1.2 THE MODERN INTERPRETATION OF RE

1.2 THE MODERN INTERPRETATION OF REGRESSION
The modern interpretation of regression is, however, quite different.
Broadly speaking, we may say
Regression analysis is concerned with the study of the dependence of one variable,
the dependent variable, on one or more other variables, the explanatory variables,
with a view to estimating and/or predicting the (population) mean or average
value of the former in terms of the known or fixed (in repeated sampling)
values of the latter.
The full import of this view of regression analysis will become clearer as
we progress, but a few simple examples will make the basic concept quite
clear.
Examples
1. Reconsider Galton’s law of universal regression. Galton was interested
in finding out why there was a stability in the distribution of heights
in a population. But in the modern view our concern is not with this explanation
but rather with finding out how the average height of sons changes,
given the fathers’ height. In other words, our concern is with predicting the
average height of sons knowing the height of their fathers. To see how this
can be done, consider Figure 1.1, which is a scatter diagram, or scatter gram. This figure shows the distribution of heights of sons in a hypothetical
population corresponding to the given or fixed values of the father’s height.
Notice that corresponding to any given height of a father is a range or distribution
of the heights of the sons. However, notice that despite the variability
of the height of sons for a given value of father’s height, the average
height of sons generally increases as the height of the father increases. To
show this clearly, the circled crosses in the figure indicate the average height
of sons corresponding to a given height of the father. Connecting these
averages, we obtain the line shown in the figure. This line, as we shall see, is
known as the regression line. It shows how the average height of sons
increases with the father’s height.3
2. Consider the scattergram in Figure 1.2, which gives the distribution
in a hypothetical population of heights of boys measured at fixed ages.
Corresponding to any given age, we have a range, or distribution, of heights.
Obviously, not all boys of a given age are likely to have identical heights.
But height on the average increases with age (of course, up to a certain age),
which can be seen clearly if we draw a line (the regression line) through thecircled points that represent the average height at the given ages. Thus,
knowing the age, we may be able to predict from the regression line the
average height corresponding to that age.
3. Turning to economic examples, an economist may be interested in
studying the dependence of personal consumption expenditure on aftertax
or disposable real personal income. Such an analysis may be helpful
in estimating the marginal propensity to consume (MPC), that is, average
change in consumption expenditure for, say, a dollar’s worth of change in
real income (see Figure I.3).