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Multiple Regression and Cost Hierarchies In some cases, a satisfactory estimation of a cost function may be based on only one independent variable, such as number of machine-hours. In many cases, however, basing the estimation on more than one independent variable (that is, multiple regression) is more economically plausible and improves accuracy. The most widely used equations to express relationships between two or more independent variables and a dependent variable are linear in the form
Example: Consider the Elegant Rugs data in Exhibit 10-19. The company’s ABC analysis indicates that indirect manufacturing labor costs include large amounts incurred for setup and changeover costs when a new batch of carpets is started. Management believes that in addition to number of machine-hours (an output unit-level cost driver), indirect manufacturing labor costs are also affected by the number of batches of carpet produced during each week (a batch-level driver). Elegant Rugs estimates the relationship between two independent variables, number of machine- hours and number of production batches of carpet manufactured during the week, and indirect manufacturing labor costs. Exhibit 10-20 presents results (in Excel) for the following multiple regression model, using data in columns B, C, and E of Exhibit 10-19: y = $42.58 + $7.60X1 + $37.77X2
where X1 is the number of machine-hours and X2 is the number of production batches. It is economically plausible that both number of machine-hours and number of production batch-es would help explain variations in indirect manufacturing labor costs at Elegant Rugs. The r2 of 0.52 for the simple regression using number of machine-hours (Exhibit 10-14) increasesto 0.72 with the multiple regression in Exhibit 10-20. The t-values suggest that the independent variable coefficients of both number of machine-hours ($7.60) and number of production batches ($37.77) are significantly different from zero (t = 2.74 is the t-value for number of machine-hours, and t = 2.48 is the t-value for number of production batches, compared to the cut-off t-value of 2.26). The multiple regression model in Exhibit 10-20 satisfies both eco-nomic plausibility and statistical criteria, and explains much greater variation (that is, r2 of 0.72 versus r2 of 0.52) in indirect manufacturing labor costs than the simple regression model using only number of machine-hours as the independent variable.8 The standard errorof the regression equation that includes number of batches as an independent variable is which is lower than the standard error of the regression with only machine-hours as the in- dependent variable, $170.54. That is, even though adding a variable reduces the degrees of freedom in the denominator, it substantially improves fit so that the numerator, Σ(Y - y)2, decreases even more. Number of machine-hours and number of production batches are bothimportant cost drivers of indirect manufacturing labor costs at Elegant Rugs.
In Exhibit 10-20, the slope coefficients—$7.60 for number of machine-hours and $37.77 for