Least Squares Method (Linear Regression)

The use of linear regression (least squares method) is the most accurate method in segregating total costs into fixed and variable components.

The total fixed cost and variable cost per unit are determined mathematically through a series of computations.

Nonetheless, like the other methods of cost segregation, the least squares method follows the same cost function:

y = a + bx

where: y = total cost; a = total fixed costs; b = variable cost per level of activity (or units); x = level of activity (or number of units).

The Normal Equations in Differential Calculus

∑y = na + b∑x
∑xy = ∑xa + b∑x²

Note that through the process of elimination, these equations can be used to determine the values of a and b. Nonetheless, formulas for total fixed costs (a) and variable cost per unit (b) can be derived from the above equations.

Variable Cost per Unit (b)

Using the normal equations above, a formula for b can be derived. The variable cost per unit or slope is computed using the following formula:

b = n∑xy – (∑x)(∑y)
  n∑x² – (∑x)²

Total Fixed Costs (a)

Once b has been determined, the total fixed cost or a can be computed using the formula:

a = ȳ - bx̄

where: ȳ = ∑y and x̄ = ∑x
    n     n

Or, it is the same as:

a = ∑y – b∑x
  n

Example

The following data was gathered for five production runs of ABC Company. Determine the cost function using the least squares method.

Batch
Units (x)
Total Cost (y)
1
680
$29,800
2
820
$34,000
3
570
$27,500
4
660
$29,000
5
750
$31,900

Solution:

Batch
Units (x)
Total Cost (y)
xy
1
680
29,800
20,264,000
462,400
2
820
34,000
27,880,000
672,400
3
570
27,500
15,675,000
324,900
4
660
29,000
19,140,000
435,600
5
750
31,900
23,925,000
562,500
3,480
152,200
106,884,000
2,457,800

Substituting the computed values in the formula, we can compute for b.

b = n∑xy – (∑x)(∑y)
  n∑x² – (∑x)²
b = (5)(106,884,000) – (3,480)(152,200)
  (5)(2,457,800) – (3,480)²

b = 26.6741 ≈ $26.67 per unit

Total fixed cost (a) can then be computed by substituting the computed b.

a = ∑y – b∑x
  n
a = 152,200 – (26.67)(3,480)
  5

a = $11,877.68

The cost function for this particular set using the method of least squares is:
y = $11,887.68 + $26.67x.

Online resource for all things accounting. more
Search this Site
Featured in the Blog
Feedback
Questions, comments and suggestions?
Contact us here.
Copyright © 2016 Accountingverse.com - Your Online Resource For All Things Accounting
Terms of Use | Home | About | Contact