Question

Question 5 (10 marks) Kira and Shin are investment managers. They use a relatively simple model as a preliminary assessment of the returns they expect if they were to invest in any given company. Their model is given by the formula Rt = do + a St + a2Ct + Et; where Rt is the daily share return on day t, St is the company's sales revenue on day t, Ct is the sales revenue for the company's main competitor on day t (assume that Kira and Shin can legally access all the necessary data), and £t is the residual. a) Kira and Shin apply their model to Lind Ltd. They calculate both R2 and adjusted R2 for their model. Give a brief explanation of the difference between these two values in terms of what they represent. [2 marks] b) For the fitted model, R2 is calculated to be 10%. Explain what this suggests in terms of the usefulness of the fitted model. [2 marks] c) State two limitations of this model. [2 marks] Kira and Shin decide to analyse Lind Ltd from a different perspective. They now build a model, given by the formula Rt = b + b Rt-1 + Zt; where Z are independent N(0,0%) error terms. d) Shin thinks that they should also include Rt-2 in this model. What calculation process can he perform to justify this assertion? Note: you do NOT need to perform any calculations for this part. [2 mark] e) Kira has calculated the autocorrelations (ACF) for the Zi's. He has found that the first five autocorrelations are 0.35, 0.15, 0.01, -0.02, and 0.04. Kira has performed further investigation, and believes that all further autocorrelations will lie between -0.01 and 0.01. Suppose that the sample size was n = 200. What do these autocorrelations suggest about the model they are using? [2 marks]

Answer 1

1) R-squared or R2 explains the degree to which the chosen input variables explain the variation of the output variable. So, if R-square is 0.8, it means 80% of the variation in the output variable is explained by the input variables. So, in simple terms, higher the R squared, the more variation is explained by the input variables and hence better is the model.

However, the problem with R-squared is that it will either stay the same or increase with addition of more variables, even if they do not have any statistically significant relationship with the output variable. This is where “Adjusted R square” comes to help. Adjusted R-square penalizes for adding variables which do not improve the existing model. So long as we are using a univariate Linear regression model both R2 and Adjusted R2 would produce the same values. But in this instance we are using multiple variables in explaining the returns to a stock, "Adjusted R2" is the useful statistic.

Hence, while building Linear regression on multiple variable, it is always suggested to use Adjusted R-squared to judge goodness of model.

Typically, the more non-significant variables are added into the model, the gap in R-squared and Adjusted R-squared increases.

b) If for the fitted model the R2 value is 10% this means that 10% of the variation in the output variable can be explained by the input variables. Such a value of R2 suggests that the model is not a very good fit. Only a small percentage of the variation on the output variable is explained by the input variables or in other words the chosen input variables does not do a very good job in explaining the variations in the output variable

c) One limitations of regression is the Independence of the input variables. The assumption is that the independent variables are not highly correlated with each other and this assumption might not hold good.

Second limitation is that the error term or the residuals is assumed to be normally distributed which also might not be the case.

d) The test that needs to be conducted to detect the presence of second order Correlation is the Durban -Watson test. The DW test is conducted using the errors of the First order Correlation.

The requirement of conducting Second Order auto correlations would depend upon the value of the DW statictic.

The Durbin-Watson statistic is interpreted as follows:

• If d is close to zero (0), then positive autocorrelation is probably present;
• If d is close to two (2), then the model is likely free of autocorrelation; and
• If d is close to four (4), then negative autocorrelation is probably present.