Economic & static Project.
Annki lee
seniorseminarpaper_1_.docx
First Draft
Senior Seminar
The chief executive officer (CEO) is the highest ranking corporate officer or executive officer of a corporation or agency.[footnoteRef:1] CEO’s salaries are rising rapidly and it is a widely debated topic today. Rapid growth in the CEOs’ pay in the past decade is primarily due to the use of stock options and additional forms of long term and incentive compensation.[footnoteRef:2] Some people believe that CEOs are overpaid, while others believe that well paid CEOs can make a difference in a company’s performance. According to USA Today, median 2005 pay of CEOs at the nation’s 100 largest companies soared to 17.9 million (a 25% increase from 2004). My paper is based on the Principle Agent Model and the Human Capital model and will use linear regression to examine the factors determining the CEO salary. [1: http://en.wikipedia.org/wiki/CEO] [2: Hall and Liebman, 1998]
Literature Review:
There has been much academic research on topics related to executive compensation. Research on executive compensation has mostly relied on the agency model. The Agency Theory is a theory concerning the relationship between a principal (shareholder) and an agent of the principal (CEO). Agents might have interests that diverge from those of the principals and might attempt to maximize their personal interests at the expense of the principals. Agents usually have more information than the principals, making the cost of monitoring high. In this situation, the principal can use short-run or long- run incentives to align an agent’s interest with that of the principal.[footnoteRef:3] [3: Deckop, Merriman, Gupta]
Stock options are the most commonly used form of long run performance incentives. They are designed to motivate the CEOs to perform because CEOs have long term stake in the future of the company, through ultimate stock ownership.
Early studies related to CEO compensation were mainly focused on examining the relationship between the CEO compensation and company performance. They were primarily focused on the firms’ size (measured by sales, revenue and total assets) and its profitability (measured by net corporate income or the rate of return on assets). Almost all of the previous research on CEO compensation has found that the firms’ size is significantly important factor in determining the level of executive compensation. As larger firms can employ better-qualified and better-paid CEOs, CEO compensation increases with the company size.[footnoteRef:4] [4: Rosen, 1982; Zhou, 2000; Agrawal, 1981. According to Rosen the elasticity of executive pay with respect to firm size is in the range of .20 to .35. Zhou’s research shows that CEO compensation increased by .25 percent for every 1 percent increase in the firms sales. ]
David Ciscel (1974) examined data on the largest industrial firms in the USA and found that executive income is positively related to sales, assets, profit and number of employees, and that the number of years the chief executive had served with the firm had no impact on compensation.
Examining large firms in the USA from 1953-59, McGuire, Chiu, and Elbing (1962) found that the executive compensation is more highly correlated with the firm’s sales than the firm’s profits. According to them, executive compensation in any one year is the result of not only the current decisions but also those of the past. The relationship between past sales and current compensation becomes of lesser significance as the time interval between the two increases.
Arch Patton (1951) used data from 411 companies selected from the SEC annual salary reports. He found that executive compensation is in direct proportion to company profits (i.e. the higher the profits, the higher the salaries). In his research, he concluded that the fastest growing companies pay relatively higher executive salaries than their slower-moving competitors.
Income, sales and assets used to determine a proxy for size are highly correlated with each other, which may lead to biased estimators. Ciscel and Carroll (1980) suggested some transformation of data in order to solve the problem of multicolinearity, heteroskedasticity and simultaneous equation bias. At first, profit was regressed against sales and then residual profit, the difference between actual profit and predicted profit, was regressed against the CEO compensation. Since residual profit is not correlated with the sales regression, analysis yields more accurate estimators.
In order to solve the principal-agent problem, CEO compensation is linked with the firm’s performance. The most common ways the firm performance has been measured are stockholders equity, stock performance (return on common stock and changes in market value), and profitability (earnings per share, return on investment, and total profits).[footnoteRef:5] Lewellen and Huntsman (1970) and Coughlan and Schmidt (1985) used stock performance in each of their studies and found that stock performance was a major factor in the determination of compensation. There is conflicting evidence as to whether the pay-performance relationship has weakened or strengthened over time. Jensen and Murphy (1990) were the first to identify that there is little relationship between executive pay and the company performance. They established that a positive statistical relationship exists between firm performance and executive pay, but the pay-performance sensitivity value was too low in magnitude to be consistent with the agency theory.[footnoteRef:6] Jensen and Murphy suggested that there is very little evidence that relative performance to other firms in the industry is an important source of managerial incentives. [5: Joyce(2001)] [6: Jensen and Murphy estimated that CEO compensation increases $3.25 for each $1000 increase in shareholder value.]
