UNIT 1E DATA-DRIVEN INTRODUCTION TO LINEAR REGRESSION
HOW TO FIND THE LINEAR REGRESSION LINE IN EXCEL:
1. Go to the second tab of our Excel sheet for today.
2. Right-click on one of the data points, and select “Add Trendline”
3. In the resulting dialog box, choose “linear,” and click the bottom two boxes
Excel gives us back both the (least squares) best fit line and a measure of how well the line fit our data:
2
r is the square of the correlation coefficient .
2
r is always a number between 0 and 1.
2
If r is close to 1, the regression line is a pretty good fit for the data points.
2
If r is close to 0, the regression line is a pretty poor fit for the data points.
HOW TO COMPARE TWO LINEAR MODELS FOR A GIVEN DATA SET: RESIDUAL SUM OF
SQUARES (RSS)
I’ll walk you through how to calculate the RSS for one data set. A small RSS means a small error in fitting the data, so if you have
two models the one with the smallest RSS “wins.”
1.
2.
3.
You should be given a table with input values 𝑥1 , 𝑥2 , … 𝑥 and corresponding output values 𝑦1 , 𝑦2 , … 𝑦 that are from
the data.
You should be given (or should have found) the equation for a linear model 𝑦 = (𝑥). You should use this model to
compute output values for each given input value: 1 = (𝑥1 ), 2 = (𝑥2 ), … , = (𝑥 ).
You should calculate the difference for each pair of output values (data from #1 versus calculated from #2), square
those differences, and add them all up:
= (𝑦1 −
1.
2
1)
+ (𝑦2 −
2
2)
+
+ (𝑦 −
)2 .
Please see the “HW #1 Car Data” tab in today’s Excel file. Let’s explore the relationship between the weight of a car and its
corresponding gas mileage.
a. We want to create a scatter plot of mpg versus weight, since we suspect that heavier cars have worse mpg.
(Imagine for a second what that plot will look like.) To do this:
i. Copy Column C (mpg) by clicking the C at the top of that column to select it, and then select home>copy.
ii. Insert that copy into Column H; do this by selecting column H, then selecting home>insert>insert copied cells.
You now have weight in column G and mpg in column H.
iii. Highlight those two columns of data, including the headings (weight and mpg).
iv. Click the Insert tab, select Scatter, and choose the top-left option.
v. Now format the chart to show the Axes labels, Chart Title, and to remove the Legend. See the instructions in
the Day 1 Activity if you’ve forgotten how.
b.
2
Find the linear regression line, and write its equation and r value here:
Equation: _________________________________
2
r value: __________________
c.
Unit 1E p. 1
Do you think there’s a strong linear relationship between a car’s weight and its gas mileage? Explain.
MA 151 Mathematical Methods for Business
Marymount University
18
d.
What gas mileage would you expect for a 3000-pound car? Use the regression line to answer.
_______________________
e.
2.
Attach a printout of your graph (AND NO spreadsheet prints), formatted as suggested above, and showing the
2
regression line and r value. The way you can print only the graph is to right click the graph and ask Excel to put it
in its own tab.
Please find in the last tab of our Excel sheet for today information about the Standard and Poor stock values on every day
for approximately the past year. We want to create a scatterplot that shows S&P value over time.
a. Note there is a column for the date and then another how many days have passed since 1/10/2014. Because date
values can be confusing to use in Excel, and because the values would otherwise be so large, we this “days”
column. You may find the formula interesting – check it out.
b. Use the days and the S&P value columns to create the scatterplot. Give the graph an appropriate title, appropriate
gridlines, and label the axes.
2
c. Use Trendline to find the linear regression line, and display this on the graph. Write the equation and the r value
here:
Equation: ____________________
2
r value: __________________
d.
What does this regression line capture from this past year’s S&P values that you think is accurate?
e.
What do you think this regression line cannot do well?
f.
Assuming there are no shocks to our economy and the stock market continues to progress in a similar fashion,
predict the value of the S&P index on February 1. Can you think of a way to also predict how accurate you will be?
g.
