Springer Texts in Business and Economics

Adding 5 to each observation of Xi, adds 5 to the sample average X and it

__ n

is now 12.5. This means that xi — Xi — X is unaffected. Hence 'Jf xi2 is the

i=i

same and since Yi, is unchanged, we conclude that "ols is still the same at

0.8095. However, aols = Y — "olsX is changed because X is changed. This is now aols = 6.5 — (0.8095/(12.5) = —3.6188. It has decreased by 5"ols since X increased by 5 while "ols and Y remained unchanged. It is easy to see that Yi = aols + "olsXi remains the same. When Xi increased by 5, with "ols the same, this increases Yi by 5"ols. But aols decreases Yi by —5"ols. The net effect on Yi is zero. Since Yi is unchanged, this means ei = Yi — Yi

n

is unchanged. Hence s2 = ei2/n — 2 is unchanged. Since xi is unchanged i=1 n

s2/ P xi2 is unchanged and seols) and the t-statistic for Ho; " = 0 are

i=i

unchanged. The

is unchanged and therefore se (aols) is unchanged. However, the t-statistic for Hb; a = 0 is now 2.4286/0.60446 = 4.018 which is now statistically significant. R2 = 1 — I J2 ei2/ J2 Уі2 ) is unchanged. Again, only aols is affected by an additive constant of 2 on Yi.

c. If each Xi is multiplied by 2, then the old X is multiplied by 2 and it is now 15. This means that each xi = Xi — X is now double what it was in

n

Table 3.1. Since yi is the same, this means that xiyi is double what it was

i=1

n

in Table 3.1 and xi2 is four times what it was in Table 3.1. Hence,

i=1

nn

"ols = J2**/£xi2 = 2 (42.5) /4 (52.5) = 0.8095/2 = 0.40475.

i=i i=i

In this case, "ols is half what it was in the numerical example. aols = Y — "olsX is the same since "ols is half what it used to be while X is double what it used to be. Also, Yi = aols + "olsXi is the same, since Xi is dou­bled while "ols is half what it used to be and aols is unchanged. Therefore,

n

Єі = Yi — Yі is unchangedand s2 = ei2/n — 2 is also unchanged.

i=1

n

Now, s2/ J2 xi2 is one fourth what it used to be and se("ols) is half what

i=i

it used to be. Since, "ols and se("ols) have been both reduced by half, the t-statistic for Ho; " = 0 remains unchanged. The

n

is unchanged since X and xi2 are now both multiplied by 4.

i=i

Hence, se(aols) and the t-statistic for H/ ; a = 0 are unchanged. R2 = 1 — I ei2/ J2 УіМ is also unchanged. Hence, only " is affected by a multiplicative constant of 2 on Xi.