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Question 1 (1) Model 6: OLS, using observations 1-177 Dependent variable: Y coefficient std. error t-ratio p-value ------------------------------------------------------------------------------------------------------------ const 619.800 65.4907 9.464 2.18e-017 X1 13.2968 [A] 2.361 0.0194 X2 0.0196671 0.0109819 [B] 0.0751 X3 [C] [D] 2.067 0.0402 Mean dependent var 865.8644 S.D. dependent var 587.5893 Sum squared resid [E] S.E. of regression 532.3780 R-squared 0.193089 Adjusted R-squared 0.179096 F([F], 173) 13.79925 P-value(F) 4.13e-08 We have the following 95% Confidence interval for X3 coefficient: VARIABLE COEFFICIENT 95% CONFIDENCE INTERVAL X3 [C] 0.0153950 0.666662 a. Calculate A, B, C, D, E, F b. Is X2 statistically significant at 5% significance level for (i) 2-sided test ? (H0: βX2 = 0 , H1 : βX2 ≠ 0 ) (ii) 1-sided test ? (H0: βX2 = 0 , H1 : βX2 > 0 ) c. Suppose now we want to add a new variable (X4) to the model (2) And get the R2 for the new model (2) equal to 0.198404. Using F-test, decide whether we should add X4 to the model (1) or not, given Fcritical = 3.896 ( ). Remember to write down the null and alternative hypothesis. Question 2 While evaluating the effect of various firm-specific factors on the returns of a sample of 200 firms, a financial analyst estimates a model with the following result (with standard errors in parenthesis): = 0.080 + 0.801Si + 0.321MBi + 0.164PEi - 0.084BETAi (0.064) (0.147) (0.136) (0.420) (0.120) Where: ri is the percentage annual return for the stock of the firm i Si is the size of the firm i measured in terms of sales revenue MBi is the market to book ratio of the firm PEi is the price/earnings (P/E) ratio of the firm BETAi is the stock CAPM beta coefficient. a. Interpret the result of each coefficient in this regression. b. Calculate t-ratio for each coefficient. c. Given tcritical=1.972 for 2-sided test at 5% of significant level, which factors have significant effect on stock returns? d. If a stock’s beta increased from 1 to 1.3, what would be the expected effect on the stock’s returns? e. Supposed we have two firms with the same MB and same P/E, but firm A has sales revenue (S) of 1 (units) higher than firm B; the Beta of firm A is 0.7, and of firm B is 1.1. What is the difference in annual returns (r) between firm A and firm B? Which firm has higher returns?
Question 1 0 1 1 2 2 3 3 Y X X X u β β β β = + + + + (1) Model 6: OLS, using observations 1-177 Dependent variable: Y coefficient std. error t-ratio p-value ------------------------------------------------------------------------------------------------------------ const 619.800 65.4907 9.464 2.18e-017 X1 13.2968 [A] 2.361 0.0194 X2 0.0196671 0.0109819 [B] 0.0751 X3 [C] [D] 2.067 0.0402 Mean dependent var 865.8644 S.D. dependent var 587.5893 Sum squared resid [E] S.E. of regression 532.3780 R-squared 0.193089 Adjusted R-squared 0.179096 F([F], 173) 13.79925 P-value(F) 4.13e-08 We have the following 95% Confidence interval for X3 coefficient: VARIABLE COEFFICIENT 95% CONFIDENCE INTERVAL X3 [C] 0.0153950 0.666662 a. Calculate A, B, C, D, E, F b. Is X2 statistically significant at 5% significance level for (i) 2-sided test ? (H 0 : β X2 = 0 , H 1 : β X2 ≠ 0 ) (ii) 1-sided test ? (H 0 : β X2 = 0 , H 1 : β X2 > 0 ) c. Suppose now we want to add a new variable (X4) to the model 0 1 1 2 2 3 3 4 4 Y X X X X u β β β β β = + + + + + (2) And get the R 2 for the new model (2) equal to 0.198404. Using F-test, decide whether we should add X4 to the model (1) or not, given F critical = 3.896 ( 0.05 (1,172) 3.896F = ). Remember to write down the null and alternative hypothesis. Question 2 While evaluating the effect of various firm-specific factors on the returns of a sample of 200 firms, a financial analyst estimates a model with the following result (with standard errors in parenthesis): ˆ i r = 0.080 + 0.801S i + 0.321MB i + 0.164PE i - 0.084BETA i (0.064) (0.147) (0.136) (0.420) (0.120) Where: r i is the percentage annual return for the stock of the firm i S i is the size of the firm i measured in terms of sales revenue MB i is the market to book ratio of the firm PE i is the price/earnings (P/E) ratio of the firm BETA i is the stock CAPM beta coefficient. a. Interpret the result of each coefficient in this regression. b. Calculate t-ratio for each coefficient. c. Given t critical =1.972 for 2-sided test at 5% of significant level, which factors have significant effect on stock returns? d. If a stock’s beta increased from 1 to 1.3, what would be the expected effect on the stock’s returns? e. Supposed we have two firms with the same MB and same P/E, but firm A has sales revenue (S) of 1 (units) higher than firm B; the Beta of firm A is 0.7, and of firm B is 1.1. What is the difference in annual returns (r) between firm A and firm B? Which firm has higher returns? Question 3 X is a population with mean μ and variance σ 2 . Draw a sample of size 100 from X we have X 1 , X 2 ,… X 100. Therefore, X 1 , X 100 are i.i.d (µ, σ 2 ). Calculate var( )X X is a population with mean μ and variance σ 2 . Draw a sample of size n from X we have X 1 , X 2 ,… X n. That means X 1 , X n are i.i.d (µ, σ 2 ). Prove that 2 var( )X n σ = Question 4 Regression without regressor (without independent variable). Suppose you have the model: i i Y u β = + Use OLS to find the estimator for β . What is the variance of this estimator Question 5 In class, you know how to retrieve the value of covariance among coefficients using Coefficient covariance matrix menu in Gretl. Supposed now you have a simple regression model ( 1 2 Y X u β β = + + ) and you want to compute 1 2 cov( , )b b using mathematical formula. Prove that 1 2 2 cov( , ) .var( )b b X b= − [Hint: using definitions, basic properties and theorems can be a good way to solve the problem Keep in mind: 1 2 .b Y b X= − ; 1 1 ( )E b β = ; 2 2 ( )E b β = ] . stock CAPM beta coefficient. a. Interpret the result of each coefficient in this regression. b. Calculate t-ratio for each coefficient. c. Given t critical. i i Y u β = + Use OLS to find the estimator for β . What is the variance of this estimator Question 5 In class, you know how to retrieve the value of covariance