Please include a graph of the data with its regression line – please follow the instructions in number 2 and feel
free to include both graphs on the same page. If you’d like to extend the graph line through the end of February,
that would help you make your point in part f well.
Unit 1E p. 2
MA 151 Mathematical Methods for Business
Marymount University
19
3.
Here’s a very simple example of some data that we want to fit with a regression line – we’ll ignore any potential application
for now:
Input 𝑥
1
2
3
4
5
Output 𝑦
1
1.8
3.1
3.8
3.5
a.
Suppose you tried to estimate this data using the model 𝑦 = 𝑥. Without using Excel (so that you’re practicing for a
test/quiz-type question), tell me the residual sum of squares (RSS). (Show your work. For your convenience, there
are a couple blank rows on the table you can use if you choose.)
b.
Sketch a graph of the data versus this linear mode. Label your axes and scale clearly, and choose a scale that
clearly shows the given data.
c.
By eyeballing this data, suggest a better equation for a linear model to fit the data, and demonstrate that this line
does fit the data better by calculating the residual sum of squares (RSS). There are many possible answers. Show
your work on the back of this sheet, but write the equation and sum here:
Unit 1E p. 3
MA 151 Mathematical Methods for Business
Marymount University
20
x
0
1
2
3
4
y
7
8
9
11
13
GUESSES
m
b
1
7
RESIDUAL CALCULATION
yguess resid resid^2
7
0
0
8
0
0
9
0
0
10
1
1
11
2
4
residual sum of squares (RSS)
5
14
12
Profit ($billion)
DATA
x y
0 7
1 8
2 9
3 11
4 13
10
8
6
4
2
0
0
1
2
3
YEAR (0=1996)
3
(0=1996)
4
5
0
4
7
11
0
1
2
3
4
Actual Data
7
8
9
11
13
GE Profits
Profit (in billions)
Year
18
16
14
12
10
8
6
4
2
0
y = 1,5x + 6,6
R² = 0,9698
0
1
2
3
Years Since 1996
4
y = 1,5x + 6,6
R² = 0,9698
5
DATA
Area BTU's
150 4900
180 5425
240 6375
250 6400
290 7025
315 7400
350 7900
370 8275
410 8750
450 9400
x
150
180
240
250
290
315
350
370
410
450
GUESSES
m
b
14,80
2736,54
RESIDUAL SUM
19769,33
RESIDUAL CALCULATION
y yguess
resid
resid^2
4900 4957,07 -57,07
3256,60
5425 5401,17 23,83
567,74
6375 6289,38 85,62
7329,96
6400 6437,42 -37,42
1400,27
7025 7029,56
-4,56
20,81
7400 7399,65
0,35
0,12
7900 7917,77 -17,77
315,90
8275 8213,84 61,16
3740,01
8750 8805,99 -55,99
3134,41
9400 9398,13
1,87
3,51
residual sum
19769,33
While shopping for an air conditioner, a buyer consulted the
following table giving room area and the number of BTUs it
takes to cool that area. What are the slope and intercept of
the best linear model?
