can make a forecast of sales look absurd. The occurrence of a war or a
change in Federal Reserve policy is a weakness of many economic models.
Least squares regression on a firm’s sales is powerless against major eco-
nomic catastrophes, but it can point a reasonable direction for the firm to
pursue.
There are four principal goals of regression and correlation analysis.
First, regression analysis provides estimates of the dependent variable for
given values of the independent variable. Second, regression analysis pro-
vides measures of the errors that are likely to be involved in using the re-
gression line to estimate the dependent variable.
Third, regression analysis provides an estimate of the effect on the
mean value of Yof a one-unit change in X. Regression analysis enables us
to estimate this slope and to test hypotheses concerning its value. Fourth,
correlation analysis provides estimates of how strong the relationship is be-
tween the two variables. The coefficient of correlation and the coefficient
of determination are two measures generally used for this purpose.
The Linear Regression Model
A model is a simplified or idealized representation of the real world. All
scientific inquiry is based to some extent on the use of models. In this sec-
tion, we describe the model—that is, the set of simplifying assumptions—
on which regression analysis is based. To begin with, the statistician
visualizes a population of all relevant pairs of observations of the indepen-
dent and dependent variables.
Holding constant the value of X(the independent variable), the statis-
tician assumes that each corresponding value of Y(the dependent variable)
is drawn at random from the population. (See Figure 5.2.)
The probability distribution of Y, given a specified value of X, is called
the conditional probability distribution of Y. The conditional probability
distribution of Y, given the specified value of X, is denoted by
P(Y|X)where Yis the value of the dependent variable and Xis the specified value
of the independent variable. The mean of this conditional probability dis-
tribution is denoted by μy⋅x, and the standard deviation of this probability
distribution is denoted by σy⋅x.
Regression analysis makes the following assumptions about the condi-
tional probability distribution of Y. First, it assumes that the mean value of