9781118041581

As the term suggests, ordinary least-squares regression computes coefficient

values that give the smallest sum of squared errors. Using calculus techniques,

statisticians have derived standard formulas for these least-squares estimates of

the coefficients.^4 Based on the airline’s 16 quarters of price and sales data, the

least-squares estimates are: a 478.6 and b 1.63. Thus, the estimated OLS

equation is

[4.2]

Table 4.3 lists Equation 4.2’s sales forecasts and prediction errors quarter by

quarter. The total sum of squared errors (SSE) is 4,847.2—significantly smaller

than the SSE (6,027.7) associated with Equation 4.1.

Q478.61.63P.

138 Chapter 4 Estimating and Forecasting Demand

TABLE 4.3

Predicted versus Actual

Ticket Sales Using

Q 478.6 1.63P

Year and Predicted Actual

Quarter Sales (Q*) Sales (Q) Q* Q(Q* Q)^2

Y1 Q1 71.1 64.8 6.3 39.7

Q2 46.6 33.6 13.0 170.3

Q3 46.6 37.8 8.9 78.3

Q4 87.4 83.3 4.1 16.8

Y2 Q1 103.7 111.7 8.0 64.0

Q2 111.8 137.5 25.7 657.9

Q3 111.8 109.5 2.3 5.5

Q4 120 96.8 23.2 538.2

Y3 Q1 103.7 59.5 44.2 1,953.6

Q2 95.5 83.2 12.3 152.5

Q3 79.3 90.5 11.2 126.6

Q4 87.4 105.5 18.1 327.6

Y4 Q1 71.1 75.7 4.6 21.2

Q2 87.4 91.6 4.2 17.6

Q3 87.4 112.7 25.3 640.1

Q4 95.5 102.2 6.7 44.2

Sum of squared errors 4,847.2

(^4) We provide the general formulas for the least-squares estimators for the interested reader.
Suppose that the estimated equation is of the form and that the data to be fitted con-
sist of n pairs of x-y observations (xi,yi),. Then the least-squares estimators are
and (Here, and are the mean values of the
variables, and the summation is over the n observations.)
b[©(yiy)(xix)] /[©(xix)^2 ] aybx. y x
i1, 2,.. ., n
yabx
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