Question

1) Suppose that you are interested in the relationship between the return rate on a stock in 2010 compared to the return rate in 2009. You believe that the return rates in both years are positively correlated. A sample of 15 stocks yields the following regression results: b0= 5.3, b1= 1.04, s= 1.79, s = 0.2163, R2 = 0.64, and MSE = 35.4.

Calculate the regression sum of squares.

What is the correlation coefficient for the stock returns of the two years? What sign does it have? Why?

What are the appropriate null and alternative hypotheses?

Test the hypothesis at α = 0.05 Test H0: β1 = 0 vs. H1 : β1 > 0 at α = 0.025

Use the F-statistic from the analysis of variance table for regression to test H0: β1 = 0 vs. H1 : β1 > 0 at α = 0.05.

Answer 1

Given,

b0= 5.3, b1= 1.04, Sb0 = 1.79, Sb1 = 0.2163, R^2 = 0.64, and MSE = 35.4.

Standard error of estimate (Se) = SQRT(MSE) = SQRT(35.4) = 5.9498

and we know that Standard error of slope (Sb1) = Se/SQRT(SSxx)

So, Se/SQRT(SSxx) = 0.2163

SQRT(SSxx) = 5.9498/0.2163

SSxx = 27.5072^2 = 756.6442

Slope = SSxy/SSxx = 1.04

SSxy = 1.04*756.6442 = 786.91

Therefore, the regression sum of squares SSR = Slope*SSxy = 1.04*786.91 = 818.3864

the correlation coefficient for the stock returns of the two years (r) = SQRT(R^2) = SQRT(0.64) = 0.8

r is positive because slope is positive

There is a Positive and Strong relationship between the return rate on a stock in 2010 compared to the return rate in 2009

F test:

H0: β1 = 0 vs. H1 : β1 > 0 at α = 0.025 and at α = 0.05

df1 = k-1 = 2-1 = 1

df2 = n-k = 15-2 = 13

Fstat = MSR/MSE

MSR = SSR/df1 = 818.3864/1 = 818.3864

F stat = 818.3864/34.5 = 23.7213

F critical at 0.025 = 6.4143

P value at 0.025 = 0.000306

F critical at 0.05 = 4.6672

P value at 0.05 = 0.000306

Reject H0, Regression is significant and positive

t test:

H0: β1 = 0 vs. H1 : β1 > 0 at α = 0.025 and at α = 0.05

t stat = b1/Sb1 = 1.04/0.2163 = 4.8081

P value = 0.0002 < 0.025 < 0.05

Reject H0, There is a significant relatioship between the return rate on a stock in 2010 compared to the return rate in 2009