Tukey’s 1-Degree of Freedom for Non-Additivity/ Yields for 8 Business Indices Over 18 Years

1. Tukey’s 1-Degree of Freedom for Non-Additivity

Yields for 8 Business Indices Over 18 Years
K.V. Smith(1969). “Stock Price and Economic Indexes for Generating Efficient
Portfolios,” The Journal of Business, Vol. 42, #3, pp. 326-336

2. Experimental Setting

• 2-Way ANOVA with one measurement per
combination of levels of factors A and B (N=a(b)(1))
• Additive Model: E(Yij)= m + ai + bj
• Interaction Model (1 df): E(Yij)= m + ai + bj + abij
Where abij = haibj/m = Daibj
Procedure Involves Estimating D and testing
whether the parameter equals 0

3. Data – 18 Years for 8 Business Indices

Year\Index
1965
1964
1963
1962
1961
1960
1959
1958
1957
1956
1955
1954
1953
1952
1951
1950
1949
1948
Average
DJIA
1.103
1.145
1.169
0.890
1.207
0.896
1.184
1.425
0.833
1.000
1.231
1.393
0.965
1.074
1.174
1.179
1.114
0.972
1.109
POOR
1.099
1.131
1.201
0.872
1.231
0.953
1.094
1.376
0.856
1.034
1.301
1.497
0.925
1.109
1.178
1.247
1.091
0.996
1.122
NYSE
1.095
1.143
1.180
0.880
1.240
0.976
1.097
1.366
0.866
1.026
1.222
1.426
0.938
1.065
1.132
1.211
1.102
0.972
1.108
GNP
1.086
1.066
1.050
1.072
1.032
1.041
1.086
1.044
1.056
1.055
1.095
0.994
1.053
1.055
1.156
1.102
0.996
1.110
1.064
Factor A: Year (a=18)
CPI
1.017
1.013
1.012
1.012
1.011
1.016
1.008
1.028
1.035
1.015
0.997
1.004
1.008
1.022
1.080
1.009
0.990
1.077
1.020
FRB
1.083
1.064
1.051
1.078
1.099
1.029
1.127
0.930
1.008
1.034
1.126
0.940
1.083
1.037
1.085
1.157
0.945
1.041
1.051
BWEEK
1.093
1.073
1.060
1.018
1.154
0.917
1.073
1.099
0.906
1.011
1.125
1.074
0.952
1.141
1.012
1.213
0.995
1.019
1.052
Factor B: Index (b=8)
MONEY
1.048
1.043
1.038
1.013
1.031
0.924
1.006
1.038
0.993
1.012
1.020
1.029
1.010
1.038
1.055
1.057
0.994
0.985
1.019
Average
1.078
1.085
1.095
0.979
1.126
0.969
1.084
1.163
0.944
1.023
1.140
1.170
0.992
1.068
1.109
1.147
1.028
1.022
1.068

4. Algorithm

• Fit the additive Model and estimate m, ai and bj
^
^
m = Y
a i = Y i Y
^
b j = Y j Y
• Fit the interaction model with abij = Daibj
Yij = m + a i + b j + Da i b j + ij
^
^
^
^
^
Yij = m + a i + b j + D a i b j + eij
• Use Least Squares to estimate D
• Obtain Sum of Squares for Interaction and Remainder
• Conduct 1-degree of freedom F-test of H0: D=0

5. OLS Estimation of D

^
^
^
^ ^
Q = e = Yij m a i b j D a i b j
i =1 j =1
i =1 j =1
a
b
a
b
2
2
ij
a
b
^
^
^
^ ^
Q
^ ^
= 2 Yij m a i b j D a i b j a i b j
D
i =1 j =1
a ^ 2 b ^
a ^
b ^ 2
a ^ 2 b ^ 2
a b ^ ^ ^ a ^ b ^
= 2 Yij a i b j m a i b j a i b j a i b j D a i b j
i =1
j =1
i =1
j =1
i =1
j =1
i =1
j =1
i =1 j =1
a ^ 2 b ^ 2
a b ^ ^
= 2 Yij a i b j 0 0 0 D a i b j
i =1
j =1
i =1 j =1
^ ^
Yij a i b j
i =1 j =1
a
^
a b ^ ^ ^ a ^ 2 b ^ 2
Q
Setting
= 0 0 = Yij a i b j D a i b j D =
D
i =1
j =1
i =1 j =1
b
a
^
Y Y
D=
b
ij
i =1 j =1
a
Y
i =1
i
i
Y
Y Y j Y
Y
2
b
j =1
j
Y
2
^ 2
a b
i
i =1
a
^ 2 b
j =1
j

