FIGURE 3.1 Least-squares criterion.
60 PART ONE: SINGLE-EQUATION REGRESSION MODELS
equal importance no matter how close or how widely scattered the individual observations are from the SRF. A consequence of this is that it is quite possible that the algebraic sum of the u is small (even zero) although the Ui are widely scattered about the SRF. To see this, let Ui, U2, U3, and U4 in Figure 3.1 assume the values of 10, —2, +2, and —10, respectively. The algebraic sum of these residuals is zero although U1 and U4 are scattered more widely around the SRF than U2 and U3. We can avoid this problem if we adopt the least-squares criterion, which states that the SRF can be fixed in such a way that
is as small as possible, where u2 are the squared residuals. By squaring Ui, this method gives more weight to residuals such as U1 and U4 in Figure 3.1 than the residuals U2 and U3. As noted previously, under the minimum J2U criterion, the sum can be small even though the Ui are widely spread about the SRF. But this is not possible under the least-squares procedure, for the larger the Ui (in absolute value), the larger the J]U2. A further justification for the least-squares method lies in the fact that the estimators obtained by it have some very desirable statistical properties, as we shall see shortly.
It is obvious from (3.1.2) that
that is, the sum of the squared residuals is some function of the estimators ft and ft. For any given set of data, choosing different values for ft and ft will give different us and hence different values of J2U. To see this clearly, consider the hypothetical data on Y and X given in the first two columns of Table 3.1. Let us now conduct two experiments. In experiment 1,
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