Question

The data in the accompanying table represent the rate of return of a certain company stock for 11 months, compared with the rate of return of a certain index of 500 stocks. Both are in percent. Test if there is a positive relationship at the α=0.01 level of significance.

Month	Rates of return of the index, x	Rates of return of the company stock, y
Apr-07	-5.94	-2.36
May-07	4.78	3.91
Jun-07	-2.02	-0.74
Jul-07	-6.13	-3.08
Aug-07	-4.27	-2.59
Sept-07	0.53	2.11
Oct-07	4.58	3.56
Nov-07	4.24	5.16
Dec-07	-3.99	-1.32
Jan-08	3.89	4.24
Feb-08	4.58	3.67

• What is the Standard Error of the regression?

• What is the left boundary of the estimation of the regression equation slope?

• What is the critical value of the test?   

• What is the test statistics of the test?

• What is the p-value of the test?

Answer 1

using excel data analysis tool for regression, following o/p is obtained

Regression Statistics
Multiple R	0.9787
R Square	0.9579
Adjusted R Square	0.9532
Standard Error	0.685
Observations	11
ANOVA
	df	SS	MS	F	Significance F
Regression	1	95.980	96	204.77	0.0000
Residual	9	4.218	0
Total	10	100.199
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	1.126	0.206	5.46	0.0004	0.456	2
X	0.677	0.047	14.31	0.0000	0.523	0.831

from obtained o/p, we find that

a)

Standard Error of the regression = 0.685

b)

slope ,    ß1 = 0.6772

n=   11
alpha=   0.01
df- n-2 = 9

t critical value=   t α/2 =    3.2498 [excel function: =t.inv.2t(0.01,9) ]

margin of error ,E=       t*estimated std error of slope = 3.2498 * 0.0473 = 0.1538
lower confidence limit =        ß̂1-E = 0.6772 - 0.1538 = 0.523  

so, left boundary of the estimation of the regression equation slope = 0.523

c)

Ho:   ß1=   0
H1:   ß1 >   0
n=   11  
alpha=   0.01  

critical value for test =   t α,df =    2.8214     [excel function: =t.inv(0.01,9) ]

d)

t stat =    ß1 /Se(ß1) =0.6772/0.0473 = 14.3098

e)

p-value =    0.0000 [excel dunction: =t.dist.rt(14.3098,9) ]