Question 1

 Consider the following model: 𝐲 = 𝐗𝛃 + 𝛆, known as the Classical Linear Regression Model (CLRM), where y is the dependent variable, X is the set of independent variables, 𝛃 is the vector of parameters to be estimated and 𝛆 is the error term. 

a) List and discuss the assumptions you need for the Ordinary Least Squares (OLS) to be a Best Linear Unbiased Estimator (BLUE). [25 marks] 

b) Derive the OLS estimator and variance and discuss where each assumption is needed for the derivation of the two parameters. [25 marks] 

c) Discuss the properties of linearity, unbiasedness, and efficiency, and what assumption you need for each of these properties to hold and show where each assumption is needed for the derivation. [25 marks]

 d) Present and discuss the R2 and the adjusted R2. Discuss pros and cons of each of the two statistics. [25 marks] 

 Question 2 

You estimate the following model: 𝐸π‘₯𝑐𝑒𝑠𝑠_π‘Ÿπ‘’π‘‘π‘’π‘Ÿπ‘›ΰ―§ = 𝛼 + 𝛽𝑒π‘₯𝑐𝑒𝑠𝑠_π‘Ÿπ‘’π‘‘π‘’π‘Ÿπ‘›_π‘šΰ―§ + πœ€ΰ―§, where 𝑒π‘₯𝑐𝑒𝑠𝑠_π‘Ÿπ‘’π‘‘π‘’π‘Ÿπ‘›ΰ―§ is the difference between return on Asset A and the risk-free rate, 𝑒π‘₯𝑐𝑒𝑠𝑠_π‘Ÿπ‘’π‘‘π‘’π‘Ÿπ‘›_π‘šΰ―§ is the difference between return on the market portfolio and the risk-free rate, and πœ€ΰ―§ is a random error term. Data are for 174 months. You estimate the model via OLS; results are reported below. OLS estimates using 174 observations Dependent variable: excess return on Asset A Variable Coefficient Std. Error t-Statistic Prob. Ξ± -0.850664 1.195838 -0.711354 0.4796 Excess_return_m 1.148796 0.339127 3.387506 0.0012 R-squared 0.158333 Mean dependent var -0.604042 Adjusted R-squared 0.144535 S.D. dependent var 10.24319 S.E. of regression 9.474063 Akaike info criterion 7.366224 Sum squared resid 5475.230 Schwarz criterion 7.434260 Log likelihood -230.0361 Hannan-Quinn criter. 7.392983 F-statistic 11.47520 Durbin-Watson stat 2.036330 Prob(F-statistic) 0.001240

 a) What theory can be associated to the model to be estimated? Discuss. [25 marks]

 b) How would you test the hypothesis that the market is in equilibrium? Perform the test at the 5% s.l. and present the logic of the test. Test the hypothesis that the stock is riskier than the market at the 5% s.l. Would you buy the stock if you aim at reducing risk in your portfolio? What issues can you see with this approach? [25 marks] 

c) Consider the following augmented CAPM model 𝐸π‘₯𝑐𝑒𝑠𝑠_π‘Ÿπ‘’π‘‘π‘’π‘Ÿπ‘›ΰ―§ = 𝛼 + 𝛽𝑒π‘₯𝑐𝑒𝑠𝑠_π‘Ÿπ‘’π‘‘π‘’π‘Ÿπ‘›_π‘šΰ―§ + 𝛾𝐷𝑒𝑏𝑑௧ + πœ€ΰ―§, where 𝐷𝑒𝑏𝑑௧ is the firm’s debt at time t. What is the sign you expect for this variable? Discuss. A regression on 300 observations gives you the value -0.022179 for the estimated parameter for the firm’s debt, and 0.011739 for its standard deviation. Is the variable statistically significant at the 10% s.l.? Are you going to adopt a passive or an active investment strategy based on your result? If your decision builds upon the OLS approach, is there a possibility that you could go for the wrong choice? Discuss. [25 marks] 

d) Is the model statistically adequate? What type of analysis would you perform to check whether this is the case? Discuss. [25 marks] 

Question 3 

a) What are the advantages of estimating the panel data models, if one is available? Would it be reasonable to estimate it using the OLS approach? Discuss. [25 marks] 

b) Consider the following fixed effects model: π‘¦ΰ―œΰ―§ = 𝛼 + 𝛽π‘₯௜௧ + πœ‡ΰ―œ + 𝜐௜௧, where πœ‡ΰ―œ is taken as a fixed effect. What approach can you adopt to estimate the parameters of interest? Discuss. [25 marks]

 c) Describe the random effects model and discuss how this model captures the cross-sectional individual heterogeneity. [25 marks]

d) How do you choose between the fixed effects and random effects panel models? Discuss. [25 marks]