Transcript Estimating and Predicting Stock Returns Using Artificial

BUCHAREST ACADEMY OF ECONOMIC STUDIES
DOCTORAL SCHOOL OF FINANCE AND BANKING-DOFIN
Estimating and Predicting
Stock Returns
Using Artificial Neural Networks
Dissertation Paper
Supervisor:
Prof. Univ. Dr. Moisa Altar
MSc Student:
Catalin-Marius Untea
Slide 2
Introduction
This paper estimates and predicts stock returns, for shares traded on the
Bucharest Stock Exchange, using both Artificial Neural Networks and Classical
Econometric Arbitrage Pricing Theory (APT) methods, and compares the results
obtained from both methods. The APT model is empirically implemented using
Two-steep Cross Sectional Regression procedure introduced by Fama and McBeth
(1973), and the One-step System of Non-linear Seemingly Unrelated Equations
procedure firstly introduced by McElroy, Burmeister and Wall (1985). Neural
network analyze, includes estimates conducted using Feedforward Neural
Networks and Elman Recurrent Neural Networks.
This paper does not try to discredit the Classical Econometric approach to the
problem of estimation and prediction of stock returns, but it tries to emphasis the
advantages brought by the new methods of estimation and prediction offered by
Artificial Neural Networks, compared to classical econometrical methods with
closed form used by many studies.
What this paper is trying to bring additionally to other empirical studies in the field,
is a complete practical approach to the problem of estimation and prediction, from
the viewpoint of both econometric methods and neural network models. It
concentrates on the shares traded on the Bucharest Stock Exchange, an emerging
market during the last years.
Slide 3
Theoretical background:
The Arbitrage Pricing Theory (APT)
Part I
The Arbitrage Pricing Theory (APT) is a theoretical model, with tries to explain
the behavior of stock returns to macroeconomic or firm specific factors.
The major difficulty in applying the APT model comes from the fact that it
shows that there is a method of predicting stock returns, but does not specify
how exactly it must be solved. The main idea of the theory is that there exists
a set of factors, so that, expected return can be expressed as a linear
combination of those factors. The APT model is based on the hypotheses of
arbitrage non-existence, which can be expressed as needing an upper
limitation to the ratio between expected return and the volatility, of the same
investment. If this ratio would not be limited, then it would be possible to obtain
positive expected return for very low levels of risks.
Slide 4
Theoretical background:
The Arbitrage Pricing Theory (APT)
Part II
The APT model can be described by two equations:
the first equation expresses the stock returns based on the set of factors
k
Rit  E[ Rit ]   bij Fjt   it , t=1,...,T i=1,...,N j=1,...,k
j 1
where E[ Ri ] represents the expected return for share i;
Ri
represents the return on share i;
Fj
represents the influence of factor j on stock return i;
bij
represents the sensitivity of the return on asset i to the fluctuations of factor j;
the second equation is for the equilibrium expected return and expresses the no arbitrage opportunity:
k
E[ Rit ]  0t   bij  jt
j 1
where 0
j
represents the free-risk rate return;
represents the risk premiums corresponding to risk factor j;
Slide 5
Two-step cross-sectional regression
procedure
Part I
Risk premium estimation for economic variables was introduced by Chen, Roll and
Ross (1986), by making used of the two-step cross-sectional regression procedure,
first introduced by Fama and McBeth. In the first stage of the procedure, the
sensitivity coefficients for independent variables are estimated by making use of
generalized method of moments (GMM).
During the first stage, the factor coefficients are estimated based on the following
regression model:
Rit   i    ij  Fjt  eit
j
where Rit
represents portfolio return i;
F jt
represents principal component j;
 ij
represents sensitivity coefficient for portfolio return i at
factor fluctuations j.
Slide 6
Estimation results for portfolio 1
2001 - 2003
2005 - 2006
Dependent Variable: RAND_PORT1
Dependent Variable: RAND_PORT1
Method: Generalized Method of Moments
Method: Generalized Method of Moments
Date: 06/24/07 Time: 16:25
Date: 06/24/07 Time: 16:27
Sample: 252 966
Sample: 1240 1714
Included observations: 715
Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iteration
Simultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5
COMP_PRIN4 COMP_PRIN5
Variable
Coefficient
Std. Error
t-Statistic
Prob.
