Question

Can a corporation's annual profit be predicted from information about the company's CEO? Forbes (May 1999) presented data (shown in TABLE 2) on company profit (y) in (millions of dollars), CEO's annual income (x1) (in thousands of dollars) and percentage of the company's stock owned by the CEO (x2). Use the data in the TABLE below and answer the following questions.

(a) Fit a multiple regression model of y on x1 and x2 (MODEL 1). Fit two simple linear regressions: (I) y on x1 (MODEL2); (II) y on x2 (MODEL 3). Discuss results on statistical significance of explanatory variables in each model. Compare the results of fitting the three models to the data using various measures of goodness of fit and model selection criteria discussed in class. Rank the models in terms of usefulness and briefly comment on their performance.

(b) Do you find any signs of multicollinearity?

(c) Use the printout to test the following hypotheses using a significance level of 5%: (I) increasing the CEO's annual income (other things constant) will increase the company profit; (II) increasing the percentage of company stock owned by the CEO will increase the company profit.

TABLE: Company profit (y), CEO's annual income (x1), and the company stock owned by the CEO (x2)

Company	Profit (y)	CEO's income (x1)	% of the company stock owned by the CEO (x2)
Gap	824.5	3743	1.71
Intel	6068.0	52598	.13
Gateway 2000	346.4	855	43.93
HJ Heinz	746.9	2916	1.63
Conseco	630.7	124579	3.64
Citicorp	5807.0	6200	.22
Cisco Systems	1362.3	560	.06
General Electric	9296.0	40626	.03
America Online	254.0	26917	.54
Computer Associates	570.0	10614	3.79
Lockheed Martin	1001.0	2533	.01
Bear Stearns	538.6	23215	3.44

Answer 1

I have used STATA to compute the following regression results:

a) I) Regression of companies profit on CEO's income and CEO's shareholding in the company.

profit	Coefficient	Standard. Error.	t- value	P>t (p-value)	95% Confidence Interval
ceo_income	0.008598	0.027214	0.32	0.759	-0.0529646	0.0701615
ceo_stock_holding	-60.309	78.78352	-0.77	0.464	-238.5298	117.9117
_cons	2372.657	1238.005	1.92	0.088	-427.9045	5173.218

Source	Sum of Squares	degrees of freedom	Mean Sum of Squares	Number of obs	=	12
				F( 2,     9)	=	0.39
Model	8006994	2	4003497	Prob > F	=	0.6855
Residual	91472467	9	10163607	R-squared	=	0.0805
				Adj R-squared	=	-0.1238
Total	99479461	11	9043587	Root MSE	=	3188

II) Regression of profit on CEO's Income

profit	Coef.	Std. Err.	t	P>t	[95% Conf. Interval]
ceo_income	0.01206	0.0263	0.46	0.656	-0.04649	0.070602
_cons	1990.38	1109.1	1.79	0.103	-480.893	4461.651

Source	SS	df	MS	Number of obs	=	12
				F( 1,    10)	=	0.21
Model	2051169.87	1	2051169.87	Prob > F	=	0.6562
Residual	97428290.8	10	9742829.08	R-squared	=	0.0206
				Adj R-squared	=	-0.0773
Total	99479460.7	11	9043587.34	Root MSE	=	3121.4

III) Regression of profit on CEO's stock holding

profit	Coef.	Std. Err.	t	P>t	[95% Conf. Interval]
ceo_stock_holding	-64.44	74.112	-0.87	0.405	-229.571	100.6904
_cons	2604.65	950.83	2.74	0.021	486.057	4723.237

Source	SS	df	MS	Number of obs	=	12
				F( 1,    10)	=	0.76
Model	6992406.5	1	6992406.5	Prob > F	=	0.405
Residual	92487054.2	10	9248705.42	R-squared	=	0.0703
				Adj R-squared	=	-0.0227
Total	99479460.7	11	9043587.34	Root MSE	=	3041.2

We observe that in all the three cases the coefficient of the explanatory variables is statistically insignificant at 5% level of significance. Also the adjusted R-square is very low which signals poor fit of the data to the model.  It is possible because the number of observations is restricted to only 12 whereas for good statistical inference in cross-section data, we need atleast 30 observations.

We should select a model  which has the least Mean Square Error (MSE) or highest R square. Model III has the least MSE whereas Model I has the highest R square.

Ranking (Best to worst) :

On the basis of R-sqaure- II, III, I

On the basis of least MSE- III, II,I

b)

We plot CEO income against CEO Stock holding . We do not observe a perfect linear relationship between them. So there is no perfect multicollinearity.

ceo_income	Coef.	Std. Err.	t	P>t	[95% Conf. Interval]
ceo_stock_holding	-480.49	902.76	-0.53	0.606	-2491.96	1530.983
_cons	26980.6	11582	2.33	0.042	1173.884	52787.35

Also, the regression coefficient of CEO Income on CEO Stock Holding is statistically insignificant. Hence there is no linear relationship between them. Hence there is no Muticollinearity in the model.

c) To comment on this, we need to use the regression results from of model I.

Since the regression coefficient of CEO's Income on Company's profit is positive, so an increase in CEO's income, given others factors remaining constant, would lead to an increase in the Company's profit, but this increase would be statistically insignificant.

Since the regression coefficient of CEO's shareholding on Company's profit is negative, so an increase in CEO's share holding, given others factors remaining constant, would lead to a decrease in the Company's profit, but this decrease would be statistically insignificant.