# Multiple regression and cost hierarchies in some

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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 + \$7.60 X 1 + \$37.77 X 2
where X 1 is the number of machine-hours and X 2 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 r 2 of 0.52 for the simple regression using number of machine-hours (Exhibit 10-14) increases to 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, r 2 of 0.72 versus r 2 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 error of 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 both important 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