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
0.010370
0.003513
2.951664
0.0033
C
0.000225
0.000222
1.012180
0.3120
COMP_PRIN1
0.001069
0.003139
0.340612
0.7335
COMP_PRIN1
6.42E-06
7.84E-06
0.819388
0.4130
COMP_PRIN2
0.000621
0.004111
0.151106
0.8799
COMP_PRIN2
0.000117
0.000119
0.987849
0.3237
COMP_PRIN3
-0.000596
0.002943
-0.202683
0.8394
COMP_PRIN3
0.000174
0.000176
0.989139
0.3231
COMP_PRIN4
0.001872
0.005899
0.317385
0.7510
COMP_PRIN4
1.18E-06
1.26E-05
0.093644
0.9254
COMP_PRIN5
-0.003121
0.004433
-0.703955
0.4817
COMP_PRIN5
-2.48E-05
2.74E-05
-0.907004
0.3649
0.002491
Mean dependent var
0.000224
-0.008143
S.D. dependent var
0.004887
R-squared
0.001508
Mean dependent var
0.010340
R-squared
-0.005533
S.D. dependent var
0.074864
Adjusted R-squared
S.E. of regression
0.075071
Sum squared resid
3.995630
S.E. of regression
0.004906
Sum squared resid
0.011290
Durbin-Watson stat
1.794584
J-statistic
7.31E-33
Durbin-Watson stat
2.006955
J-statistic
2.40E-28
Adjusted R-squared
Slide 7
Estimation results for portfolio 2
2001 - 2003
2005 - 2006
Dependent Variable: RAND_PORT2
Dependent Variable: RAND_PORT2
Method: Generalized Method of Moments
Method: Generalized Method of Moments
Date: 06/24/07 Time: 16:22
Date: 06/24/07 Time: 16:23
Sample: 252 966
Sample: 1240 1714
Included observations: 715
Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iteration
Simultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5
COMP_PRIN4 COMP_PRIN5
Variable
Coefficient
Std. Error
t-Statistic
Prob.
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
0.008733
0.003082
2.833567
0.0047
C
-2.12E-33
6.68E-19
-3.17E-15
1.0000
COMP_PRIN1
-0.002338
0.003964
-0.589960
0.5554
COMP_PRIN1
3.08E-33
6.03E-19
5.11E-15
1.0000
COMP_PRIN2
-0.004659
0.002904
-1.604640
0.1090
COMP_PRIN2
-1.00E-32
1.62E-18
-6.20E-15
1.0000
COMP_PRIN3
-0.002547
0.002832
-0.899318
0.3688
COMP_PRIN3
4.62E-33
1.02E-18
4.53E-15
1.0000
COMP_PRIN4
-0.001160
0.004485
-0.258574
0.7960
COMP_PRIN4
1.39E-32
2.64E-18
5.26E-15
1.0000
COMP_PRIN5
-0.002081
0.002495
-0.834141
0.4045
COMP_PRIN5
1.16E-33
1.79E-18
6.45E-16
1.0000
R-squared
0.011744
Mean dependent var
0.008567
Mean dependent var
0.000000
S.D. dependent var
0.000000
Adjusted R-squared
0.004774
S.D. dependent var
0.056587
S.E. of regression
1.74E-32
Sum squared resid
1.41E-61
S.E. of regression
0.056451
Sum squared resid
2.259404
Durbin-Watson stat
1.715121
J-statistic
0.049420
Durbin-Watson stat
1.427165
J-statistic
1.55E-31
Slide 8
Estimation results for portfolio 3
2001 - 2003
2005 - 2006
Dependent Variable: RAND_PORT3
Dependent Variable: RAND_PORT3
Method: Generalized Method of Moments
Method: Generalized Method of Moments
Date: 06/24/07 Time: 16:18
Date: 06/24/07 Time: 16:20
Sample: 252 966
Sample: 1240 1714
Included observations: 715
Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iteration
Simultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5
COMP_PRIN4 COMP_PRIN5
Variable
Coefficient
Std. Error
t-Statistic
Prob.
