1. Regression and the ANOVA table. A question of interest in finance is how retu
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Question
1. Regression and the ANOVA table.
A question of interest in finance is how returns on common stocks in overseas markets are related to returns in US markets. To address this question a simple linear regression model was built by regressing the annual rate of return on the Morgan Stanley Europe, Australasia, Far East (EAFE) index on the annual rate of return on the Standard & Poor’s 500 stock index for the years 1993 to 2012. Both the EAFE index and the S&P 500 index are recorded in percentages. The fitted regression line obtained is given by
EAFE = - 0.168 + 0.845 S&P.
a. Part of the ANOVA table from the regression output is provided below. Complete the missing information.
Source SS df MS
Model 4947.2
Residual
Total 8251.5 19
b. Using the information from the ANOVA table determine the value of s, the estimated standard deviation for the model. Compute the value of R2. What percentage of variation in EAFE is explained by the linear model?
c. Using the Goodness of Fit F-test, test the hypothesis that the linear model does a good job of explaining the relationship between EAFE and S&P. State the explicit hypotheses that you are testing, the value of the test statistic, the p-value and the conclusion to your test.
d. Interpret the coefficients of the linear regression equation in the context of this model.
Explanation / Answer
Answers to the questions is as below:
EAFE = - 0.168 + 0.845 S&P.
a.
We complete the ANOVA table as below:
SSresidual = SStotala - SSmodel = 8251.5 - 4947.2 = 3304.2
dfresidual = total df - 1 = 19-1 = 18
dfmodel = 1
MSmodel = SSmodel/dfmodel = 4947.2/1 = 4947.2
MSresidual = SSresidual/dfresidual = 3304.2/18 = 183.56
F = MSmodel/MSresidual = 4947.2/183.56 = 26.95
p-value for p(F,df1,df2) = p(26.95,1,18)
The p-value is .000061. The result is significant at p < .05.
b.
s = sqrt(SSE/n-k-1) =sqrt(3304.2/(19-1-1)) = 13.94
Rsquare = 4947.2/8251.5 = .5995 or 60%
Percentage of variation in EAFW is explained by the linear model is .5995 ot 59.95%
c.
The hypotheses of interest in this ANOVA are as follows:
H0: eafe = s&p
H1: Means are not all equal
c.
The value of test statistic is F-stat is 26.95
p-value is .000061.
The conclusion - the fit of the regression model is significant, the model fits the points
d. The EAFE has a positive linear relation with S&P. Per increase in S&P points EAFE increases by .845
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