$[26$ marks$]$ Investment firms provide estimates of the systematic risks of stocks called betas. A

stock's beta measures the relationship between a stock's rate of return and the rate of return for

the stock market overall. A stock's beta can be determined from the estimated slope in a linear

model in which a stock's rate of return for a year is the response variable. A stock can be deemed

aggressive if it has a beta greater than $1,$ neutral $($or market$-$like$)$ if its beta equals $1,$ or

defensive if its beta is less than $1.$

We wish to estimate the beta for two stocks $($IBM and General Mills$),$ based on $10$ years worth of data

for the stock returns. The data are included in the table below, and are also posted on LEARN in the

file beta.txt$.$ The variable names are Year, IBM, GM$,$ Market.

a$.[5]$ Let $Y_{i,\text{I}\text{B}\text{M}}$ represent the rate of return for the IBM stock in year $i,$ and let $x_i$ be the

market return for year i$.$ Fit the model: $Y_{i,\text{I}\text{B}\text{M}}={}_{0,\text{I}\text{B}\text{M}}+{}_{1,\text{I}\text{B}\text{M}}{x}_i+R_{i,\text{I}\text{B}\text{M}},$ where

$R_{i,\text{I}\text{B}\text{M}}G(0,{}_{\text{I}\text{B}\text{M}}),i=1,2,$dots,$10$; Independent. Determine the maximum likelihood

estimates of ${}_{0,\text{I}\text{B}\text{M}},{}_{1,\text{I}\text{B}\text{M}},$ and the estimate of ${}_{\text{I}\text{B}\text{M}},$ using the unbiased estimate of

${}_{\text{I}\text{B}\text{M}}^2.$

b$.[3]$ Using the standardized residual plots from this model, examine the model

assumptions using residual plots. Comment on the suitability of the assumptions.

Include all necessary plots with your work for this part of the question.

Note: Residual plot analysis will allow you to check for Linearity, Independence, and

Constant variance. We will check Normality in the next part of this question.

c$.[2]$ Using R$,$ you can conduct a Shapiro$-$Wilk test for Normality of the residuals. Obtain

the residuals $($either by hand and load them into R$,$ or directly in $R$ if you loaded the

data in$),$ then use the command shapiro.test in $R$ to test for Normality of the residuals.

In this case, $H_0$ : Residuals are Normal.

Calculate a $p-$value and use the guidelines on page $128$ of your Course Notes to assess

the amount of evidence against $H_0,$ based on the data. What would you conclude here?

Note: You can also reference a QQ plot, or you may also include the QQ plot in your

work for this part of the question.