Regression analysis

Learning outcome statements

The learning outcome statements relevant for this section are:

1. demonstrate an understanding of a simple regression equation
2. define a multiple regression equation
3. calculate the result of a simple regression equation
4. identify the benefits and shortcomings of regression analysis

Regression analysis discussion

Forecasting is the process of using historical data and analytical techniques to predict future outcomes, such as future sales, costs, and operational activities. Accurate forecasts give businesses the information they need to budget effectively - allocating resources, setting realistic goals, and monitoring performance. By using forecasting techniques such as regression analysis, companies can identify trends and relationships between variables, which leads to budgets that better align with their strategic goals.

Definitions

Regression analysis

This is a statistical method used to understand the relationship between two or more variables.

Regression equation

A mathematical formula that describes the relationship between the dependent and independent variables.

In the context of forecasting, regression helps predict a dependent variable (such as costs or sales) based on the value of one or more independent variables (such as production volume or advertising expenditure). This is particularly useful for making financial projections and evaluating trends in business data.

Simple regression equation

A simple linear regression is a model that describes the relationship between one independent variable and one dependent variable. In the context of business, if you’re modeling costs as a function of production volume, the equation helps determine how costs change as production increases.

The general form of the simple regression equation is:

$$\text{Y}=\text{a}+\text{bX}$$

Where:

• Y = dependent variable (e.g., total costs)
• X = independent variable (e.g., production volume)
• a = intercept (the fixed cost if we are modeling costs)
• b = slope (the variable cost per unit if we are modeling costs)

Example: Modeling total cost

Suppose a company wants to model its total cost (Y) based on production volume (X). The company’s analysis shows that the fixed cost is $10,000, and the variable cost per unit is $50. The regression equation would be:

$$\text{Y}=10,000+50\text{X}$$

This means that for each additional unit produced, total costs increase by $50, with a base cost of $10,000 when no units are produced. The intercept (fixed cost) and slope (variable cost per unit) work together as a prediction tool: plug in any production volume for X, and the equation forecasts total cost.

For example, at a production volume of 200 units, predicted total cost is $10,000 + ($50 × 200) = $20,000.

Answer: at X = 200 units, predicted total cost Y = $20,000.

CMA pitfall: Don’t treat the intercept as the company’s true fixed cost unless X = 0 actually falls within the range of data used to build the regression. Outside that relevant range, the equation’s predictions - including the intercept - may not hold.

Multiple regression equation

Multiple regression is a statistical technique used to explain the relationship between a dependent variable and two or more independent variables. It extends simple regression (which uses only one independent variable) by allowing multiple factors to be considered simultaneously.

For the CMA exam, candidates are not required to perform calculations for multiple regression. The exam only requires that students be able to define multiple regression and understand its purpose and applications in decision-making.

The general form of the multiple regression equation is:

$$\text{Y}=\text{a}+{\text{b}}_1{\text{X}}_1+{\text{b}}_2{\text{X}}_2+\cdots +{\text{b}}_n{\text{X}}_n+\epsilon$$

Where:

• Y = dependent variable (the outcome being explained)
• a = intercept (value of Y when all X’s are zero)
• b1, b2, … bn = coefficients (slopes showing the impact of each independent variable on Y)
• X1, X2, … Xn = independent variables (factors that influence Y)
• ε = error term (variation not explained by the model)

The simple regression equation from the prior section has an error term too; it’s just usually dropped once the equation is being used to predict a single value of Y.

In practice, this equation helps identify how much each factor contributes to changes in the dependent variable, while holding the others constant.

Example: Multiple regression in production costs

A manufacturing company wants to understand what drives changes in its monthly production costs. Instead of analyzing one factor in isolation, management considers production volume, labor hours, and machine maintenance spending together.

Using multiple regression, the company can estimate how each factor influences production costs and make better budgeting and cost control decisions. This illustrates how multiple regression provides more realistic insights when multiple variables affect business performance.

Benefits and shortcomings of regression analysis

Benefits of regression analysis

• Predictive power: Regression analysis allows companies to forecast outcomes, such as sales or costs, based on different levels of input (independent) variables.
• Decision-making: By understanding the relationship between variables, businesses can make informed decisions about how changes in one factor (e.g., production volume) will impact others (e.g., costs or revenue).
• Identifying trends: Regression can help identify underlying trends in historical data, allowing organizations to project future performance.

Shortcomings of regression analysis

• Assumption of linearity: Simple regression assumes a linear relationship between variables, which may not always be the case in real-world situations.
• Data limitations: Regression analysis depends on the quality of data. Inaccurate or incomplete data can lead to unreliable predictions.
• Overfitting: Using too many variables in a regression model can lead to overfitting, where the model becomes too tailored to historical data and fails to predict future outcomes accurately.
• Relevant range: Predictions are only reliable within the range of the historical data used to build the equation; the intercept may not represent a realistic fixed cost if X = 0 falls outside that range.
• Correlation is not causation: A strong statistical relationship between variables doesn’t prove that one causes the other.
• Stability of relationships: Regression assumes that the relationship between variables observed in the past will continue into the future, which may not hold.