Referencing Styles : Open Q1 (1 mark): In lectures we saw that the ‘total deviation about the mean’ is always zero, i.e. ∑= − = n i i x x 1 ( ) 0 . Expand the summation across the brackets and apply this result to the i y to prove that: ( )( ) ( ) 1 1 x x y y x y y i n i i i n i ∑ i − − = ∑ − = = . Ans: Q2 (1 mark) Prove that � (xi − x�)2 𝑛 𝑖=1 = � xi (xi − x�) 𝑛 𝑖=1 Ans: Q3 (2 marks): By differentiating the summation, show that ( 0 , 1 ) 0 0 = ∂ ∂ f b b b when b y b x 0 0 1 ˆ ≡ β = − . Ans: Q4 (1 mark): Differentiate the summation with respect to 1 b to get a summation expression for ( , ) 0 1 1 f b b ∂b ∂ . Ans: Q5 (2 marks): Prove that by choosing b0 and b1 to minimize ∑= − − n i i i y b b x 1 2 0 1 ( ) you obtain the least squares estimators, namely: b y b x x x x x y y b n i i i n i i 0 0 1 1 2 1 1 1 ˆ ( ) ( )( ) ˆ ≡ = − − − − ≡ = ∑ ∑ = = β β Ans:Q6. (1 marks): Read the supplied data into Eviews. Generate two new variables 𝑟𝐵𝐻𝑃_𝑟𝑓 and 𝑟𝑚_𝑟𝑓 , which are the stock and market‘excess returns’ respectively, assuming 𝑟𝑓 = 0.005. Use these to estimate the model given by Equation (1): 𝑟𝐵𝐻𝑃 − 𝑟𝑓 = 𝐵0 + 𝐵1( 𝑟𝑚 − 𝑟𝑓) + 𝑢𝑡 Paste your Eviews output below Ans: Q7. (1 mark) Comment on the sign of the estimated coefficient 𝐵1, and state whether this is what you expect. Ans: Q8 (1 mark) Formulate and carry out an appropriate hypothesis test, to test whether the excess market returns explain the excess returns of BHP shares. Use the t-statistic approach, at the α=0.05 level. Assume the large sample approximation applies. Ans: Q9 (1 mark) Formulate and carry out an appropriate hypothesis test for testing whether BHP’s ‘beta’ is greater than one at the α=0.05 level Q10. (1 mark): Use the data to estimate the Fama-French 3-Factor model given by Equation (2): 𝑟𝐵𝐻𝑃 − 𝑟𝑓 = 𝐵0 + 𝐵1( 𝑟𝑚 − 𝑟𝑓) + 𝐵2 𝑆𝑀𝐵 + 𝐵3 𝐻𝑀𝐿 + 𝑢𝑡 Paste your Eviews output below Ans: Q11 (2 marks) In the Fama-French 3-factor model you estimated, test the following hypotheses about the coefficients B2 and B3. Clearly specify the rejection region if you are using critical values, and clearly state your conclusions. When using p-values, calculate and compare your p-values to the test size then state your conclusion. (Hint, assume the Central Limit Theorem Holds) (a) H0: 𝐵2 = 0, H1: 𝐵2 > 0, with α=0.05 using the critical-value approach. (b) H0: 𝐵2 = 0, H1: 𝐵2 < 0, with α=0.05 using the critical-value approach. (c) H0: 𝐵3 = 0, H1: 𝐵3 > 0, with α=0.05 using the p-value approach. (d) H0: 𝐵3 = 0, H1: 𝐵3 < 0, with α=0.05 using the p-value approach. Q12 (3 marks) Formulate a joint-hypothesis test to test whether the Fama-French 3-Factor model explains the stock returns better than the model given by Equation (1). Perform the hypothesis test by calculating the homoskedasticity-consistent F-Statistic, using the relevant formula. Verify your conclusion by performing the Wald test in Eviews and considering p-values. What is your conclusion? Q13 (3 marks) A Financial Analyst believes that the effect of book-to-market values (HML) on stock returns is twice as great as the effect of market capitalization (SMB). Formulate an appropriate hypothesis test and use re-parametrisation to convert it to a simple t-test to test the assertion. Perform the required regression and paste your Eviews output below. State your conclusion at the 5% level.