4. Answer all parts (a) -(c) of this question. (a) [10 marks] Define the R" of a fitted simple linear regression Y = a + b.X, with residuals denoted u, taking care to define also each component that arises in the formula for R-. What does the R- measure? (b) Consider two regression models: Y = a+BX+u, (1) Y = y+ ologX +v. (2) where u and v are unknown random disturbances. (i) [10 marks] How do the interpretations of the slope coefficients in (1) and (2) differ? (ii) [15 marks] A researcher fits (1) and obtains an R- of 0.34. They then fit (2) and obtain an R- of 0.76. A colleague tells them that specification (2) fits the data better because it has a higher R", but the researcher is adamant that the R2 of (1) and (2) are not comparable. You are called upon to mediate. Use the definition of R2 to determine who is right. (iii) [5 marks] The researcher now fits a third model logY = + + vlogX + w, (3) where w is an unknown random disturbance, and finds that the R2 from this is 0.67. They now state that this R- can be compared to that obtained from fitting (2). Are they right? Explain briefly. (c) [10 marks] R- can also be defined for a multiple linear regression model. In this con- text, discuss one shortcoming of the R- as a measure of goodness of fit and propose an alternative, explaining how it corrects the