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 machinehours. 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 1019. 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 machinehours (an output unitlevel cost driver), indirect
manufacturing labor costs are also affected by the number of batches of carpet produced
during each week (a batchlevel 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 1020 presents results (in Excel) for the following multiple regression model, using
data in columns B, C, and E of Exhibit 1019:
y
= $42.58 + $7.60
X
1
+ $37.77
X
2
where
X
1
is the number of machinehours and
X
2
is the number of production batches. It is
economically plausible that both number of machinehours 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 machinehours (Exhibit 1014) increases
to 0.72 with the multiple regression in Exhibit 1020. The
t
values suggest that the
independent variable coefficients of both number of machinehours ($7.60) and number of
production batches ($37.77) are significantly different from zero (
t
= 2.74 is the
t
value for
number of
machinehours, and
t
= 2.48 is the
t
value for number of production batches, compared to
the cutoff
t
value of 2.26). The multiple regression model in Exhibit 1020 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 machinehours 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 machinehours 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 machinehours and number of production batches are both
important cost drivers of indirect manufacturing labor costs at Elegant Rugs.
In Exhibit 1020, the slope coefficients—$7.60 for number of machinehours and $37.77 for