The variables are x=SP500 market monthly log return and y = monthly return of Ap

Question

The variables are x=SP500 market monthly log return and y = monthly return of Apple for 48 months beginning in January 2009.
For input into R, the data vectors for monthly market return and monthly stock return are
x=c(-0.08955, -0.116457, 0.081953, 0.089772, 0.051721, 0.000196, 0.071522, 0.033009, 0.0351, -0.01996, 0.055779, 0.017615, -0.037675, 0.028115, 0.057133, 0.014651, -0.085532, -0.055388, 0.066516, -0.048612, 0.083928, 0.036193, -0.002293, 0.063257, 0.022393, 0.031457, -0.001048, 0.028097, -0.013593, -0.018426, -0.021708, -0.058467, -0.074467, 0.102307, -0.005071, 0.008497, 0.04266, 0.039787, 0.030852, -0.007526, -0.064699, 0.038793, 0.012519, 0.019571, 0.023947, -0.019988, 0.002843, 0.007043)
and
y=c(0.054521, -0.009844, 0.163178, 0.180219, 0.075986, 0.047628, 0.1374, 0.028859, 0.097099, 0.016933, 0.058762, 0.05271, -0.09252, 0.063101, 0.13851, 0.105141, -0.0162, -0.020846, 0.022278, -0.056502, 0.15454, 0.058929, 0.033429, 0.036004, 0.050494, 0.04018, -0.013426, 0.004635, -0.006538, -0.035537, 0.151108, -0.01443, -0.009222, 0.05975, -0.057437, 0.057982, 0.119578, 0.172546, 0.100109, -0.02637, -0.010644, 0.01077, 0.044731, 0.089729, 0.00286, -0.113904, -0.012387, -0.095123)

For the questions below, use 3 decimal places.

Part a) The coefficients of the least square regression line are Part b) Suppose we want to get a prediction interval for each of the next 10 months (beginning January 2013; when the SP500 returns are values in the following R vector xnext=C(0.0491 98, 0.011, 0.035355, 0.01 7924, 0.02055,-0.015113, 0.048278,-0.031798, 0.029316, 0.04363) The t critical value for the 95% prediction interval is Using the fitted regression equation for January 2009 to December 2012, the lower endpoint of the 95% prediction interval for January 2013 (SP500 return 0.049198) is The upper endpoint of this 95% prediction interval is The lower endpoint of the 95% prediction interval for October 2013 (SP500 return 0.04363) is The upper endpoint of this 95% prediction interval is Part c) Get the 10 prediction intervals for January to October 2013 from part (b) of which you were asked to enter two intervals. The actual values of the monthly stock returns for Apple are in the following vector ynext-c(-0.155568, -0.02563, 0.002789, 0.000328, 0.022193, -0.126007, 0.132236, 0.080422, -0.021832, 0.092029) How many of these observed values (not used in the regression equation) are contained in the corresponding prediction intervals. (The response here is an integer between 0 and 10; theoretically it is close to 9.)

Explanation / Answer

a)
> x=c(-0.08955, -0.116457, 0.081953, 0.089772, 0.051721, 0.000196, 0.071522, 0.033009, 0.0351, -0.01996, 0.055779, 0.017615, -0.037675, 0.028115, 0.057133, 0.014651, -0.085532, -0.055388, 0.066516, -0.048612, 0.083928, 0.036193, -0.002293, 0.063257, 0.022393, 0.031457, -0.001048, 0.028097, -0.013593, -0.018426, -0.021708, -0.058467, -0.074467, 0.102307, -0.005071, 0.008497, 0.04266, 0.039787, 0.030852, -0.007526, -0.064699, 0.038793, 0.012519, 0.019571, 0.023947, -0.019988, 0.002843, 0.007043)

> y=c(0.054521, -0.009844, 0.163178, 0.180219, 0.075986, 0.047628, 0.1374, 0.028859, 0.097099, 0.016933, 0.058762, 0.05271, -0.09252, 0.063101, 0.13851, 0.105141, -0.0162, -0.020846, 0.022278, -0.056502, 0.15454, 0.058929, 0.033429, 0.036004, 0.050494, 0.04018, -0.013426, 0.004635, -0.006538, -0.035537, 0.151108, -0.01443, -0.009222, 0.05975, -0.057437, 0.057982, 0.119578, 0.172546, 0.100109, -0.02637, -0.010644, 0.01077, 0.044731, 0.089729, 0.00286, -0.113904, -0.012387, -0.095123)

> model=lm(y~x) #fitting linear regression model
> coefficients(model)
(Intercept) x
0.03025609 0.84611481

Estimated Beta0 = 0.03025609
Estimated Beta1 = 0.84611481

b)

>newdata=data.frame(x=c(0.049198,0.011,0.035355,0.017924,0.02055,-0.015113,0.048272,-0.031798,0.029316,0.04363) )
> predict(model,newdata,interval="predict")
fit lwr upr
1 0.071883251 -0.04728894 0.1910554
2 0.039563357 -0.07882298 0.1579497
3 0.060170483 -0.05854906 0.1788900
4 0.045421856 -0.07299882 0.1638425
5 0.047643754 -0.07080251 0.1660900
6 0.017468761 -0.10122024 0.1361578
7 0.071099748 -0.04803626 0.1902358
8 0.003351336 -0.11588667 0.1225893
9 0.055060796 -0.06352085 0.1736424
10 0.067172083 -0.05179526 0.1861394

c) 8 ( Except the 1st and 6th observation in ynext, all other observations fall into the prediction interval.)

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