energy (BTUs)
10000
8000
6000
4000
2000
0
0
Area
150
180
240
250
290
315
350
370
410
450
BTU's
4900
5425
6375
6400
7025
7400
7900
8275
8750
9400
100
200
300
area (sq feet)
400
500
BTU's
10000
9000
y = 14,804x + 2736,5
R² = 0,9989
8000
7000
6000
5000
4000
3000
2000
1000
0
0
50
100
150
200
250
300
350
400
450
500
150 4957
450 9398
make
AMC Concord
AMC Pacer
AMC Spirit
Buick Century
Buick Electra
Buick LeSabre
Buick Opel
Buick Regal
Buick Riviera
Buick Skylark
Cad. Deville
Cad. Eldorado
Cad. Seville
Chev. Chevette
Chev. Impala
Chev. Malibu
Chev. Monte Carlo
Chev. Monza
Chev. Nova
Dodge Colt
Dodge Diplomat
Dodge Magnum
Dodge St. Regis
Ford Fiesta
Ford Mustang
Linc. Continental
Linc. Mark V
Linc. Versailles
Merc. Bobcat
Merc. Cougar
Merc. Marquis
Merc. Monarch
Merc. XR-7
Merc. Zephyr
Olds 98
Olds Cutl Supr
Olds Cutlass
Olds Delta 88
Olds Omega
Olds Starfire
Olds Toronado
Plym. Arrow
Plym. Champ
Plym. Horizon
Plym. Sapporo
Plym. Volare
price
4099
4749
3799
4816
7827
5788
4453
5189
10372
4082
11385
14500
15906
3299
5705
4504
5104
3667
3955
3984
4010
5886
6342
4389
4187
11497
13594
13466
3829
5379
6165
4516
6303
3291
8814
5172
4733
4890
4181
4195
10371
4647
4425
4482
6486
4060
mpg
rep78
22
17
22
20
15
18
26
20
16
19
14
14
21
29
16
22
22
24
19
30
18
16
17
28
21
12
12
14
22
14
15
18
14
20
21
19
19
18
19
24
16
28
34
25
26
18
3
3
3
4
3
3
3
3
3
2
3
3
4
3
2
2
3
5
2
2
2
4
3
3
3
3
4
4
3
3
4
3
4
3
3
4
3
1
3
3
5
3
2
headroom trunk
weight
mpg
2,5
11
2930
3
11
3350
3
12
2640
4,5
16
3250
4
20
4080
4
21
3670
3
10
2230
2
16
3280
3,5
17
3880
3,5
13
3400
4
20
4330
3,5
16
3900
3
13
4290
2,5
9
2110
4
20
3690
3,5
17
3180
2
16
3220
2
7
2750
3,5
13
3430
2
8
2120
4
17
3600
4
17
3600
4,5
21
3740
1,5
9
1800
2
10
2650
3,5
22
4840
2,5
18
4720
3,5
15
3830
3
9
2580
3,5
16
4060
3,5
23
3720
3
15
3370
3
16
4130
3,5
17
2830
4
20
4060
2
16
3310
4,5
16
3300
4
20
3690
4,5
14
3370
2
10
2730
3,5
17
4030
2
11
3260
2,5
11
1800
4
17
2200
1,5
8
2520
5
16
3330
22
17
22
20
15
18
26
20
16
19
14
14
21
29
16
22
22
24
19
30
18
16
17
28
21
12
12
14
22
14
15
18
14
20
21
19
19
18
19
24
16
28
34
25
26
18
length
186
173
168
196
222
218
170
200
207
200
221
204
204
163
212
193
200
179
197
163
206
206
220
147
179
233
230
201
169
221
212
198
217
195
220
198
198
218
200
180
206
170
157
165
182
201
Pont. Catalina
Pont. Firebird
Pont. Grand Prix
Pont. Le Mans
Pont. Phoenix
Pont. Sunbird
Audi 5000
Audi Fox
BMW 320i
Datsun 200
Datsun 210
Datsun 510
Datsun 810
Fiat Strada
Honda Accord
Honda Civic
Mazda GLC
Peugeot 604
Renault Le Car
Subaru
Toyota Celica
Toyota Corolla
Toyota Corona
VW Dasher
VW Diesel
VW Rabbit
VW Scirocco
Volvo 260
5798
