Question

We are interested in the relationship between the compensation of Chief Executive Officers (CEO) of firms and the return on equity of their respective firm, using the dataset below. The variable salary shows the annual salary of a CEO in thousands of dollars, so that y = 150 indicates a salary of $150,000. Similarly, the variable ROE represents the average return on equity (ROE)for the CEO’s firm for the previous three years. A ROE of 20 indicates an average return of 20%.

obs	salary	roe
1	1095	14.1
2	1001	10.9
3	1122	23.5
4	578	5.9
5	1368	13.8
6	1145	20
7	1078	16.4
8	1094	16.3
9	1237	10.5
10	833	26.3
11	567	25.9
12	933	26.8
13	1339	14.8
14	937	22.3
15	2011	56.3
16	1585	12.6
17	905	20.4
18	1058	1.9
19	922	19.9
20	1220	15.4
21	1022	38.7
22	759	16.4
23	1414	24.4
24	1041	15.6
25	1688	14.4
26	2983	19
27	1160	16.1
28	3844	12.1
29	476	16.2
30	1492	18.4
31	1024	14.2
32	1593	14.9
33	427	12.4
34	829	17.1
35	797	16.9
36	577	18.1
37	1342	10.9
38	1774	19.3
39	709	18.3
40	860	18.4
41	1336	13.8
42	516	13.7
43	931	12.7
44	815	15.1
45	1681	16.5
46	568	10.2
47	775	19.6
48	1188	12.8
49	782	15.9
50	1170	17.3
51	1469	8.5
52	916	16.4
53	1070	19.5
54	894	19.2
55	829	15.9
56	780	19.9
57	2327	28.1
58	717	25
59	1368	15
60	2028	12.6
61	1195	20.3
62	256	22.7
63	775	14.8
64	1407	13.2
65	543	10.3
66	874	17.7
67	1287	10
68	1248	15.6
69	875	6.8
70	925	12.4
71	798	13.1
72	760	15.8
73	600	12.8
74	991	15.3
75	1570	0.5
76	911	16.5
77	1360	15.1
78	700	13
79	741	11.1
80	1097	8.9
81	953	17.5
82	441	15.9
83	595	14.2
84	1067	9.3
85	1298	9.5
86	1798	15.5
87	4143	14.4
88	1336	11.1
89	1750	15.9
90	912	16.4
91	1892	8.6
92	833	24.6
93	1142	15.4
94	1159	16.9
95	1283	7.2
96	2109	11.6
97	1039	26.4
98	992	21.4
99	1253	19.2
100	721	15.1
101	1351	9
102	1391	9.4
103	1245	19
104	1550	3.5
105	2150	22.1
106	1846	10.9
107	573	15.1
108	6640	10.2
109	959	17.3
110	612	33.3
111	1820	22.8
112	1411	11.1
113	1026	12.4
114	1287	20.9
115	800	6.7
116	1115	7.1
117	1631	11.8
118	1910	14
119	996	10.1
120	918	6.4
121	1261	12.4
122	1053	17.6
123	1221	15.1
124	1738	23.6
125	3142	35.7
126	1900	23.2
127	427	12.4
128	1700	44.4
129	360	2.1
130	459	18.4
131	1340	16.1
132	729	15.1
133	223	22.7
134	2101	23.4
135	1082	25.7
136	1781	27
137	791	19.9
138	2092	43.7
139	1573	16.4
140	1045	11.6
141	1694	24.8
142	453	26.2
143	1130	44.5
144	1334	22.3
145	1344	22.3
146	1585	35.1
147	1946	13.1
148	1619	11
149	1620	19.4
150	967	28.5
151	1431	43.9
152	1231	26.8
153	770	15.7
154	1594	15
155	1568	28.2
156	995	15.4
157	1077	20
158	1161	42.2
159	1401	19.6
160	1127	16.2
161	3068	21.5
162	730	29.5
163	729	22.6
164	11233	22.9
165	949	13
166	3646	7.8
167	1502	48.1
168	807	18
169	713	18
170	1489	21.7
171	736	21.3
172	1226	26.9
173	543	30.5
174	14822	19.4
175	890	15.6
176	1627	19.4
177	2408	29.1
178	2248	40.8
179	787	13.7
180	474	11.1
181	439	10.8
182	465	5.1
183	594	12.3
184	688	7.4
185	607	6.2
186	634	12.7
187	532	10.6
188	441	7.4
189	694	12.6
190	520	12.8
191	757	2.9
192	668	13.5
193	803	10.7
194	500	11.9
195	552	12.9
196	412	10.1
197	1100	7.3
198	959	14.6
199	333	13.8
200	503	8.9
201	448	14
202	732	12.9
203	720	14.5
204	808	14.7
205	930	9
206	525	15.5
207	658	12.1
208	555	13.7
209	626	14.4

a) Draw a boxplot and a histogram of the salary of CEO. Are there any apparent outliers in the data? Are there high leverage points?

b) Use your software to estimate the relationship and report your results.

??????? = ?0 + ?1???? + ??

c) Looking at a plot of the residuals against predicted values and at the normal probability plot of residuals, does the estimated model appear satisfactory?

Answer 1

a)

Outliers are those which have a high value of residual; i.e.- the difference between predicted Y_i and the given Y_i . (marked in red)

High Leverage Points are those points which do follow the regression line trend however they are far away from most of the data set points (marked in green)

b) beta₀= 963.1913275 ; beta₁=18.50118685

c)

Looking at the residual vs predicted salary plot we can see that there is no clear pattern that the residuals follow. Which means that linear assumption for the model is apt. Also it means that "variance of residual is same " assumption is true(homoscedastic).

However, there are a few outliers in the figure( high residual value points).

From the normal probability plot, we can see that residuals vs sample percentile is following a linear trend apart from a few points with high values of residuals( outliers). Therefore, the assumption that the residuals are normally distributed is true.

So, overall the model is satisfactory but it can be better if the outlying points are removed