Question

2.48 The “January effect.’’ Some people think that the behavior of the stock

market in January predicts its behavior for the rest of the year. Take the explana-
tory variable x to be the percent change in a stock market index in January and

the response variable y to be the change in the index for the entire year. We
expect a positive correlation between x and y because the change during January
contributes to the full year’s change. Calculation based on 38 years of data gives

x = 1.75% sx = 5.36% r = 0.596
y = 9.07% sy = 15.35%

(a) What percent of the observed variation in yearly changes in the index is
explained by a straight-line relationship with the change during January?
(b) What is the equation of the least-squares line for predicting the full-year
change from the January change?

(c) The mean change in January is x = 1.75%. Use your regression line to pre-
dict the change in the index in a year in which the index rises 1.75% in January.

Why could you have given this result (up to roundoff error) without doing the
calculation?

Answer 1

(a)

Given r = 0.596

r^2 = 0.3552 = 35.52%

35.52% percent of the observed variation in yearly changes in the index is explained by a straight-line relationship with the change during January

b)

Given

x bar = 1.75% sx = 5.36% r = 0.596
y bar = 9.07% sy = 15.35%

Let least-squares line Y = mx + b

Slope (m) = r(Sy/Sx) = 0.596 (15.35/5.36)

Slope (m) = 1.7068

Intercept (b) = Y bar - m * X bar

Intercept (b) = 9.07 - 1.7068 * 1.75

Intercept (b) = 6.0831

the equation of the least-squares line for predicting the full-year change from the January change

Y = 1.7068 * X + 6.0831

c)

If X = 1.75

Y = 1.7068 * X + 6.0831

Y = 1.7068 * 1.75 + 6.0831 = 9.07%

So, We know that the regression equation is going through mean values of x and y