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
0.006710
0.002460
2.727974
0.0065
C
0.002804
0.004118
0.680892
0.4963
COMP_PRIN1
-0.000543
0.002928
-0.185520
0.8529
COMP_PRIN1
0.001672
0.001025
1.631485
0.1035
COMP_PRIN2
0.003382
0.003659
0.924428
0.3556
COMP_PRIN2
-0.011449
0.003852
-2.971858
0.0031
COMP_PRIN3
-0.004555
0.003514
-1.296118
0.1954
COMP_PRIN3
-0.008726
0.003318
-2.629798
0.0088
COMP_PRIN4
0.002028
0.005067
0.400208
0.6891
COMP_PRIN4
0.005482
0.003542
1.547695
0.1224
COMP_PRIN5
-0.002333
0.004169
-0.559522
0.5760
COMP_PRIN5
0.006124
0.004981
1.229481
0.2195
R-squared
0.006025
Mean dependent var
0.006937
R-squared
0.034714
Mean dependent var
0.003431
-0.000985
S.D. dependent var
0.066368
Adjusted R-squared
0.024423
S.D. dependent var
0.098178
S.E. of regression
0.066401
Sum squared resid
3.126022
S.E. of regression
0.096972
Sum squared resid
4.410230
Durbin-Watson stat
1.975762
J-statistic
1.10E-31
Durbin-Watson stat
2.066803
J-statistic
5.30E-31
Adjusted R-squared
Slide 9
Estimation results for portfolio 4
2001 - 2003
2005 - 2006
Dependent Variable: RAND_PORT4
Dependent Variable: RAND_PORT4
Method: Generalized Method of Moments
Method: Generalized Method of Moments
Date: 06/24/07 Time: 16:13
Date: 06/24/07 Time: 16:15
Sample: 252 966
Sample: 1240 1714
Included observations: 715
Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iteration
Simultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5
COMP_PRIN4 COMP_PRIN5
Variable
Coefficient
Std. Error
t-Statistic
Prob.
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
0.013891
0.004260
3.261272
0.0012
C
-0.002295
0.004776
-0.480618
0.6310
COMP_PRIN1
-4.93E-05
0.004573
-0.010774
0.9914
COMP_PRIN1
-0.001392
0.001519
-0.916460
0.3599
COMP_PRIN2
-0.005801
0.004750
-1.221195
0.2224
COMP_PRIN2
-0.012073
0.003925
-3.076084
0.0022
COMP_PRIN3
-0.017844
0.003964
-4.501326
0.0000
COMP_PRIN3
-0.010983
0.003934
-2.791577
0.0055
COMP_PRIN4
0.000319
0.005538
0.057647
0.9540
COMP_PRIN4
0.001190
0.004629
0.257030
0.7973
COMP_PRIN5
0.005738
0.006927
0.828380
0.4077
COMP_PRIN5
0.004451
0.002542
1.751189
0.0806
R-squared
0.035663
Mean dependent var
0.014011
R-squared
0.035129
Mean dependent var
-0.001873
Adjusted R-squared
0.028863
S.D. dependent var
0.102160
Adjusted R-squared
0.024843
S.D. dependent var
0.104259
S.E. of regression
0.100675
Sum squared resid
7.186013
S.E. of regression
0.102956
Sum squared resid
4.971373
Durbin-Watson stat
1.883555
J-statistic
4.66E-31
Durbin-Watson stat
2.068589
J-statistic
7.70E-32
Slide 10
Estimation results for portfolio 5
2001 - 2003
2005 - 2006
Dependent Variable: RAND_PORT5
Dependent Variable: RAND_PORT5
Method: Generalized Method of Moments
Method: Generalized Method of Moments
Date: 06/24/07 Time: 16:06
Date: 06/24/07 Time: 16:08
Sample: 252 966
Sample: 1240 1714
Included observations: 715
Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iteration
Simultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5
COMP_PRIN4 COMP_PRIN5
Variable
Coefficient
Std. Error
t-Statistic
Prob.
