1. Given the bivariate sample regression model Yi = βˆ0 + βˆ1Xi + ˆei derive the OLS parameter estimates, i.e., derive the critical values of βˆ0 and βˆ1 that minimize the residual sum of squares (RSS).
2. For the sample regression model Yi = ˆα0 + ˆα1(Xi - X¯) + ˆei derive the critical values of ˆα0 and ˆα1 that minimize the RSS.
3. In the simple linear regression model yi = βˆ1 + βˆ2xi + ˆei, where E(ˆei) 6= 0. Show that the model can always be written with the same slope, but a new intercept and error term that does not violate the property that E( ˆei) = 0.
For the questions below you must use STATA. The Do-File must perform every task below neatly, and correctly. Please attach a print-out of your Do file to the back of your answers.
4. The data set you need for this question is HW02-01.dta.
(a) Using the data set, use the derivations of βˆ0 and βˆ1 from question 1 in order to get their respective values.
(b) Generate the predicted values of Y .
(c) What is the mean of the predicted value of Y ? Is it the same as the mean of the true value of Y ?
(d) Generate the sample error term. What is its mean?
(e) Derive the value of R2
(f) How much of the variation in Y is explained by X? How much of the variation in Y is explained by variables other than X?
(g) Use the derivations ˆα0 and ˆα1 from question 2 in order to get their respective values.
5. The data set you need for this questions is HW02-02.dta. This data set contains information on CEOs of US Corporations. The salary variable corresponds to annual CEO compensation, in thousands of dollars, and the ceoten variable corresponds to a CEO's tenure, in years, as company executive.
(a) Find the average salary and average tenure in the sample.
(b) How many CEOs are in their first year as CEO?
(c) For the simple regression salaryi = βˆ0 + βˆ1ceoteni + ˆei estimate the values of the parameters βˆ 0 and βˆ1.
(d) On average, what is the predicted increase in salary given one additional year as CEO?
(e) How much of the variation in salary is explained by years as CEO?
(f) Is years as CEO a good predictor of the variation in CEO salary? Why or Why not?
Attachment:- assign.rar