Budgeting Basics and Beyond

16.1. Least-Squares Method

The least-squares method is widely used in regression analysis for estimating the parameter values in a regression equation. The regression method includes all the observed data and attempts to find a line of best fit. To find this line, a technique called the least-squares method is used. Exhibit 16.1 shows the regression relationship.

To explain the least-squares method, we define the error as the difference between the observed value and the estimated one and denote it with u.

Symbolically, u =Y - Y′

where Y= observed value of the dependent variab]

Y′= estimated value based on Y′ = a + bX

The least-squares criterion requires that the line of best fit be such that the sum of the squares of the errors (or the vertical distance in Exhibit 16.1 from the observed data points to the line) is a minimum, that is,

Minimum: Σu² = Σ(Y - a - bX)²

Using differential calculus we obtain these equations, called normal equations:

ΣY = na +bσX

ΣXY = aσX +bσX²

Solving the equations for b and a yields

Example 1

To illustrate the computations of b and a, we will refer to the data in Exhibit 16.2. All the sums required are computed and shown in the exhibit.

From Exhibit 16.2:

Substituting these values into the formula for b first:

Thus, Y' = 10.5836 + 0.5632 X