Hall and Liebman (1998) examined a data set of CEOs in the largest publicly traded U.S.A. firms. They found that CEO compensation is closely related to the firm’s performance and that sensitivity of compensation to the firm’s performance has increased due to holding of stock and stock options by CEOs. They found a strong relationship between the wealth of CEOs and the fortune of the companies they manage.[footnoteRef:7] Unlike Hall and Liebman, Conyon and Leech found a very small pay elasticity with respect to firm performance. [7: Hall and Liebman found that between 1980 and 1994 the direct compensation of CEOs increased by 136% at the median and 209% at the mean in real terms.]
William Joyce (2001) researched executive compensation and firm performance with a sample of publicly traded banks and savings and loans. Using return on assets, a profitability measure, and other individual-related variables as independent variables, the study found CEO compensation positively correlated with firm profitability, tenure, and stock ownership.
The Human Capital Theory suggests that education or training raises the productivity of workers by increasing knowledge and skills, hence raising workers future income by increasing lifetime earnings. Human capital is a stock of individual knowledge, capability and skills that are economically usable. Compensation grows over time as an individual accumulates human capital through additional formal education or informal on the job training. Better educated and more experienced executives can be expected to perform better and hence lead their organization to a higher performance level. An executive with more experience, more education and more general and firm specific training should perform better on the job and earn a higher salary. [footnoteRef:8] [8: Becker,1962]
Harry Holtzer (1990) examined the effect of productivity on earnings (using age, tenure, training, sex, education, and industrial dummies). He found that previous experience and current job tenure has significant positive effects on wage and productively levels.
Hogan and McPheters (1980) in addition to performance variables added experience and education as predictors of an executive’s future performance. They found that age, on the job training and experience are statistically significant in determining CEO compensation. They found that the age of a CEO is positively related to the executive compensation and years spent as a chief executive is positively related to compensation level and concluded that time served as chief executive constitutes a type of specific training which commands a wage premium. Hogan and McPheters did not find administrative background and graduate studies statistically significant in explaining the executive compensation.
Grey and Benson (2003) examined 114 directors of small business development centers in the United States. Like Hogan and McPheters they found that coefficients of educational level and tenure in office are positive and statistically significant..
Even though both the human capital model and the agency model help in explaining CEO compensation, Scott and Tiessen[footnoteRef:9] found that changes in cash compensation of established CEOs respond more strongly to firm performance than to changes in the human capital of the executive. [9: http://www.irpp.org/events/archive/may00/scott.pdf]
Variable Definition:
Dependent Variable
Logarithm of the total compensation will be used as dependent variable.
Independent Variables
I am going to use profit and sales as proxies to measure the size of the firm. Since sales and profit are highly correlated to each other, I am going to use residual profit, transformation variable used by Ciscel and Carroll (1980) discussed before in this paper. Since, larger firms have more employees, the complexity of jobs to be done by the CEO increases. In order to stop CEOs from moving to another firm the CEO will have to paid more. I predict that CEO compensation is positively related to the residual profit variable and number of Employees they have to look after.
CEO compensation is primarily determined by the firm’s performance. I am going to use earnings per share to measure the firm’s performance. Since good performance leads to higher compensation I predict that CEO compensation to be positively related to the earnings per share.
According to the Human Capital model an executive with more experience, more education, and more general and firm specific training should perform better on the job and get higher compensation. So, I expect education Level, number of years spent as CEO, number of years spent in the company and the age of the CEO to be positively related to the CEO compensation. According to the Human Capital model, women are relatively paid less than their male counterpart. I predict that women will be paid less relative to men in my sample data.
The Model:
I was thinking about including sales and residual profits from the year 2004 and 2003 and return on equity for each firms but I could not find data for more than half of my sample CEOs. I ended up with nine independent variables.