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
0.011629
0.004888
2.379364
0.0176
C
0.004861
0.004364
1.114030
0.2658
COMP_PRIN1
0.006290
0.005295
1.188006
0.2352
COMP_PRIN1
-0.002308
0.001453
-1.588477
0.1129
COMP_PRIN2
-0.010741
0.005006
-2.145833
0.0322
COMP_PRIN2
-0.022491
0.005128
-4.386342
0.0000
COMP_PRIN3
-0.012818
0.004205
-3.048595
0.0024
COMP_PRIN3
-0.040446
0.007050
-5.737089
0.0000
COMP_PRIN4
-0.003786
0.006712
-0.564091
0.5729
COMP_PRIN4
-0.000495
0.004523
-0.109453
0.9129
COMP_PRIN5
-0.001225
0.005709
-0.214617
0.8301
COMP_PRIN5
-0.001721
0.004894
-0.351652
0.7253
R-squared
0.025255
Mean dependent var
0.011390
R-squared
0.179157
Mean dependent var
0.004630
Adjusted R-squared
0.018380
S.D. dependent var
0.108759
Adjusted R-squared
0.170406
S.D. dependent var
0.127867
S.E. of regression
0.107754
Sum squared resid
8.232218
S.E. of regression
0.116464
Sum squared resid
6.361403
Durbin-Watson stat
1.688630
J-statistic
4.69E-30
Durbin-Watson stat
2.323714
J-statistic
3.19E-31
Slide 11
Estimation results for portfolio 6
2001 - 2003
2005 - 2006
Dependent Variable: RAND_PORT6
Dependent Variable: RAND_PORT6
Method: Generalized Method of Moments
Method: Generalized Method of Moments
Date: 06/24/07 Time: 16:00
Date: 06/24/07 Time: 16:04
Sample: 252 966
Sample: 1240 1714
Included observations: 715
Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iteration
Simultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5
COMP_PRIN4 COMP_PRIN5
Variable
Coefficient
Std. Error
t-Statistic
Prob.
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
0.018845
0.004709
4.002243
0.0001
C
0.009514
0.005311
1.791333
0.0739
COMP_PRIN1
0.005747
0.006621
0.867955
0.3857
COMP_PRIN1
0.001340
0.001369
0.978250
0.3285
COMP_PRIN2
-0.013967
0.006309
-2.213911
0.0272
COMP_PRIN2
-0.030951
0.004950
-6.253226
0.0000
COMP_PRIN3
-0.028096
0.006492
-4.327917
0.0000
COMP_PRIN3
-0.061180
0.006443
-9.495897
0.0000
COMP_PRIN4
-0.009366
0.008081
-1.159018
0.2468
COMP_PRIN4
0.004193
0.004279
0.979856
0.3277
COMP_PRIN5
0.004671
0.008355
0.559061
0.5763
COMP_PRIN5
0.006695
0.003997
1.675108
0.0946
R-squared
0.070940
Mean dependent var
0.018974
R-squared
0.295558
Mean dependent var
0.009086
Adjusted R-squared
0.064388
S.D. dependent var
0.123354
Adjusted R-squared
0.288048
S.D. dependent var
0.146438
S.E. of regression
0.119317
Sum squared resid
10.09372
S.E. of regression
0.123560
Sum squared resid
7.160256
Durbin-Watson stat
1.830613
J-statistic
7.01E-31
Durbin-Watson stat
2.095171
J-statistic
2.32E-32
Slide 12
Estimation results for portfolio 7
2001 - 2003
2005 - 2006
Dependent Variable: RAND_PORT7
Dependent Variable: RAND_PORT7
Method: Generalized Method of Moments
Method: Generalized Method of Moments
Date: 06/24/07 Time: 15:50
Date: 06/24/07 Time: 15:52
Sample: 252 966
Sample: 1240 1714
Included observations: 715
Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iteration
Simultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5
COMP_PRIN4 COMP_PRIN5
Variable
Coefficient
Std. Error
t-Statistic
Prob.
