Thus, for the variable M,

The simple is just the square of the correlation of average cost per case and the proportion of the cases in that category. The two are presented together to illustrate how the partial rs can be lower or higher than the simple rs.

The regression coefficients in this example have to be interpreted in a different way. In the usual case, each regression coefficient measures the change in the dependent variable for unit increase in that independent variable, holding other variables constant. In this example, since the independent variables are proportions in that category; an increase in one variable holding other variables constant does not make sense (in fact, it is impossible to do it). The proper interpretation of the coefficients in this case is this: Put the value of M = 1, all others 0. Then the estimated value of the dependent variable = (constant + coefficient of M) = 69.51 + 44.97 = 114.48. This is the average cost of treating a case in M. Similarly, constant + coefficient of P = 69.51 - 44.54 = 24.97 is the average cost of treating a case in P. Finally, putting M, P, . . . , Ob all = 0, we get the constant term = 69.51 as the average cost of treating a case in the "Other" category. These coefficients are all presented in the last column of Table 4.2. What the regression equation enables us to estimate in this case is the average cost of treating a case in each category. The standard errors of these estimates can be calculated as SE(a + .) if we know the covariance between the constant term & and the other regression coefficient -

An important point to note is that we have not estimated a coefficient for the category "other." An alternative procedure in which we estimate coefficients for all the categories directly would be to include all the variables and delete the constant term. We have to delete the constant term because the sum of all the explanatory variables (which are proportions of cases treated in each category) is equal to 1 for all observations. Since the constant term corresponds to an explanatory variable that is identically equal to 1 for all observations, we cannot include it in the regression equation. If we do, we have a situation where one of the explanatory variables is identically the sum of the other explanatory variables. In this case we cannot estimate all the coefficients.

To see what the problem is, consider the equation

= a + P,x, + 22 + - + "

where = xi + xj. We can easily see that we cannot get separate estimates of 1, 2 and since the equation reduces to

= a + , :, + 22 + (1 + ) + = + ( , + ) 1 + ( + + "

such situation is known as "perfect multicollinearity" and is discussed in greater detail in Chapter 7.

Example 2: Demand for Food

Table 4.3 presents data on per capita food consumption, price of food, and per capita income for the years 1927-1941 and 1948-1962.* Two equations are estimated for these data:

Equation 1: log = a + , log p + P2 log

Equation 2: log = a -f 1 log P + 2 log + log p log

The last term \n Equation 2 is an interaction term that allows the price and income elasticities to vary. Equation 1 implies constant price and income elasticities.

*The data are from Frederick V. Waugh, Demand and Price Analysis: Some Examples from Agriculture, U.S.D.A. Technical BulleUn 1316, November 1964, p. 16. The value of j for 1955 has been corrected to 96.5 from 86.5 that table.

A regression equation with no constant term is also called "regression through the origin" because if the x-variables are all zero, the y-variable is also zero. The estimation of such an equation proceeds as before and the normal equations are defined as in equations (4.7) and (4.8) except that we do not apply the "mean corrections." That is, we define

511 = X xi and not X (xi, - xjf

512 = S Xi,X2, and not X (xi, - X)(x2, - Xj) etc.

After these changes are made, the solution of the normal equations proceeds as before. The definition of RSS and also proceeds as before, with the change that in 5 as well, no mean correction be applied.

Cautionary Note: Many regression programs allow the option of including or excluding the constant term. However, some of them may not give the correct R when running a regression equation with no constant term. If the R is computed as 1 - RSS/5„ and 5„ is computed using the mean correction, we might get a very small and sometimes even a negative R.

Note that with two explanatory variables, and no constant term, RSS = 2 >? - Pi(5] y.xu) - 02(2 . ) and if S„ is computed (wrongly) with a mean correction, so that 5„ = X ? ~ 3 happen that < RSS and thus we get negative R.

The correct R in this case is 1 - RSS/X ?- Of course, if is very high X y? will be very high compared with RSS and we might get a high R. But one should not infer from this that the equation with no constant term is better than the one with a constant term. The two R values are not comparable. With a constant term, the R measures the proportion of XCVi ~ explained by the explanatory variables. Without a constant term that R explains the proportion of 2 y explained by the explanatory variables. Thus we are talking of proportions of two different things, and hence we cannot compare them.

Retail prices of Bureau of Labor Statistics, deflated by dividing by Consumer Price Index. Per capita disposable income, deflated by dividing by Consumer Price Index

The estimates of a, , and for 1927-1942 and 1948-1962 separately and for both periods combined are presented in Table 4.4. Note that we have presented the t-ratios in parentheses because this is more convenient to interpret. If we are interested in significance tests, the /-ratios are more convenient and if we are interested in obtaining confidence intervals, it is convenient to have the standard errors.

Table 4.3 Indexes of Food Consumption, Food Price, and Consumer Income"