The model that is to be used for the research is as follows:
Where:
LNCOMP: Log of total compensation of CEO in the year 2005
LNEMPL: Log of no of employees in 2005
EPS: Earnings per share for a particular firm
RES05: Difference between the predicted and actual profit for a firm in the year 2005
LNSLS05: Log of sales of a firm in the year 2004.
AGE: Age of the CEO
TEN: Number of years spent by CEO in a particular firm.
CEO: Number of years spent in a particular firm as CEO.
EDU: A dummy variable that takes the value 1 if CEO has finished master and 0 otherwise.
SEX: A dummy variable that takes the value 1 if male and 0 otherwise
Predictions for the Values of the β:
β2 : I expect that an increase in LNEMPL should have a positive effect on LNCOMP and therefore the predicted sign is positive.
β3: I expect that an increase in EPS should have a positive effect on LNCOMP and therefore the predicted sign is positive.
β4 : I expect that an increase in RES05 should have a positive effect on LNCOMP and therefore the predicted sign is positive.
β5: I expect that an increase in LNSLS05 should have a positive effect on LNCOMP and therefore the predicted sign is positive.
β6: I expect that an increase in AGE should have a positive effect on LNCOMP and therefore the predicted sign is positive.
β7: I expect that an increase in TEN should have a positive effect on LNCOMP and therefore the predicted sign is positive.
β8: I expect that an increase in CEO should have a positive effect on LNCOMP and therefore the predicted sign is positive.
β9: I expect that an increase in EDU should have a positive effect on LNCOMP and therefore the predicted sign is positive.
β10: As males should be paid more than females, according to the human capital model
I predict this to be positive.
Data:
I started with 200 CEOs from large publicly traded companies that had filed proxies by March 31 for the year 2005. Data was complied by Pearl Meyer & Partners and was published in the New York Times on 9th April 2006.
Data Sources
CEO total compensation: Obtained from the www.pearlmeyer.com[footnoteRef:10], and was published in The New York Times on 9th April 2006. [10: www.pearlmeyer.com/knowledgecenter/research/ceopay/CEOPaySheet-j.pdf]
Earnings per share, net income, sales, number of employee, and sex: obtained from “SEC disclosure” database, accessed through LexisNexis.
CEO Age, time as CEO in the company, time in company, and education: obtained from www.spencerstuart.com [footnoteRef:11]. It has profile of all the CEOs in S&P 500 lists and any missing CEO profiles were obtained from the www.Forbes.com [footnoteRef:12] [11: http://content.spencerstuart.com/sswebsite/pdf/lib/2005_SP_500_Profile_Data_FIN.pdf] [12: http://www.forbes.com/2005/04/20/05ceoland.html]
I ended up deleting 26 CEOs as I could not find data for one or more than one independent variable for them.
Result and Analysis:
Descriptive statistics
Descriptive statistics are given in the table below. The mean CEO total compensation for 174 CEOs in my sample is 11,721,000 and it ranges from 1,009,100 to 63,086,000. On average CEOs were 55.72 years old, their firms’ earnings per share was 3,272 and they had spent 7.06 years as CEO in a particular company.
|
NAME |
N |
MEAN |
ST. DEV |
VARIANCE |
MINIMUM |
MAXIMUM |
|
TOTALCO |
174 |
0.11721E+08 |
0.92484E+07 |
0.85532E+14 |
0.10091E+07 |
0.63086E+08 |
|
EMPL |
174 |
0.10037E+06 |
0.56584E+06 |
0.32018E+12 |
131.00 |
0.74675E+07 |
|
EPS05 |
174 |
3.2720 |
2.6541 |
7.0442 |
-5.0900 |
13.140 |
|
RES05 |
174 |
0.10909E-03 |
0.22221E+10 |
0.49377E+19 |
-0.59305E+10 |
0.16740E+11 |
|
SALES05 |
174 |
0.25180E+11 |
0.63402E+21 |
0.13161E+08 |
0.19364E+12 |
0.20492E+11 |
|
AGE |
174 |
55.724 |
6.2421 |
38.964 |
41.000 |
73.000 |
|
TEN |
174 |
19.555 |
12.036 |
144.87 |
0.50000 |
46.000 |
|
CEO |
174 |
7.0632 |
6.9550 |
48.372 |
0.50000 |
38.000 |
Result
Running Statistical software “SHAZAM- Professional Edition” I obtained the following equation:
LNCOMP = 12.583 + 0.033137 LNEMPL +0.091735 EPS+ 0.000000000019346 RES05
(9.867) (0.6498) (4.364) (0.8086)
+ 0.13160 LNSLS05 + 0.00066712 AGE - 0.0072089 TEN + 0.010859 CEO+
(2.457) (0.06970) (-1.441) (1.159)
0.20014 EDU - 0.35765 SEX
(1.917) (-0.7361)
- As expected coefficient for LNEMPL was positive but statistically insignificant. For every 1% increase in employee, CEOs compensation increased by 3.31 %.