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
0.015913
0.004237
3.755909
0.0002
C
0.009633
0.004151
2.320428
0.0207
COMP_PRIN1
-0.001386
0.005520
-0.251057
0.8018
COMP_PRIN1
-0.000167
0.001006
-0.165679
0.8685
COMP_PRIN2
-0.013094
0.004741
-2.761637
0.0059
COMP_PRIN2
-0.023277
0.004432
-5.252230
0.0000
COMP_PRIN3
-0.025616
0.005611
-4.565651
0.0000
COMP_PRIN3
-0.038118
0.005367
-7.102547
0.0000
COMP_PRIN4
-0.002776
0.006131
-0.452871
0.6508
COMP_PRIN4
0.001651
0.003284
0.502787
0.6153
COMP_PRIN5
-0.009707
0.006174
-1.572245
0.1163
COMP_PRIN5
-3.72E-05
0.003682
-0.010109
0.9919
R-squared
0.073087
Mean dependent var
0.015806
R-squared
0.238383
Mean dependent var
0.009630
Adjusted R-squared
0.066550
S.D. dependent var
0.109234
Adjusted R-squared
0.230264
S.D. dependent var
0.106029
S.E. of regression
0.105537
Sum squared resid
7.896851
S.E. of regression
0.093024
Sum squared resid
4.058460
Durbin-Watson stat
1.845098
J-statistic
2.88E-30
Durbin-Watson stat
2.096640
J-statistic
3.21E-31
Slide 13
Estimation results for portfolio 8
2001 - 2003
2005 - 2006
Dependent Variable: RAND_PORT8
Dependent Variable: RAND_PORT8
Method: Generalized Method of Moments
Method: Generalized Method of Moments
Date: 06/24/07 Time: 15:19
Date: 06/24/07 Time: 15:20
Sample: 252 966
Sample: 1240 1714
Included observations: 715
Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iteration
Simultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5
COMP_PRIN4 COMP_PRIN5
Variable
Coefficient
Std. Error
t-Statistic
Prob.
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
0.022262
0.005429
4.100448
0.0000
C
0.021635
0.005345
4.047977
0.0001
COMP_PRIN1
-0.001124
0.008180
-0.137423
0.8907
COMP_PRIN1
-0.000399
0.001227
-0.324855
0.7454
COMP_PRIN2
-0.028633
0.007732
-3.703067
0.0002
COMP_PRIN2
-0.045252
0.004896
-9.242509
0.0000
COMP_PRIN3
-0.061396
0.007952
-7.720936
0.0000
COMP_PRIN3
-0.089708
0.007388
-12.14312
0.0000
COMP_PRIN4
-0.012017
0.010598
-1.133904
0.2572
COMP_PRIN4
-0.004756
0.003832
-1.241197
0.2152
COMP_PRIN5
0.001534
0.008002
0.191727
0.8480
COMP_PRIN5
0.009463
0.003924
2.411639
0.0163
R-squared
0.207651
Mean dependent var
0.022447
R-squared
0.506522
Mean dependent var
0.020567
Adjusted R-squared
0.202063
S.D. dependent var
0.153207
Adjusted R-squared
0.501261
S.D. dependent var
0.163423
S.E. of regression
0.136856
Sum squared resid
13.27921
S.E. of regression
0.115412
Sum squared resid
6.247043
Durbin-Watson stat
1.864400
J-statistic
3.95E-32
Durbin-Watson stat
2.023208
J-statistic
2.08E-31
Slide 14
Estimation results for portfolio 9
2001 - 2003
2005 - 2006
Dependent Variable: RAND_PORT9
Dependent Variable: RAND_PORT9
Method: Generalized Method of Moments
Method: Generalized Method of Moments
Date: 06/24/07 Time: 15:18
Date: 06/24/07 Time: 15:16
Sample: 252 966
Sample: 1240 1714
Included observations: 715
Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iteration
Simultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5
COMP_PRIN4 COMP_PRIN5
Variable
Coefficient
Std. Error
t-Statistic
Prob.
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
0.012672
0.004324
2.930734
0.0035
C
0.012620
0.004426
2.851511
0.0045
COMP_PRIN1
0.009091
0.006160
1.475942
0.1404
COMP_PRIN1
-0.006055
0.005933
-1.020475
0.3080
COMP_PRIN2
-0.028994
0.005571
-5.204247
0.0000
COMP_PRIN2
-0.057778
0.008377
-6.897398
0.0000
COMP_PRIN3
-0.055857
0.006506
-8.585827
0.0000
COMP_PRIN3
-0.090673
0.005644
-16.06432
0.0000
COMP_PRIN4
-0.010915
0.007077
-1.542251
0.1235
COMP_PRIN4
-0.018203
0.015571
-1.169005
0.2430
COMP_PRIN5
-0.010612
0.006769
-1.567750
0.1174
COMP_PRIN5
0.024597
0.016501
1.490665
0.1367
R-squared
0.265682
Mean dependent var
0.012540
R-squared
0.590284
Mean dependent var
0.011881
Adjusted R-squared
0.260503
S.D. dependent var
0.125231
Adjusted R-squared
0.585916
S.D. dependent var
0.168003
S.E. of regression
0.107691
Sum squared resid
8.222594
S.E. of regression
0.108109
Sum squared resid
5.481453
Durbin-Watson stat
1.874611
J-statistic
2.93E-31
Durbin-Watson stat
2.089196
J-statistic
6.06E-31
Slide 15
Two-step cross-sectional regression
procedure
Part II
At the second stage, the estimated sensitivity coefficients are used as
independent variables in the cross-sectional regression in order to estimate the
risk premium of the observed variables.