4934
5222
4723
4424
4172
9690
6295
9735
6229
4589
5079
8129
4296
5799
4499
3995
12990
3895
3798
5899
3748
5719
7140
5397
4697
6850
11995
18
18
19
19
19
24
17
23
25
23
35
24
21
21
25
28
30
14
26
35
18
31
18
23
41
25
25
17
4
1
3
3
2
5
3
4
4
5
4
4
3
5
4
4
3
5
5
5
5
4
5
4
4
5
4
1,5
2
3,5
3,5
2
3
2,5
2,5
1,5
2
2,5
2,5
2,5
3
2,5
3,5
3,5
3
2,5
2,5
3
2
2,5
3
3
2
2,5
20
7
16
17
13
7
15
11
12
6
8
8
8
16
10
5
11
14
10
11
14
9
11
12
15
15
16
14
3700
3470
3210
3200
3420
2690
2830
2070
2650
2370
2020
2280
2750
2130
2240
1760
1980
3420
1830
2050
2410
2200
2670
2160
2040
1930
1990
3170
18
18
19
19
19
24
17
23
25
23
35
24
21
21
25
28
30
14
26
35
18
31
18
23
41
25
25
17
214
198
201
199
203
179
189
174
177
170
165
170
184
161
172
149
154
192
142
164
174
165
175
172
155
155
156
193
turn
displacement
40
40
35
40
43
43
34
42
43
42
44
43
45
34
43
31
41
40
43
35
46
46
46
33
43
51
48
41
39
48
44
41
45
43
43
42
42
42
43
40
43
37
37
36
38
44
gear_ratio
121
258
121
196
350
231
304
196
231
231
425
350
350
231
250
200
200
151
250
98
318
318
225
98
140
400
400
302
140
302
302
250
302
140
350
231
231
231
231
151
350
156
86
105
119
225
3,58
2,53
3,08
2,93
2,41
2,73
2,87
2,93
2,93
3,08
2,28
2,19
2,24
2,93
2,56
2,73
2,73
2,73
2,56
3,54
2,47
2,47
2,94
3,15
3,08
2,47
2,47
2,47
2,73
2,75
2,26
2,43
2,75
3,08
2,41
2,93
2,93
2,73
3,08
2,73
2,41
3,05
2,97
3,37
3,54
3,23
foreign
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
45
40
35
30
25
20
15
10
5
0
0
1000
2000
42
42
45
40
43
41
37
36
34
35
32
34
38
36
36
34
33
38
34
36
36
35
36
36
35
35
36
37
231
231
231
231
231
151
131
97
121
119
85
119
146
105
107
91
86
163
79
97
134
97
134
97
90
89
97
163
2,73
3,08
2,93
2,93
3,08
2,73
3,2
3,7
3,64
3,89
3,7
3,54
3,55
3,37
3,05
3,3
3,73
3,58
3,72
3,81
3,06
3,21
3,05
3,74
3,78
3,78
3,78
2,98
Domestic
Domestic
Domestic
Domestic
Domestic
Domestic
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
Foreign
mpg
3000
4000
5000
6000
Effective date
1/10/2013
1/11/2013
1/14/2013
1/15/2013
1/16/2013
1/17/2013
1/18/2013
1/22/2013
1/23/2013
1/24/2013
1/25/2013
1/28/2013
1/29/2013
1/30/2013
1/31/2013
2/1/2013
2/4/2013
2/5/2013
2/6/2013
2/7/2013
2/8/2013
2/11/2013
2/12/2013
2/13/2013
2/14/2013
2/15/2013
2/19/2013
2/20/2013
2/21/2013
2/22/2013
2/25/2013
2/26/2013
2/27/2013
2/28/2013
3/1/2013
3/4/2013
3/5/2013
3/6/2013
3/7/2013
3/8/2013
3/11/2013
3/12/2013
3/13/2013
3/14/2013
3/15/2013
3/18/2013
3/19/2013
3/20/2013
Days
0
1
4
5
6
7
8
12
13
14
15
18
19
20
21
21
24
25
26
27
28
31
32
33
34
35
39
40
41
42
45
46
47
48
51
54
55
56
57
58
61
62
63
64
65
68
69
70
S&P 500
1472,12
1472,05
1470,68
1472,34
1472,63
1480,94
1485,98
1492,56
1494,81
