Question

UESTION 7 Most investment firms provide estimates of systematic risks of securities, called betas. A stock's beta measures the relationship between the stock's rate of return and the rate of return for the market as a whole. The term beta derives its name from the beta coefficient for the slope in simple linear regression, where the dependent variable is the stock's rate of return and the independent variable i:s the market rate of return. Stocks with beta values greater than 1 are considered "aggressive" securities since their rates of return are expected to move (upward or downward) faster than the market asa whole. By contrast, stocks with beta values less than 1 are called "defensive" securities since their rates of return move slower than the market. A stock with a beta value of 1 is called a "neutral security for its rate of return mirrors the market Below are the annual rates of return (in %) for General Mills stock for nine randomly selected years The market rate of return, as measured by Standard & Poor's 500 Composite Index is also given (in %) for each of these years. Assume that the simple linear regression model is appropriate for these data ear 6 Market 19.0 14.7 -26.5 37.223.8 7.2 6.6 18.4 32.4 General Mills: 57.4-13.0 -24.4 50.4 17.3 -9.3 0.7 -11.8 13.1 (a) Determine the least squares line y-b b1Xi. Use the estimate that you obtain for β1 to categorize General Mills stock as aggressive, defensive, or neutral (b) Compute an estimate of the error term variance (c) Determine a 95% confidence interval for the mean rate of return on General Mills stock in years (d) Calculate and interpret the coefficient of determination, and classify the strength of the linear (e) If the market rate of return is forecast to be 10% for next year, determine a 95% prediction interval (f) If the market rate of return is forecast to be 80% for next year, would you still be inclined to where the market rate of return is 10%, and interpret your result component here as strong, moderate, or weak for the rate of return on General Mills stock, and interpret your result determine a 95% prediction interval for the rate of return on General Mills stock? Explain

Answer 1

x <- c(19,-14.7,-26.5,37.2,23.8,-7.2,6.6,18.4,32.4)
y <- c(57.4,-13,-24.4,50.4,17.3,-9.3,0.7,-11.8,13.1)
model <- lm(y~x)
summary(model)

Call:
lm(formula = y ~ x)

Residuals:
    Min      1Q  Median      3Q     Max 
-28.946  -5.060  -1.744   1.793  39.675 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept)  -0.6086     7.6636  -0.079   0.9389  
x             0.9649     0.3349   2.881   0.0236 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 20.73 on 7 degrees of freedom
Multiple R-squared:  0.5426,    Adjusted R-squared:  0.4772 
F-statistic: 8.302 on 1 and 7 DF,  p-value: 0.02361

a)

y^= -0.6086 + 0.9649 x

b1 = 0.9649

This is defensive

b)

error term variance = standard error ^2 = 20.73^2 =

429.9 

c)

y^= -0.6086 + 0.9649 *10

= 9.0404

d)

coefficient of determination = R^2 = 0.5426

it means 54.26 % of variation is explained by this model

strength - moderate

e)

predict(model, data.frame(x =10),interval = "prediction")
       fit       lwr      upr
1 9.040546 -42.63862 60.71971

95% prediction interval = (-42.63862, 60.71971)

f)

No, because 80% is outside sample data

Extrapolation is not trustworthy in linear regression