The previously estimated sensitivity coefficients β, in the first stage, are used in the
cross-section regression as independent variables, and portfolios mean returns are
used as dependent variables. Each coefficient obtained by estimating the crosssection regression, represents an estimation for the risk premium associated to
the exposure to unexpected variation in one of the factors.
mean _ portfolio _ return  ˆ1  ˆCOMP _ PRIN1  ˆCOMP _ PRIN1 
ˆCOMP _ PRIN 2  ˆCOMP _ PRIN 2  ˆCOMP _ PRIN3  ˆCOMP _ PRIN3  ˆCOMP _ PRIN 4  ˆCOMP _ PRIN 4 
ˆCOMP _ PRIN5  ˆCOMP _ PRIN5  u
Slide 16
Cross-section regression
2001 - 2003
2005 - 2006
Dependent Variable: MEDII_PORT_01_03
Dependent Variable: MEDII_PORT_05_06
Method: Generalized Method of Moments
Method: Generalized Method of Moments
Date: 06/24/07 Time: 18:17
Date: 06/24/07 Time: 18:51
Sample: 1 9
Sample: 1 9
Included observations: 9
Included observations: 9
Kernel: Bartlett, Bandwidth: Fixed (2), No prewhitening
Kernel: Bartlett, Bandwidth: Fixed (2), No prewhitening
Simultaneous weighting matrix & coefficient iteration
Simultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Convergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: BETA_COMP_PRIN1_01_03 BETA_COMP_PRIN2_01_
Instrument list: BETA_COMP_PRIN1_05_06 BETA_COMP_PRIN2_05_
03 BETA_COMP_PRIN3_01_03 BETA_COMP_PRIN4_01_03
06 BETA_COMP_PRIN3_05_06 BETA_COMP_PRIN4_05_06
BETA_COMP_PRIN5_01_03
BETA_COMP_PRIN5_05_06
Variable
Coefficient
Std. Error
t-Statistic
Prob.
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
0.010591
0.001742
6.081121
0.0089
C
0.000309
0.000184
1.674079
0.1927
BETA_COMP_PRIN1
_01_03
-0.359591
0.249104
-1.443537
0.2446
BETA_COMP_PRIN1
_05_06
4.283085
0.201847
21.21945
0.0002
BETA_COMP_PRIN2
_01_03
-0.086729
0.227678
-0.380926
0.7286
BETA_COMP_PRIN2
_05_06
-0.879158
0.071276
-12.33456
0.0011
BETA_COMP_PRIN3
_01_03
-0.064155
0.098361
-0.652237
0.5608
BETA_COMP_PRIN3
_05_06
0.195435
0.035902
5.443565
0.0122
BETA_COMP_PRIN4
_01_03
-0.414972
0.395612
-1.048936
0.3713
BETA_COMP_PRIN4
_05_06
-1.391730
0.093100
-14.94884
0.0006
BETA_COMP_PRIN5
_01_03
0.315264
0.137419
2.294187
0.1055
BETA_COMP_PRIN5
_05_06
-0.834989
0.047847
-17.45122
0.0004
R-squared
0.783210
Mean dependent var
0.013446
R-squared
0.992682
Mean dependent var
0.006397
Adjusted R-squared
0.421893
S.D. dependent var
0.004979
Adjusted R-squared
0.980485
S.D. dependent var
0.007140
S.E. of regression
0.003786
Sum squared resid
4.30E-05
S.E. of regression
0.000997
Sum squared resid
2.98E-06
Durbin-Watson stat
1.422391
J-statistic
6.18E-29
Durbin-Watson stat
2.211269
J-statistic
3.59E-25
Slide 17
Risk premiums associated with factors
C
γˆCOMP_PRIN5
COMP_PRIN1
COMP_PRIN2
COMP_PRIN3
COMP_PRIN4
C
γˆCOMP_PRIN1
γˆCOMP_PRIN2
γˆCOMP_PRIN3
γˆCOMP_PRIN4
γˆCOMP_PRIN5
Interval 2001 – 2003
0.010591
-0.359591
-0.086729
-0.064155
-0.414972
0.315264
Interval 2005 – 2006
