(i) Let ßo and ß, be the intercept and slope from the regression of y; on X;, using n observations. Let C? and c2, with c2 + 0, be constants. Let Bo and ß be the intercept and slope from the regression of c?y; on C2x;. Show that Bi (C1/c2), and B. = cißo, thereby verifying the claims on units of measurement in Section 2-4. [Hint: To obtain B1, plug the scaled versions of x and y into (2.19). Then, use (2.17) for Bo, being sure to plug in the scaled x and y and the correct slope.] =C Therefore, provided that §(: - 1) (x; – x)2 > 0, [2.18] the estimated slope is $(x; – x)(y; - y) [2.19] Σα - ) (x; – x2 Equation (2.19) is simply the sample covariance between and yi divided by the sample variance of x;. Using simple algebra we can also write Ê, as X; ß = Pay . X where êxy is the sample correlation between x; and y; and ?x, ô, denote the sample standard devia- tions. (See Math Refresher C for definitions of correlation and standard deviation. Dividing all sums by n - 1 does not affect the formulas.) An immediate implication is that if x; and y; are positively cor- related in the sample then Ê> 0; if x; and y; are negatively correlated then Ñ, <0. Not surprisingly, the formula for ß, in terms of the sample correlation and sample standard devia- tions is the sample analog of the population relationship o Bi = Pxy () x where all quantities are defined for the entire population. Recognition that B, is just a scaled version of P.xy highlights an important limitation of simple regression when we do not have experimental data: in effect, simple regression is an analysis of correlation between two variables, and so one must be careful in inferring causality. Although the method for obtaining (2.17) and (2.19) is motivated by (2.6), the only assumption needed to compute the estimates for a particular sample is (2.18). This is hardly an assumption at all: (2.18) is true, provided the x; in the sample are not all equal to the same value. If (2.18) fails, then X; n Using the basic properties of the summation operator from Math Refresher A, equation (2.14) can be rewritten as y = ßo + Bix, [2.16] where y n-'X=1y; is the sample average of the y; and likewise for x. This equation allows us to write B, in terms of B1, y, and x: Bo = y - Bx. [2.17] Therefore, once we have the slope estimate ß1, it is straightforward to obtain the intercept estimate ßo, given y and x. Dropping the n in (2.15) (because it does not affect the solution) and plugging (2.17) into (2.15) yields Ex[y: - (– B17) – B?x;] = 0, which, upon rearrangement, gives ) Ex(); – 5) = B. Xx(x; – 7). From basic properties of the summation operator (see (A-7) and (A-8) in Math Refresher A], n Σ(α; x) = - ) - 5) = $ (x; – 7)? and Šx(); – 5) = $(x; – 7)(y – 5). Σ( - )) = - i- y) i= i=1

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