THE MODERN INTERPRETATION OF REGRESSION

The modern interpretation of regression is , however, quite different, Broadly speaking, we may say

Regression analysis is concerned whit the study of the dependence of one variable , the dependent variable , on one or more other variable , the explanatory variable, with a view to estimating and / or predicting ( 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 universal regression. Galton was interested in finding pot 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 father's height. In other words, our concern is with predicting the average height of sons knowing the height of their fatherts, To see how this can be done, consider Figure 1.1 , Wich is a scatter diagram, or scatter gram.

 This figure shows the distribution of height of son in a hypothetical population corresponding to given or fixed vaues 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 given value of father's height. the average show this clearly, the circled crosses in the figure indicate the average height average , of son corresponding to a given height of the father's.Connecting these average, 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 Figure1.2 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 given age are likely to have identical height. But height on the average increases with age ( of corse, up to a certain age ), which can be seen clearly if we draw aline ( the regression line ) though the circled point that represent the average height at the given ages. thus, knowing the age , we may by able to predict from the regression line the average height corresponding that age.
 

FIGURE 1.2

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

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 helppfuk in estimating the marginal propensity to consume ( MPC ) , that is, average change in consumption exspenditure  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 mos 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 Philips curve relating change 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 rat. 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 price.

FIGURE  1.3

FIGURE1.4

 

 7. The marketing director of a company may want to know how the demand for company's product is related to,say ,advertising expenditure. Such study will be of considerable help in finding out elasticity of demand with respect to advertising expenditure, that is, the percent change in demand in response to ,say, a1 percent change in advertising budget.

This  Knowledge may be helpful in determining the " optimum" advertising budget.

8. Finally, an agronomist may be interested in studying the dependence of crop tied, say,of wheat, on temperature, rainfall, amount of sunshine, and fertilizer. Such a dependence analysis may enable the prediction  or forecasting of the average crop yield, given information about the explanatoy variable.

The reader can supply scores of such examples  of the dependence of one variable on one or more other variable. The techniques of regression analysis discussed in this text are specially designed to study such dependence among variables.