110) (1, 362)}{\sqrt{[10 (127, 400) - {(1, 100)}^{2}] 10 (190, 140) - (1, 363)^{2}]}} \\ r = .9783 \\ r = .9571 \\ r = \frac{n \sum x y - \sum x \sum y}{\sqrt{[n \sum x^{2} - {(\sum x)}^{2}] [n \sum y^{2} - {(\sum y)}^{2}]}} \\ r = \frac{10 (155, 365) - (1, 110) (1, 362)}{\sqrt{[10 (127, 400) - {(1, 100)}^{2}] 10 (190, 140) - (1, 363)^{2}]}} \\ r = \frac{10 (155, 365) - (1, 110) (1, 362)}{\sqrt{[10 (127, 400) - {(1, 100)}^{2}] 10 (190, 140) - (1, 363)^{2}]}} \\ r = \frac{41, 830}{\sqrt{1, 828, 138, 900}} \\ r = \frac{41, 830}{42, 756} \\ r = .9783 \\ r^{2} = .9571 \end{array}

$\begin{array}{l} r = \frac{n \sum x y - \sum x \sum y}{\sqrt{[n \sum x^{2} - {(\sum x)}^{2}] [n \sum y^{2} - {(\sum y)}^{2}]}} \\ r = \frac{10 (155, 365) - (1, 110) (1, 362)}{\sqrt{[10 (127, 400) - {(1, 100)}^{2}] 10 (190, 140) - (1, 363)^{2}]}} \\ r = .9783 \\ r = .9571 \\ r = \frac{n \sum x y - \sum x \sum y}{\sqrt{[n \sum x^{2} - {(\sum x)}^{2}] [n \sum y^{2} - {(\sum y)}^{2}]}} \\ r = \frac{10 (155, 365) - (1, 110) (1, 362)}{\sqrt{[10 (127, 400) - {(1, 100)}^{2}] 10 (190, 140) - (1, 363)^{2}]}} \\ r = \frac{10 (155, 365) - (1, 110) (1, 362)}{\sqrt{[10 (127, 400) - {(1, 100)}^{2}] 10 (190, 140) - (1, 363)^{2}]}} \\ r = \frac{41, 830}{\sqrt{1, 828, 138, 900}} \\ r = \frac{41, 830}{42, 756} \\ r = .9783 \\ r^{2} = .9571 \end{array}$

The results suggest there is a strong relationship between the number of meals served and the quantity (in lbs.) of beef ordered. Therefore, this restaurant can feel confident that its forecast will be accurate.

Impact

Causal forecasts help business professionals plan for the future by measuring the relationship between two types of variables—dependent and independent. As demand conditions change, so too should the amount of product produced. In other words, the value (size, quantity, amount) of the dependent variable is directly influenced by the independent variable (market demand). A change in a product or marketing program can affect buyer behaviors (a price reduction might lead to increased purchases, albeit such a tactic often has short-lived benefits). Or, an emerging trend may signal a market opportunity, changing the performance of the business as a result. Causal forecasting enables managers to measure the possible impact to their business (and/or customers or other value chain participants) from these changes.

For example, companies such as Nike or Adidas, both of which make athletic footwear, would be interested in forecasting how many basketball shoes to produce to sell to teen basketball players worldwide over the next three years. By reviewing census data of the teen population and surveys of growth trends in teen basketball, the companies project the potential demand for their respective products. Assuming the teen population is forecast to grow (independent variable), as is the interest in basketball, then it is plausible to project an increase in sales (dependent variable).

Other examples include:

–travel to ski resorts increases in winter months if weather permits;

–toll roads collect more tolls during peak commute times;

–demand increases for air conditioning during summer months;

–increases/decreases of ice cream sales due to temperature changes; and

–more workers needed at restaurants on busy nights.

The result also suggests that the costs of the product can be reasonably projected. By extension, the final price offered to the customer can be determined as well. Prices should be set based on the strategic objectives for the positioning of this restaurant, its image (premium, mass market, value), cost factors, and the projected amount of business in the future. For sales people, causal forecasts are useful, particularly with controllable activities such as short-term promotions, where the outcome can be reasonably anticipated. Causal forecasting is not useful in every situation. It works best when the correlation between the dependent and independent variables is strong.

Data is gathered from market research conducted by the company directly and/or an independent research firm. Historical data can be used to build a trend analysis, which serves as a guide to future potential as well. Of course, no prediction is ironclad, and as business school students know well, a company with a ten year track record of consistent double-digit growth is not guaranteed to grow at the same pace in year eleven, even despite its track record.

iCausal-Based Forecasting: Relevance Behind the Screens, Accenture. Retrieved May 2, 2017 from https://www.accenture.com/sg-en/insight-interactive-causal-forecasting-relevance-summary; Rob J. Hyndman and Anne B. Koehler, “Another Look at Measures of Forecast Accuracy” (October–December 2006); L. Lapide, “New Developments in Business Forecasting,” Journal of Business Forecasting Methods & Systems 18, no. 2 (Summer 1999); J. Scott Armstrong (Ed.). Principles of Forecasting, A Handbook for Researchers and Practitioners, University of Pennsylvania. Retrieved May 2, 2017 from http://morris.wharton.upenn.edu/forecast; G. Cachon and C. Terwiesch, Matching Supply with Demand: An Introduction to Operations Management, International Edition New York: McGraw-Hill, 2006); Forecasting. Retrieved May 9, 2017 from www.uoguelph.ca/~dsparlin/forecast.htm

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Table of Contents for Chapter 16: Causal Forecast

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Chapter 16 Causal Forecast

Measurement Need

Solutioni

Impact

Table of Contents for
Chapter 16: Causal Forecast

Chapter 16
Causal Forecast