To economize on >ace. we are not presenting the residuals here. We present residuals for some multiple regression equations in CSiiter 12.

lUustratfve Example

In Table 3.11 we present data on output, labor input, and capital input for the United States few the period 1929-1967. The variables are

X = index of gross national product in constant dollars

jL, = labor input index (number of persons adjusted for hours of work and educational level)

L2 = persons engaged

Kt = captal input index (capital stock adjusted for rates of utilization) K2 = capital stock in constant ddlm

Since output depends on both labor and capital inputs we have to estimate a regression equation of Z on L and K. This is what we will illustrate in Chapter 4. For the present let us study the relationship between output and labor input, and output and capital input. We will use L, for labor input and A, for capital input.

From the data it is clear that X has risen faster than L, or Ki. Hence we have a situation depicted in Figure 3.7(b). This suggests a semilogarithmic form or a double-logarithmic form. The advantage with the latter is that we can interpret the coefficients as elasticities. Moreover, even though X has been rising faster than £, ( Ki), it has not been rising that fast to justify a regression of log X on L, (or K. We will therefore estimate the equation in the double-logarithmic form:

logX = a - pbgl.,

A regression of X on £, ive the following results (figures in parentheses are stand-d errors):

1 = -338.01 + 3.052 r2 = 0.9573

.») «.106»

When \re examined the residuals, they were positive for the first 12 observations, negative for the middle 17 observations, and positive again for the last 10 observations. This is what we would expect if the relationship is as shown in Figure 3.7(b) and we estimate a linear function.

The results of a regression of log X on log L, gave the following results (figures ia i»ientheses are standard errors):

log 1 = -5.480 - 2.084 log L, = 0.9851

«0 226)

This equation still exhibited some pattern in the residuals, although not to the same extent as the linear form.

Table 3.11 Output, Labor Input, and Capital Input for the United States, 1929-1967

Source: L. R. Christensen and D. W. Jorgenson, "U.S. Real Product and Real Factor Input 1929-67," Review of Income and Wealth, March 1970.

*Optional section.

The log-linear form is more easy to interpret. The elasticity of output with respect to the labor input is about 2. This is, of course, very high, but this is because of the omission of the capital input. In Chapter 4 we estimate the production function with the capital input included.

In this particular example, estimation in the linear form is not very meaningful. A linear production function assumes perfect substitutability between capital and labor inputs. However, it is given here as an example illustrating Figure 3.7(b).

Note that is higher in the log-linear form than in the linear form. But this comparison does not mean anything because the dependent variables in the two equations are different. (This point is discussed in greater detail in Section 5.6 of Chapter 5.)

Finally, in the case of a simple regression it is always easy to plot the points on a graph and see what functional form is the most appropriate for the problem. This is not possible in the case of multiple regression with several explanatory variables. Hence greater use is made of the information from the residuals.

* Inverse Prediction in the Least Squares Regression Model

At times a regression model of on x is used to make a prediction of the value of X which could have given rise to a new observation y. As an illustration suppose that we have data on sales and advertising expenditures for a number of firms in an industry. There is a new firm whose sales are known but advertising expenditures are not. In this case we would want to get an estimate of the advertising expenditures of this firm.

In this example we have the estimated regression equation

= a + px

and given a new value of we have to estimate the value Xq of x that could have given rise to and obtain a confidence interval for Xo.

To simplify the algebra we will write x = x - x and y = - y. Then the estimated regression equation can be written as

y = px (3.12)

We are given the value of and we are asked to estimate the value Xo of x that could have given rise to . Define yo = - and xo = Xq - x. The estimate of xo given from equation (3.12) is

x\ = (3.13)