Question

Download the dataset returns.xlsx. This dataset records 83 consecutive monthly returns on the stock of Philip Morris (MO) and on Standard & Poor’s 500 stock index, measured in percent. Investors might be interested to know if the return on MO stock is influenced by the movement of the S&P 500 index. Please be aware that return is defined as new price − old price old price × 100%, so it is always reported as a percentage.

1. What is the response and explanatory variable for this dataset?

2. Create a scatterplot between the two variables and describe the form, direction, and strength of the linear relationship between the two variables.

3. Create a residual plot (residuals on y-axis, explanatory variable on x-axis).

4. Based on your answers to parts 2 and 3, are the assumptions for the regression model met? Address the linearity and constant variance assumptions.

5. Verify the values of the following: the sample means of the monthly returns of MO stocks and S&P stocks are 1.8783 and 1.3036 respectively; the sample standard deviations of the monthly returns of MO stocks and S&P stocks are 7.5539 and 3.3915 respectively.

MO	S&P
-5.7	-9
1.2	-5.5
4.1	-0.4
3.2	6.4
7.3	0.5
7.5	6.5
18.6	7.1
3.7	1.7
-1.8	0.9
2.4	4.3
-6.5	-5
6.7	5.1
9.4	2.3
-2	-2.1
-2.8	1.3
-3.4	-4
19.2	9.5
-4.8	-0.2
0.5	1.2
-0.6	-2.5
2.8	3.5
-0.5	0.5
-4.5	-2.1
8.7	4
2.7	-2.1
4.1	0.6
-10.3	0.3
4.8	3.4
-2.3	0.6
-3.1	1.5
-10.2	1.4
-3.7	1.5
-26.6	-1.8
7.2	2.7
-2.9	-0.3
-2.3	0.1
3.5	3.8
-4.6	-1.3
17.2	2.1
4.2	-1
0.5	0.2
8.3	4.4
-7.1	-2.7
-8.4	-5
7.7	2
-9.6	1.6
6	-2.9
6.8	3.8
10.9	4.1
1.6	-2.9
0.2	2.2
-2.4	-3.7
-2.4	0
3.9	4
1.7	3.9
9	2.5
3.6	3.4
7.6	4
3.2	1.9
-3.7	3.3
4.2	0.3
13.2	3.8
0.9	0
4.2	4.4
4	0.7
2.8	3.4
6.7	0.9
-10.4	0.5
2.7	1.5
10.3	2.5
5.7	0
0.6	-4.4
-14.2	2.1
1.3	5.2
2.9	2.8
11.8	7.6
10.6	-3.1
5.2	6.2
13.8	0.8
-14.7	-4.5
3.5	6
11.7	6.1
1.3	5.8

Answer 1

As we can see the independent variable or the predictor variable is the S&P index. The dependent variable, response variable or the predicted variable is the returns on the stock of Philip Morris (MO).

So first we calculate the summary statistics:

Regression Statistics
Multiple R	0.525090166
R Square	0.2757196825
Adjusted R Square	0.2667779502
Standard Error	6.468314267
Observations	83

We can see the R square value is very low hence the variables are not strongly correlated.

Now seeing the scatter plot with the blue line as the line of best fit and the red points the predicted fit.

Now seeing the residual plot;

Based on the above we can see the residual values for our predicted model;