1494,82
1502,96
1500,18
1507,84
1501,96
1498,11
1513,17
1495,71
1511,29
1512,12
1509,39
1517,93
1517,01
1519,43
1520,33
1521,38
1519,79
1530,94
1511,95
1502,42
1515,6
1487,85
1496,94
1515,99
1514,68
1518,2
1525,2
1539,79
1541,46
1544,26
1551,18
1556,22
1552,48
1554,52
1563,23
1560,7
1552,1
1548,34
1558,71
3/21/2013
3/22/2013
3/25/2013
3/26/2013
3/27/2013
3/28/2013
4/1/2013
4/2/2013
4/3/2013
4/4/2013
4/5/2013
4/8/2013
4/9/2013
4/10/2013
4/11/2013
4/12/2013
4/15/2013
4/16/2013
4/17/2013
4/18/2013
4/19/2013
4/22/2013
4/23/2013
4/24/2013
4/25/2013
4/26/2013
4/29/2013
4/30/2013
5/1/2013
5/2/2013
5/3/2013
5/6/2013
5/7/2013
5/8/2013
5/9/2013
5/10/2013
5/13/2013
5/14/2013
5/15/2013
5/16/2013
5/17/2013
5/20/2013
5/21/2013
5/22/2013
5/23/2013
5/24/2013
5/28/2013
5/29/2013
5/30/2013
5/31/2013
6/3/2013
6/4/2013
6/5/2013
6/6/2013
6/7/2013
71
72
75
76
77
78
81
82
83
84
85
88
89
90
91
92
95
96
97
98
99
102
103
104
105
106
109
110
111
112
113
116
117
118
119
120
123
124
125
126
127
130
131
132
133
134
138
139
140
141
143
144
145
146
147
1545,8
1556,89
1551,69
1563,77
1562,85
1569,19
1562,17
1570,25
1553,69
1559,98
1553,28
1563,07
1568,61
1587,73
1593,37
1588,85
1552,36
1574,57
1552,01
1541,61
1555,25
1562,5
1578,78
1578,79
1585,16
1582,24
1593,61
1597,57
1582,7
1597,59
1614,42
1617,5
1625,96
1632,69
1626,67
1633,7
1633,77
1650,34
1658,78
1650,47
1667,47
1666,29
1669,16
1655,35
1650,51
1649,6
1660,06
1648,36
1654,41
1630,74
1640,42
1631,38
1608,9
1622,56
1643,38
6/10/2013
6/11/2013
6/12/2013
6/13/2013
6/14/2013
6/17/2013
6/18/2013
6/19/2013
6/20/2013
6/21/2013
6/24/2013
6/25/2013
6/26/2013
6/27/2013
6/28/2013
7/1/2013
7/2/2013
7/3/2013
7/5/2013
7/8/2013
7/9/2013
7/10/2013
7/11/2013
7/12/2013
7/15/2013
7/16/2013
7/17/2013
7/18/2013
7/19/2013
7/22/2013
7/23/2013
7/24/2013
7/25/2013
7/26/2013
7/29/2013
7/30/2013
7/31/2013
8/1/2013
8/2/2013
8/5/2013
8/6/2013
8/7/2013
8/8/2013
8/9/2013
8/12/2013
8/13/2013
8/14/2013
8/15/2013
8/16/2013
8/19/2013
8/20/2013
8/21/2013
8/22/2013
8/23/2013
8/26/2013
150
151
152
153
154
157
158
159
160
161
164
165
166
167
168
171
172
173
175
178
179
180
181
182
185
186
187
188
189
192
193
194
195
196
199
200
201
201
202
205
206
207
208
209
212
213
214
215
216
219
220
221
222
223
226
1642,81
1626,13
1612,52
1636,36
1626,73
1639,04
1651,81
1628,93
1588,19
1592,43
1573,09
1588,03
1603,26
1613,2
1606,28
1614,96
1614,08
1615,41
1631,89
1640,46
1652,32
1652,62
1675,02
1680,19
1682,5
1676,26
1680,91
1689,37
1692,09
1695,53
1692,39
1685,94
1690,25
1691,65
1685,33
1685,96
1685,73
1706,87
1709,67
1707,14
1697,37
1690,91
1697,48
1691,42
1689,47
1694,16
1685,39
1661,32
1655,83
