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# Economics 257 Assignment 3 (due March 20,2013) V.Zinde-Walsh 1. CEO salary and return on equity. The following table provides 7 observations of...

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Economics 257 Assignment 3 (due March 20,2013) V.Zinde-Walsh 1. CEO salary and return on equity. The following table provides 7 observations of salaries of CEO (in thou- sands of dollars) and the average return on equity of their &rm for the previous three years (return on equity is de&ned as percentage of net income on common equity). return CEO salary 14 : 1 1095 10 : 9 1001 23 : 5 1122 5 : 9 578 13 : 8 1368 20 1145 16 : 4 1078 (a) If one wishes to evaluate how the &rm performance is re±ected in CEO compensation which variable would one consider to be the dependent variable, and which the explanatory variable? (b) Computation: compute (b1) the covariance and the correlation coe¢ cient for the two variables; (b2) the regression coe¢ cients; (b3) the &tted value for each observation and the residual for each observa- tion; (b4) total sum of squares (TSS), sum of squared residuals (SSresid) and sum of squares explained by regression (regr. SS); write down and verify numerically the relation between these quantities. (b5) the coe¢ cient of determination; verify its relation to the correlation coe¢ cient; (b6) the standard errors of the regression coe¢ cients. (c) Interpretation: (c1) What does the covariance indicate about the relationship you are inter- ested in? The correlation coe¢ cient? (c2) Interpret the regression coe¢ cients. Plot the data and the regression line. (c3) Comment on the value of the sum of errors, on the sum of squared errors as measures of &t of the regression line. (c4) What is the interpretation of the coe¢ cient of determination here? (d) Inference: (d1) Construct a 90% and a 95% con&dence interval for the slope coe¢ cient; 1
(d2) Construct a 90% con&dence interval for the intercept; (d3) Does the sample evidence support the claim that CEO compensation re±ects returns on the equity of the &rm? Give an answer based on your answer to (d1,2) and brie±y comment. (e) Forecasting: (e1) Forecast the expected salary of a CEO if the average return on equity was at 10. How would this di/er from the forecast of he actual salary for this CEO? (e2) Construct the con&dence intervals for the two forecasts. (e3) Comment on the validity of predicting CEO salary based on this model. 2. The data on performance of the &rm and CEO salaries (from which a subsample of 7 observations was used in question 1) for 209 &rms was published in Business Week (5²6²91). The following results were obtained from bivarite regressions. Notation: y = salary; x = return on equity; z = sales (annual &rm sales measured in millions of dollars). ^ y = 963 : 191 + 18 : 501 x ; R 2 = : 0132 (1) Constant elasticity model for impact of sales ln ^ y = 4 : 822 + : 257 ln z ; R 2 = : 211 : (2) (a) Compare the results from your regression in question 1 and in (1). Com- ment on the di/erences. (b) Interpret the estimated coe/cients in (2). (c) How would a change in measurement scale, speci&cally, measuring y in dollars (rather than thousands of dollars), switching from x to x= 100 and measuring z in thousands (rather than millions) of dollars a/ect the results in ( 1) and in (2)? 3*. Write down the linear regression model. Suppose that you have some linear estimators for the slope and the intercept. Derive the conditions under which the estimators are unbiased. 2

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(a) If one wish to evaluate how the .firm performance is refected in CEO compensation then CEO salary is our
dependent variable and firm return is explanatory variable. (b) (b1)...

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