- As expected coefficient for EPS was positive and statistically significant at any level. For every 1 unit increase in earnings per share, CEOs compensation increased by 9.17 %.
- As expected coefficient for RES05 was positive but statistically insignificant. For every one unit increase in residual profit, CEOs compensation increased by .00000000193 %.
- As expected coefficient for LNSLS05 was positive and statistically significant at any α≥0.0075. For every 1% increase in sales, CEOs compensation increased by 13.16 %.
-As expected coefficient for AGE was positive but statistically insignificant. For every 1 year increase in age, CEOs compensation increased by .066 %.
- Coefficient of TEN was negative (completely opposite of what we expected) and statistically significant at α≥ .076. For every 1 more years spent in the company CEOs salary decreased by 7.2 %.
- As expected coefficient for CEO was positive and statistically insignificant. For every 1 more years spent as CEO in a company CEOs salary increased by 1.08 %.
- As expected coefficient for EDU was positive and statistically significant at any α≥0.0285. CEOs with graduate degree earned 20.0 % more.
- Coefficient of SEX was negative (completely opposite of what we expected) and statistically insignificant. Males made 35.5 % less than female.
According to SHAZAM output only 21.3 % of variation in dependent variable is explained by my model. Since my model has log dependent variable this is irrelevant.
Test for overall fitness
Since, p-value for F statistics is 0.000 we can conclude that our model does have significant explanatory power at any reasonable levels of significance.
Testing for Multicollinearity
At first I used rule of thumb test of simple multicollinearity to check multicollinearity. Since absolute value of r was less than .8 for all the pairs, there is no evidence in support of multicollinearity. Then I used Klein’s rule of thumb to check severe multicollinearity. Since highest value of auxiliary R2 (0.3776) was greater than R2 for our model (.2136) we conclude that there is severe multicollinearity.
Testing for Autocorrelation
Since my sample size was large I used large sample run test (Normal Statistics =1.2855 with P-value=0.1988). Therefore at any α ≥ .1988 we can conclude that we do not have problem with first-order autocorrelation. Using Durban Watson test at n=174 and k=9, d= 2.0134 we find that d is greater than dU and less than 4-dU at α=.02 and α=.01. So, we can conclude that there is no problem with first-order autocorrelation in our model. Therefore we can be pretty much sure that there is no first-order autocorrelation at reasonable level of significance.
Testing for Heteroskedasticity
With a p-value of 0.08684 at any reasonable level of significance we can conclude that the variance of the error term is not a linear function of the predicted value of the dependent variable.
Conclusion:
In my research earnings per share, sales of 2005, time in the company, and education are the only variables that are statistically significant in explaining variation in CEOs compensation. Since earnings per share and sales of 2005 were statistically significant, we can conclude that CEO compensation in my sample of CEOs was determined by their firm’s performance. Human capital (time spent in the company and education) also played significant role in determining CEO compensation.
