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.

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-

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Father's height, inches

FIGURE 1.1 Hypothetical distribution of sons' heights corresponding to given heights of fathers.

CHAPTER ONE: THE NATURE OF REGRESSION ANALYSIS 19

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 the

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Age, years

FIGURE 1.2 Hypothetical distribution of heights corresponding to selected ages.

3At this stage of the development of the subject matter, we shall call this regression line simply the line connecting the mean, or average, value of the dependent variable (son's height) corresponding to the given value of the explanatory variable (father's height). Note that this line has a positive slope but the slope is less than 1, which is in conformity with Galton's regression to mediocrity. (Why?)

20 PART ONE: SINGLE-EQUATION REGRESSION MODELS

circled 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).

4. A monopolist who can fix the price or output (but not both) may want to find out the response of the demand for a product to changes in price. Such an experiment may enable the estimation of the price elasticity (i.e., price responsiveness) of the demand for the product and may help determine the most profitable price.

5. A labor economist may want to study the rate of change of money wages in relation to the unemployment rate. The historical data are shown in the scattergram given in Figure 1.3. The curve in Figure 1.3 is an example of the celebrated Phillips curve relating changes in the money wages to the unemployment rate. Such a scattergram may enable the labor economist to predict the average change in money wages given a certain unemployment rate. Such knowledge may be helpful in stating something about the inflationary process in an economy, for increases in money wages are likely to be reflected in increased prices.

Unemployment rate, %

FIGURE 1.3 Hypothetical Phillips curve.

CHAPTER ONE: THE NATURE OF REGRESSION ANALYSIS 21

Money Income

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