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STAT 371 S24 Assignment #2
(Submission deadline: 11:59 pm Fri. June 21st)
1) Here, we will investigate the distributional properties of the parameter estimators by repeating the procedure in 2a) of Assignment #1 1000 times and examining the 1000 sets of parameter estimates to determine whether our empirical findings from the simulation is consistent with the theoretical results on the distribution of βˆ derived in lectures (remember to set the seed to your ID # before you carryout the procedure).
One way to carry this out in R:
> betahat_mat = array(0,c(5,1000)) #creates an array to store sets of estimates
> set.seed(your ID#)
> for (i in 1:1000){
+ y<-X%*%Beta+rnorm(24,0,15000) #X and Beta vector as defined in A1.
+ A1sim.lm<-lm(y~x1+x2+x3+x4)
+ betahat_mat [,i]<-coef(A1sim.lm) #yields estimates for each iteration
+ }
The resulting (5 x 1000) matrix contains the beta estimates for the 1000 repetitions.
a) Consider the set of βˆ2 values.
i) Plot a histogram of the 1000 values.
ii) Calculate the (sample) mean.
iii) Calculate the (sample) standard deviation.
iv) Compare the empirical results from i)-iii) above with the distribution ofβ2(ˆ) derived in lectures:
b) Calculate the estimate of Cov(βˆ3, βˆ4) obtained from the data. Compare this with the value of Cov(βˆ3, βˆ4) obtained from the distribution of βˆ (due to the relatively large variance associated with a sample covariance, there maybe some discrepancy between the theoretical and empirical results).
2) The ceocomp dataset contains CEO information for 100 U.S. companies. We wish to fit a regression model to assess the relationship between CEO compensation (COMP) in thousands of dollars (includes salary and bonus, but not stock gains) and the following variates:
AGE: The CEOs age, in years
EDUCATN: The CEO’s education level (0 = no college degree; 1 = college/undergrad. degree; 2 = grad.
degree)
BACKGRD: Background type(1= banking/financial; 2 = sales/marketing; 3 = technical; 4 = legal; 5 = other)
TENURE: Number of years employed by the firm
EXPER: Number of years as the firm CEO
SALES: Sales revenues, in millions of dollars
VAL: Market value of the CEO's stock, in natural logarithm units
PCNTOWN: Percentage of firm's market value owned by the CEO
PROF: Profits of the firm, before taxes, in millions of dollars
a) Exploratory analysis suggests that, not surprisingly, companies with negative profit do not appear to exhibit the same relationship with compensation than do companies with positive profit. Here, we will only consider firms with positive profit in our analysis by removing firms with a negative profit from the dataset. One way this can be carried out in R is by using the subset function. Note that this will reduce the scope of our inference to only those firms in the population with a positive profit.
Once the companies with a negative profit are removed, select a random (sub)sample of 70 of the remaining companies for study, being sure to set the seed to your ID #, as in Assignment #1 . One way to accomplish this in R:
> set.seed(your ID#)
> my.ceo_pos = ceo_pos [sample(nrow(ceo_pos),70),] #selects a random sample of 70 observations by selecting 70 random row and selecting all columns in these rows.
Fit a linear regression model to your revised dataset to investigate the relationship between compensation and the other variates. Be sure to fit BACKGRD as a categorical variate. One way to achieve this is with the as.factor function. For this and for all remaining questions, fit education level as a single (quantitative) variate (given that there are only one or two CEOs with no college degree, there will be little difference in modelling education level as either a quantitative or categorical variate)
Briefly comment on the fit of the model, both in terms of the overall fit (by referring to the R-squared value) and on the significance of the individual parameter estimators.
b) Confirm the test statistic and p-value associated with H0 : β1 = β2 = ... β12 = 0 given in the output in the same way as presented in lessons (you can use the pf function to obtain the p-value)
c) Consider the Background variable.
i) Interpret the parameter estimate associated with a legal background.
ii) What can you conclude from the p-value associated with the sales/marketing background parameter?
ii) Does CEO background have an effect on compensation, after accounting for the other variates? Answer this question by performing an additional sum of squares test. Do not use the anova function. Perform the test using the summary output as shown in the lessons. Use the pf function in R to obtain the p-value.
iv) Now use the anova function to confirm the test statistic and p-value from the test in iii).
Use the reduced model (i.e. the model fit without the Background variable) for the remaining questions:
d) Consider the three variables, age (yrs), tenure (yrs), and experience (yrs).
i) What is it about the relationship between these three variables that might be problematic?
ii) Use the pairs function to plot the pairwise scatterplots of these variates. Do you suspect multicollinearity maybe an issue?
iii) Assess whether multicollinearity is a problem with the age variate by calculating the VIF.
e) The CEO of a company with 3.1 billion dollars in sales and 172 million dollars in profits is 48 year old, with an undergraduate degree, has been with the company for 14 years (including 3 years as the CEO), has a stock value of 2.1, and owns 1.1% of the company. Is their compensation of 1.5 million dollars consistent with other companies in the population? Answer this question by obtaining an appropriate 95% prediction interval. Use the predict function as described in lectures.