RESIDUAL OUTPUT
Observation	Predicted MO	Residuals	Standard Residuals
1	-10.17216881	4.472168807	0.6956511264
2	-6.078781251	7.278781251	1.132222998
3	-0.1141308108	4.214130811	0.6555130121
4	7.838736442	-4.638736442	-0.7215609183
5	0.938454561	6.361545439	0.9895458872
6	7.955690373	-0.4556903726	-0.07088317429
7	8.657413954	9.942586046	1.546580972
8	2.341901723	1.358098277	0.2112537868
9	1.406270282	-3.206270282	-0.4987391195
10	5.382703908	-2.982703908	-0.4639631067
11	-5.4940116	-1.0059884	-0.1564826808
12	6.31833535	0.3816646501	0.05936838594
13	3.043625304	6.356374696	0.988741572
14	-2.102347624	0.1023476241	0.0159202935
15	1.874086003	-4.674086003	-0.727059584
16	-4.324472298	0.9244722978	0.143802755
17	11.46430828	7.735691722	1.20329596
18	0.1197770496	-4.91977705	-0.7652771158
19	1.757132072	-1.257132072	-0.1955483748
20	-2.570163345	1.970163345	0.3064612292
21	4.447072467	-1.647072467	-0.2562040625
22	0.938454561	-1.438454561	-0.2237533015
23	-2.102347624	-2.397652376	-0.3729576516
24	5.031842118	3.668157882	0.5705862798
25	-2.102347624	4.802347624	0.7470108303
26	1.055408491	3.044591509	0.4735897959
27	0.7045467006	-11.0045467	-1.711770203
28	4.330118537	0.4698814634	0.07309061517
29	1.055408491	-3.355408491	-0.5219377436
30	2.107993863	-5.207993863	-0.8101095806
31	1.991039933	-12.19103993	-1.896330623
32	2.107993863	-5.807993863	-0.9034402875
33	-1.751485833	-24.84851417	-3.865215651
34	3.511441025	3.688558975	0.5737596938
35	0.002823119415	-2.902823119	-0.4515375559
36	0.4706388402	-2.77063884	-0.4309761355
37	4.797934257	-1.297934257	-0.2018952028
38	-1.166716183	-3.433283817	-0.5340513424
39	2.809717444	14.39028256	2.238425404
40	-0.8158543919	5.015854392	0.7802220596
41	0.5875927704	-0.08759277039	-0.01362515862
42	5.499657839	2.800342161	0.4355965221
43	-2.804071205	-4.295928795	-0.6682367848
44	-5.4940116	-2.9059884	-0.4520299191
45	2.692763514	5.007236486	0.7788815342
46	2.224947793	-11.82494779	-1.83938456
47	-3.037979066	9.037979066	1.405868291
48	4.797934257	2.002065743	0.3114236848
49	5.148796048	5.751203952	0.8946065499
50	-3.037979066	4.637979066	0.7214431074
51	2.926671374	-2.726671374	-0.4241369444
52	-3.973610507	1.573610507	0.2447769682
53	0.35368491	-2.75368491	-0.4283389317
54	5.031842118	-1.131842118	-0.1760593748
55	4.914888188	-3.214888188	-0.5000796449
56	3.277533165	5.722466835	0.8901364575
57	4.330118537	-0.7301185366	-0.1135707985
58	5.031842118	2.568157882	0.399479984
59	2.575809584	0.6241904163	0.09709355458
60	4.213164606	-7.913164606	-1.230902077
61	0.7045467006	3.495453299	0.5437218785
62	4.797934257	8.402065743	1.306951224
63	0.35368491	0.54631509	0.08497995583
64	5.499657839	-1.299657839	-0.2021633078
65	1.172362421	2.827637579	0.4398423564
66	4.330118537	-1.530118537	-0.2380117409
67	1.406270282	5.293729718	0.8234458939
68	0.938454561	-11.33845456	-1.763709964
69	2.107993863	0.5920061371	0.09208725203
70	3.277533165	7.022466835	1.092352989
71	0.35368491	5.34631509	0.8316256105
72	-4.792288019	5.392288019	0.8387767536
73	2.809717444	-17.00971744	-2.645881587
74	6.43528928	-5.13528928	-0.7988002971
75	3.628394955	-0.7283949554	-0.1133026934
76	9.242183605	2.557816395	0.3978713535
77	-3.271886926	13.87188693	2.157788353
78	7.604828582	-2.404828582	-0.3740739189
79	1.289316352	12.51068365	1.94605158
80	-4.909241949	-9.790758051	-1.522963949
81	7.370920722	-3.870920722	-0.6021262784
82	7.487874652	4.212125348	0.65520106
83	7.137012861	-5.837012861	-0.9079542268