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SHAZAM output
NAME N MEAN ST. DEV VARIANCE MINIMUM MAXIMUM
TOTALCO 174 0.11721E+08 0.92484E+07 0.85532E+14 0.10091E+07 0.63086E+08
EMPL 174 0.10037E+06 0.56584E+06 0.32018E+12 131.00 0.74675E+07
EPS05 174 3.2720 2.6541 7.0442 -5.0900 13.140
RES05 174 0.10909E-03 0.22221E+10 0.49377E+19 -0.59305E+10 0.16740E+11
SALES05 174 0.20492E+11 0.25180E+11 0.63402E+21 0.13161E+08 0.19364E+12
AGE 174 55.724 6.2421 38.964 41.000 73.000
TEN 174 19.555 12.036 144.87 0.50000 46.000
CEO 174 7.0632 6.9550 48.372 0.50000 38.000
CORRELATION MATRIX OF VARIABLES - 174 OBSERVATIONS
TOTALCO 1.0000
EMPL -0.19250E-01 1.0000
EPS05 0.42473 -0.28363E-01 1.0000
RES05 0.11704 0.19427E-01 0.12720 1.0000
SALES05 0.29277 0.47407E-02 0.12066 0.18979E-03 1.0000
AGE 0.16706 0.86378E-02 0.15749 -0.16116E-01 0.57104E-01
1.0000
TEN 0.56559E-01 0.69487E-01 0.17052 0.39042E-01 0.12220
0.32483 1.0000
CEO 0.21902 -0.44601E-01 0.14942 -0.11950 0.10097E-01
0.47567 0.48474 1.0000
TOTALCO EMPL EPS05 RES05 SALES05
AGE TEN CEO
|_ols lncomp lnempl eps05 res05 &
| lnsale age ten ceo edu sex/AUXRSQR
REQUIRED MEMORY IS PAR= 36 CURRENT PAR= 4000
OLS ESTIMATION
174 OBSERVATIONS DEPENDENT VARIABLE= LNCOMP
...NOTE..SAMPLE RANGE SET TO: 1, 174
R-SQUARE OF LNEMPL ON OTHER INDEPENDENT VARIABLES = 0.2381
R-SQUARE OF EPS05 ON OTHER INDEPENDENT VARIABLES = 0.1504
R-SQUARE OF RES05 ON OTHER INDEPENDENT VARIABLES = 0.0642
R-SQUARE OF LNSALE ON OTHER INDEPENDENT VARIABLES = 0.2089
R-SQUARE OF AGE ON OTHER INDEPENDENT VARIABLES = 0.2592
R-SQUARE OF TEN ON OTHER INDEPENDENT VARIABLES = 0.2708
R-SQUARE OF CEO ON OTHER INDEPENDENT VARIABLES = 0.3776
R-SQUARE OF EDU ON OTHER INDEPENDENT VARIABLES = 0.0340
R-SQUARE OF SEX ON OTHER INDEPENDENT VARIABLES = 0.0197
R-SQUARE OF CONSTANT ON OTHER INDEPENDENT VARIABLES = 0.0000
R-SQUARE = 0.2136 R-SQUARE ADJUSTED = 0.1704
VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.45753
STANDARD ERROR OF THE ESTIMATE-SIGMA = 0.67641
SUM OF SQUARED ERRORS-SSE= 75.034
MEAN OF DEPENDENT VARIABLE = 16.013
LOG OF THE LIKELIHOOD FUNCTION = -173.719
MODEL SELECTION TESTS - SEE JUDGE ET AL. (1985,P.242)
AKAIKE (1969) FINAL PREDICTION ERROR - FPE = 0.48382
(FPE IS ALSO KNOWN AS AMEMIYA PREDICTION CRITERION - PC)
AKAIKE (1973) INFORMATION CRITERION - LOG AIC = -0.72617
SCHWARZ (1978) CRITERION - LOG SC = -0.54461
MODEL SELECTION TESTS - SEE RAMANATHAN (1998,P.165)
CRAVEN-WAHBA (1979)
GENERALIZED CROSS VALIDATION - GCV = 0.48542
HANNAN AND QUINN (1979) CRITERION = 0.52073
RICE (1984) CRITERION = 0.48724
SHIBATA (1981) CRITERION = 0.48080
SCHWARZ (1978) CRITERION - SC = 0.58007
AKAIKE (1974) INFORMATION CRITERION - AIC = 0.48376
ANALYSIS OF VARIANCE - FROM MEAN
SS DF MS F
REGRESSION 20.375 9. 2.2639 4.948
