2.1 (E Sections 2.1.2, 2.3.1) Let two data sets (xi, yi) and (xi , yi ) be related by xi c1 c2xi and

2.1 (E Sections 2.1.2, 2.3.1)

Let two data sets (xi, yi) and (x
i , y
i ) be related by x
i ¼ c1 þ c2xi and y
i ¼ c3 þ c4yi for all i ¼ 1,    , n. This means that the only differences between the two data sets are the location and the scale of measurement. Such data transformations are often applied in economic studies. For example, yi may be the total variable production costs in dollars of a firm in month i and y
i the total production costs in millions of dollars. Then c3 are the total fixed costs (in millions of dollars) and c4 ¼ 106.

a. Derive the relation y
¼ a
þ b
x
between y

and x
if y and x would satisfy the linear relation y ¼ a þ bx.

b. For arbitrary data (xi, yi), i ¼ 1,    , n, derive the relation between the least squares estimators

(a,

b) for the original data and (a
, b
) for the transformed data.

c. Which of the statistics R2, s2, sb, and tb are invariant with respect to this transformation?

d. Check the results in b and c with the excess returns data of Example 2.1 on stock market returns. Perform two regressions, one with the original data

(in percentages) and the other with transformed data with the actual excess returns—that is, with c1 ¼ c3 ¼ 0 and c2 ¼ c4 ¼ 0:01.