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200 PART ONE: SINGLE-EQUATION REGRESSION MODELS

that is, the mean of actual Y values need not be equal to the mean of the estimated Y values; the two mean values are identical for the intercept-present model, as can be seen from (3.1.10).

It was noted that, for the zero-intercept model, r2 can be negative, whereas for the conventional model it can never be negative. This condition can be shown as follows.

Now for the conventional, or intercept-present, model, Eq. (3.3.6) shows that

unless p2 is zero (i.e., X has no influence on Y whatsoever). That is, for the conventional model, RSS < TSS, or, r2 can never be negative. For the zero-intercept model it can be shown analogously that

(Note: The sums of squares of Y and X are not mean-adjusted.) Now there is no guarantee that this RSS will always be less than Y yf = Yf — NY2 (the TSS), which suggests that RSS can be greater than TSS, implying that r2, as conventionally defined, can be negative. Incidentally, notice that in this case RSS will be greater than TSS if ¡i^YX2 < NY2.

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