Regression Analysis

Regression analysis is a statistical technique for estimating the relationship between a dependent variable and one or more independent variables. In accounting and finance, the dependent variable is often a cost, such as monthly utilities or maintenance spending, and the independent variable is an activity that drives it, such as machine hours, units produced or customer orders. The result is an equation that can be used to predict the cost at different levels of activity.

The technique is a core tool of managerial accounting, where it is used to understand cost behavior, prepare budgets and set prices. It is also used in finance to estimate a share's beta, in auditing to set expectations for analytical review, and more broadly in econometrics.

The most common form is simple linear regression by the least-squares method. It fits a straight line through a set of past observations so that the sum of the squared vertical distances between the points and the line is as small as possible. For a cost, the line takes this form:

• $Y$ is the total cost being estimated, the dependent variable.
• $a$ is the intercept, the estimated fixed cost per period that is incurred even at zero activity.
• $b$ is the slope, the estimated variable cost per unit of activity.
• $X$ is the activity level, the independent variable.

Spreadsheet functions and accounting software calculate $a$ and $b$ automatically. They also report the coefficient of determination, $R^{2}$, which shows the share of the variation in cost explained by the activity. An $R^{2}$ of 0.90 means 90 percent of the variation is explained; a low figure suggests the chosen driver is a poor one.

A print shop records 12 months of electricity costs against machine hours. Its mixed costs range from $3,900 in a quiet month of 500 hours to $7,300 in a busy month of 1,400 hours. Running a regression on all 12 points gives an intercept of $2,050 and a slope of $3.70 per machine hour, with an $R^{2}$ of 0.94. The cost formula is therefore:

If the shop expects 1,100 machine hours next month, the predicted cost is 2,050 plus 3.70 times 1,100, or $6,120. The $2,050 is treated as part of fixed costs and the $3.70 as a variable cost per hour. These figures can then feed a flexible budget and cost-volume-profit analysis.

The simpler high-low method uses only the highest and lowest activity months to draw a line. It is quick but can be distorted if either of those months was unusual. Regression uses every observation, so it is generally more accurate and also gives a measure of how good the fit is. A scatter graph, plotting cost against activity, is often drawn first to check that the relationship looks roughly linear and to spot outliers.

Multiple regression extends the method to more than one driver, for example estimating shipping costs from both the number of orders and total weight. This can produce better predictions where costs depend on several activities, as in activity-based costing.

A regression line describes past relationships and is reliable only within the range of activity that was observed. Predicting far outside that range, for instance at double the busiest month, assumes cost behavior will not change, which is often untrue because new equipment or extra shifts would be needed. Correlation also does not prove cause: a cost may move with an activity simply because both rise in the same season. Inflation, changes in prices or processes during the data period, and too few observations can all weaken the estimate. Accountants therefore combine regression results with judgment about how the business actually operates, and they update the analysis as conditions change. It is one input to cash forecasting and budgeting, not a substitute for it.