1646,06
1652,35
1642,8
1656,96
1663,5
1656,78
8/27/2013
8/28/2013
8/29/2013
8/30/2013
9/3/2013
9/4/2013
9/5/2013
9/6/2013
9/9/2013
9/10/2013
9/11/2013
9/12/2013
9/13/2013
9/16/2013
9/17/2013
9/18/2013
9/19/2013
9/20/2013
9/23/2013
9/24/2013
9/25/2013
9/26/2013
9/27/2013
9/30/2013
10/1/2013
10/2/2013
10/3/2013
10/4/2013
10/7/2013
10/8/2013
10/9/2013
10/10/2013
10/11/2013
10/14/2013
10/15/2013
10/16/2013
10/17/2013
10/18/2013
10/21/2013
10/22/2013
10/23/2013
10/24/2013
10/25/2013
10/28/2013
10/29/2013
10/30/2013
10/31/2013
11/1/2013
11/4/2013
11/5/2013
11/6/2013
11/7/2013
11/8/2013
11/11/2013
11/12/2013
227
228
229
230
233
234
235
236
239
240
241
242
243
246
247
248
249
250
253
254
255
256
257
260
261
262
263
264
267
268
269
270
271
274
275
276
277
278
281
282
283
284
285
288
289
290
291
291
294
295
296
297
298
301
302
1630,48
1634,96
1638,17
1632,97
1639,77
1653,08
1655,08
1655,17
1671,71
1683,99
1689,13
1683,42
1687,99
1697,6
1704,76
1725,52
1722,34
1709,91
1701,84
1697,42
1692,77
1698,67
1691,75
1681,55
1695
1693,87
1678,66
1690,5
1676,12
1655,45
1656,4
1692,56
1703,2
1710,14
1698,06
1721,54
1733,15
1744,5
1744,66
1754,67
1746,38
1752,07
1759,77
1762,11
1771,95
1763,31
1756,54
1761,64
1767,93
1762,97
1770,49
1747,15
1770,61
1771,89
1767,69
11/13/2013
11/14/2013
11/15/2013
11/18/2013
11/19/2013
11/20/2013
11/21/2013
11/22/2013
11/25/2013
11/26/2013
11/27/2013
11/29/2013
12/2/2013
12/3/2013
12/4/2013
12/5/2013
12/6/2013
12/9/2013
12/10/2013
12/11/2013
12/12/2013
12/13/2013
12/16/2013
12/17/2013
12/18/2013
12/19/2013
12/20/2013
12/23/2013
12/24/2013
12/26/2013
12/27/2013
12/30/2013
12/31/2013
1/2/2014
1/3/2014
1/6/2014
1/7/2014
1/8/2014
1/9/2014
1/10/2014
303
304
305
308
309
310
311
312
315
316
317
319
322
323
324
325
326
329
330
331
332
333
336
337
338
339
340
343
344
346
347
350
351
357
358
361
362
363
364
365
1782
1790,62
1798,18
1791,53
1787,87
1781,37
1795,85
1804,76
1802,48
1802,75
1807,23
1805,81
1800,9
1795,15
1792,81
1785,03
1805,09
1808,37
1802,62
1782,22
1775,5
1775,32
1786,54
1781
1810,65
1809,6
1818,32
1827,99
1833,32
1842,02
1841,4
1841,07
1848,36
1831,98
1831,37
1826,77
1837,88
1837,49
1838,13
1842,37
Source: S&P Dow Jones Indices.
All information presented prior to the index launch date is back-tested. Back-tested performance is not actual performance, but
calculations are based on the same methodology that was in effect when the index was officially launched. Past performance is
results. Please see the Performance Disclosure at http://www.spindices.com/regulatory-affairs-disclaimers/ for more information
limitations associated with back-tested performance.