6. Sum of Squares for Interaction (abij = Daibj )

a
^ 2
b
a
b
^ 2 ^ 2 ^ 2
SSAB* = ab ij = D a i b j
i =1 j =1
i =1 j =1
a b
Yij Y i Y Y j Y
i =1 j =1
= a
b
2
2
Y
Y
Y
Y
i
j
j =1
i =1
a b
Yij Y i Y Y j Y
i =1 j =1
= a
b
2
2
Y
Y
Y
Y
i j
i =1
2
Y
a
i =1
i Y
Y
2
b
j
Y
j =1
2
j =1
" Remainder" SS : SSREM * = SSTO SSA SSB SSAB *
Under H 0 : D = 0 (No interactio ns exist of the form Da i b j )
SSAB* ~ 12
F* =
SSREM * ~ (2a 1)( b 1) 1
SSAB* SSREM *
SSAB * 1
~ F1,( a 1)( b 1) 1
SSREM * (a 1)(b 1) 1
2

7. Business Index Example

mu
1.0679
Y
18
Business Index Example
i Y
Y
2
= 0.081676
i =1
i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
alpha_i
0.0101
0.0169
0.0273
-0.0885
0.0578
-0.0989
0.0165
0.0954
-0.1237
-0.0445
0.0718
0.1018
-0.0761
-0.0002
0.0411
0.0790
-0.0395
-0.0464
j
1
2
3
4
5
6
7
8
Y
8
beta_j
0.0407
0.0539
0.0398
-0.0040
-0.0482
-0.0169
-0.0159
-0.0493
Y
j
2
= 0.011447
j =1
Y Y
18
8
i =1 j =1
ij
i
Y Y j Y = 0.022112
0.022112
D=
= 23.65
(0.081676)(0.011447)
^
1.103
1.145
1.169
0.890
1.207
0.896
1.184
1.425
0.833
1.000
1.231
1.393
0.965
1.074
1.174
1.179
1.114
0.972
1.099
1.131
1.201
0.872
1.231
0.953
1.094
1.376
0.856
1.034
1.301
1.497
0.925
1.109
1.178
1.247
1.091
0.996
1.095
1.143
1.180
0.880
1.240
0.976
1.097
1.366
0.866
1.026
1.222
1.426
0.938
1.065
1.132
1.211
1.102
0.972
1.086
1.066
1.050
1.072
1.032
1.041
1.086
1.044
1.056
1.055
1.095
0.994
1.053
1.055
1.156
1.102
0.996
1.110
1.017
1.013
1.012
1.012
1.011
1.016
1.008
1.028
1.035
1.015
0.997
1.004
1.008
1.022
1.080
1.009
0.990
1.077
1.083
1.064
1.051
1.078
1.099
1.029
1.127
0.930
1.008
1.034
1.126
0.940
1.083
1.037
1.085
1.157
0.945
1.041
1.093
1.073
1.060
1.018
1.154
0.917
1.073
1.099
0.906
1.011
1.125
1.074
0.952
1.141
1.012
1.213
0.995
1.019
1.048
1.043
1.038
1.013
1.031
0.924
1.006
1.038
0.993
1.012
1.020
1.029
1.010
1.038
1.055
1.057
0.994
0.985

8. Sums of Squares & F-Test for Interaction

Sums of Squares & F-Test for Interaction
SSTO = Y
a
b
ij
i =1 j =1
a
Y
SSA = b Y i Y
2
2
= 1.78991
= 0.653406
i =1
b
SSB = a Y j Y
2
= 0.206039
j =1
2
a b
Yij Y i Y Y j Y
2
0.022112
i
=
1
j
=
1
=
SSAB* = a
= 0.522991
b
2
2
(0.081676)(0.011447)
Y
Y
Y
Y
i
j
i =1
j =1
SSREM * = SSTO SSA SSB SSAB* = 0.407475
Test Statistic for Testing H 0 : D = 0 :
SSAB * 1
SSAB *
0.522991
F* =
=
=
= 151.45
SSREM * (18 1)(8 1) 1 SSREM * 118 0.407475 118
F (.05,1,118) = 3.92 P( F F *) = .0000

9.

Plot of Residuals From additive Model vs (a_i)(b_j)
0,3
0,2
e_ij
0,1
0
-0,1
-0,2
-0,3
-0,008
-0,006
-0,004
-0,002
0
(a_i)(b_j)
0,002
0,004
0,006
0,008