ERROR 75.034 164. 0.45753 P-VALUE
TOTAL 95.410 173. 0.55150 0.000
ANALYSIS OF VARIANCE - FROM ZERO
SS DF MS F
REGRESSION 44635. 10. 4463.5 9755.637
ERROR 75.034 164. 0.45753 P-VALUE
TOTAL 44710. 174. 256.95 0.000
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 164 DF P-VALUE CORR. COEFFICIENT AT MEANS
LNEMPL 0.33137E-01 0.5099E-01 0.6498 0.517 0.051 0.0516 0.0217
EPS05 0.91735E-01 0.2102E-01 4.364 0.000 0.323 0.3279 0.0187
RES05 0.19346E-10 0.2392E-10 0.8086 0.420 0.063 0.0579 0.0000
LNSALE 0.13160 0.5356E-01 2.457 0.015 0.188 0.1913 0.1911
AGE 0.66712E-03 0.9572E-02 0.6970E-01 0.945 0.005 0.0056 0.0023
TEN -0.72089E-02 0.5003E-02 -1.441 0.152-0.112 -0.1168 -0.0088
CEO 0.10859E-01 0.9373E-02 1.159 0.248 0.090 0.1017 0.0048
EDU 0.20014 0.1044 1.917 0.057 0.148 0.1351 0.0065
SEX -0.35765 0.4859 -0.7361 0.463-0.057 -0.0515 -0.0221
CONSTANT 12.583 1.275 9.867 0.000 0.610 0.0000 0.7858
DURBIN-WATSON = 2.0124 VON NEUMANN RATIO = 2.0241 RHO = -0.00755
RESIDUAL SUM = -0.19241E-11 RESIDUAL VARIANCE = 0.45753
SUM OF ABSOLUTE ERRORS= 89.397
R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.2136
RUNS TEST: 96 RUNS, 93 POS, 0 ZERO, 81 NEG NORMAL STATISTIC = 1.2855
COEFFICIENT OF SKEWNESS = -0.1966 WITH STANDARD DEVIATION OF 0.1841
COEFFICIENT OF EXCESS KURTOSIS = 0.8137 WITH STANDARD DEVIATION OF 0.3662
JARQUE-BERA NORMALITY TEST- CHI-SQUARE(2 DF)= 5.2479 P-VALUE= 0.073
GOODNESS OF FIT TEST FOR NORMALITY OF RESIDUALS - 15 GROUPS
OBSERVED 2.0 1.0 3.0 6.0 15.0 17.0 24.0 32.0 27.0 24.0 16.0 3.0 3.0 0.0 1.0
EXPECTED 0.8 1.6 3.8 7.8 13.6 20.1 25.5 27.6 25.5 20.1 13.6 7.8 3.8 1.6 0.8
CHI-SQUARE = 10.0267 WITH 3 DEGREES OF FREEDOM, P-VALUE= 0.018
|_DIAGNOS / HET
REQUIRED MEMORY IS PAR= 141 CURRENT PAR= 4000
DEPENDENT VARIABLE = LNCOMP 174 OBSERVATIONS
REGRESSION COEFFICIENTS
0.331372995924E-01 0.917350668273E-01 0.193455014357E-10 0.131596992552
0.667123073046E-03 -0.720890621515E-02 0.108593022563E-01 0.200143310217
-0.357653328859 12.5831825002
HETEROSKEDASTICITY TESTS
CHI-SQUARE D.F. P-VALUE
TEST STATISTIC
E**2 ON YHAT: 2.932 1 0.08684
E**2 ON YHAT**2: 2.748 1 0.09736
E**2 ON LOG(YHAT**2): 3.126 1 0.07703
E**2 ON LAG(E**2) ARCH TEST: 0.029 1 0.86569
LOG(E**2) ON X (HARVEY) TEST: 7.137 9 0.62286
ABS(E) ON X (GLEJSER) TEST: 11.225 9 0.26059
E**2 ON X TEST:
KOENKER(R2): 15.166 9 0.08647
B-P-G (SSR) : 20.901 9 0.01310
...MATRIX INVERSION FAILED IN ROW 18
...RESULTS MAY BE UNRELIABLE
E**2 ON X X**2 (WHITE) TEST:
KOENKER(R2): ********** 18 *********
B-P-G (SSR) : ********** 18 *********
...MATRIX INVERSION FAILED IN ROW 18
...RESULTS MAY BE UNRELIABLE
E**2 ON X X**2 XX (WHITE) TEST:
KOENKER(R2): ********** 54 *********
B-P-G (SSR) : ********** 54 *********
3
ˆ
b
4
ˆ
b
5
ˆ
b
6
ˆ
b
7
ˆ
b
8
ˆ
b
9
ˆ
b
10
ˆ
b
e
1
ˆ
b
2
ˆ
b