Copyright © 2013 by S&P Dow Jones Indices LLC, a part of McGraw Hill Financial. All rights reserved. Redistribution, reproduct
whole or in part are prohibited without the written permission of S&P Dow Jones Indices. Standard & Poor’s and S&P are registe
Poor’s Financial Services LLC (“S&P”), a part of McGraw Hill Financial, Inc. Dow Jones is a registered trademark of Dow Jones
(“Dow Jones”). S&P Dow Jones Indices LLC, Dow Jones, S&P and their respective affiliates (“S&P Dow Jones Indices”) make n
express or implied, as to the ability of any index to accurately represent the asset class or market sector that it purports to repre
Indices shall have no liability for any errors, omissions, or interruptions of any index or the data included therein. Past performan
indication of future results. This document does not constitute an offer of any services. All information provided by S&P Dow Jon
and not tailored to the needs of any person, entity or group of persons. It is not possible to invest directly in an index. S&P Dow
compensation in connection with licensing its indices to third parties. Exposure to an asset class represented by an index is ava
instruments offered by third parties that are based on that index. S&P Dow Jones Indices does not sponsor, endorse, sell, prom
Copyright © 2013 by S&P Dow Jones Indices LLC, a part of McGraw Hill Financial. All rights reserved. Redistribution, reproduct
whole or in part are prohibited without the written permission of S&P Dow Jones Indices. Standard & Poor’s and S&P are registe
Poor’s Financial Services LLC (“S&P”), a part of McGraw Hill Financial, Inc. Dow Jones is a registered trademark of Dow Jones
(“Dow Jones”). S&P Dow Jones Indices LLC, Dow Jones, S&P and their respective affiliates (“S&P Dow Jones Indices”) make n
express or implied, as to the ability of any index to accurately represent the asset class or market sector that it purports to repre
Indices shall have no liability for any errors, omissions, or interruptions of any index or the data included therein. Past performan
indication of future results. This document does not constitute an offer of any services. All information provided by S&P Dow Jon
and not tailored to the needs of any person, entity or group of persons. It is not possible to invest directly in an index. S&P Dow
compensation in connection with licensing its indices to third parties. Exposure to an asset class represented by an index is ava
instruments offered by third parties that are based on that index. S&P Dow Jones Indices does not sponsor, endorse, sell, prom
fund or other investment vehicle that seeks to provide an investment return based on the performance of any Index. S&P Dow J
investment advisor, and S&P Dow Jones Indices makes no representation regarding the advisability of investing in any such inv
investment vehicle. For more information on any of our indices please visit www.spdji.com.
rmance is not actual performance, but is hypothetical. The back-test
officially launched. Past performance is not a guarantee of future
affairs-disclaimers/ for more information regarding the inherent
hts reserved. Redistribution, reproduction and/or photocopying in
Standard & Poor’s and S&P are registered trademarks of Standard &
s a registered trademark of Dow Jones Trademark Holdings LLC
tes (“S&P Dow Jones Indices”) make no representation or warranty,
r market sector that it purports to represent and S&P Dow Jones
e data included therein. Past performance of an index is not an
l information provided by S&P Dow Jones Indices is general in nature
o invest directly in an index. S&P Dow Jones Indices may receive
et class represented by an index is available through investable
does not sponsor, endorse, sell, promote or manage any investment
hts reserved. Redistribution, reproduction and/or photocopying in
Standard & Poor’s and S&P are registered trademarks of Standard &
s a registered trademark of Dow Jones Trademark Holdings LLC
tes (“S&P Dow Jones Indices”) make no representation or warranty,
r market sector that it purports to represent and S&P Dow Jones
e data included therein. Past performance of an index is not an
l information provided by S&P Dow Jones Indices is general in nature
o invest directly in an index. S&P Dow Jones Indices may receive
et class represented by an index is available through investable
does not sponsor, endorse, sell, promote or manage any investment
performance of any Index. S&P Dow Jones Indices LLC is not an
advisability of investing in any such investment fund or other
m.
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