0.000309
4.283085
-0.879158
0.195435
-1.391730
-0.834989
Slide 18
Predictions using sensitivity coefficients estimated through Two-step
cross sectional regression procedure
2001 - 2003
Root Mean
Squared Error
Statistic
2005 - 2006
Success ratio
sign prediction
Root Mean
Squared Error
Statistic
Success ratio
sign prediction
Portfolio 1
0.022827449
0.35
Portfolio 1
0.00028515
0
Portfolio 2
0.05469349
0.4
Portfolio 2
8.81318E-33
0
Portfolio 3
0.02152143
0.3
Portfolio 3
0.056057293
0.2
Portfolio 4
0.051632161
0.4
Portfolio 4
0.08000203
0.6
Portfolio 5
0.075466939
0.4
Portfolio 5
0.136708046
0.6
Portfolio 6
0.08583762
0.75
Portfolio 6
0.114273376
0.65
Portfolio 7
0.062730021
0.65
Portfolio 7
0.145340356
0.5
Portfolio 8
0.081078654
0.7
Portfolio 8
0.079147186
0.7
Portfolio 9
0.075674544
0.65
Portfolio 9
0.068176612
0.7
Slide 19
One-step system of non-linear
seemingly unrelated equations
An alternative method to the two-step procedure of estimating risk premium for
economic observed variables was introduced by McElroy, Burmeister and Wall
(1985), who demonstrated that the APT model can be expressed as a system of
non-linear seemingly unrelated equations, in which factor loading and risk premium
are estimated in one single step.
The APT model has to parts: the procedure that generates returns and another
equation for expected returns. By substituting expected returns in the returns
generating equation, is resulting a single equation for APT:
k
k
j 1
j 1
Rit  0t   bij  jt   bij Fjt  it
or by passing to the left side the risk-free rate of return, the last equation becomes:
k
k
j 1
j 1
Rit  0t   bij  jt   bij Fjt  it
The risk-free rate of return is known, and considered for the purpose of this paper
equal to the return on average annual interest rate for all existent deposits.
Slide 20
IN-SAMPLE statistics in interval
2001 – 2003
Coefficient estimates for the system of
equations in interval 2001 – 2003
Variable
Coeff
A1
-0.003233423
0.001693366
-1.90946
0.05620213
A2
-3.613009384
1.94176553
-1.86068
0.06278901
Portfolio 1
4.3905947134
-0.008163
A3
-0.004907741
0.001710068
-2.86991
0.00410589
Portfolio 2
2.3835046056
0.013809
A4
0
0
0
0
Portfolio 3
3.3469906452
-0.010630
A5
-0.012755416
0.001426302
-8.943
0
Portfolio 4
7.3023258879
0.050090
A6
0
0
0
0
Portfolio 5
8.5221651357
0.031992
A7
0.00022876
0.002508174
0.09121
0.92732906
Portfolio 6
10.680690442
0.063338
A8
0
0
0
0
Portfolio 7
8.1580819435
0.074072
A9
-0.005935964
0.002616418
-2.26874
0.02328435
Portfolio 8
15.654610604
0.089775
0
0
0
0
Portfolio 9
10.298273372
0.108219
A10
Std Error
T-Stat
Signif
A1,A3,A5,A7,A9 represent sensitivity coefficients
A2,A4,A6,A8,A10 represents risk premiums
Sum of Squared
Residuals
R-squared
Slide 21
Coefficient estimates for the system of
equations in interval 2005 – 2006
Variable
Coeff
Std Error
T-Stat
IN-SAMPLE statistics in interval
2005 – 2006
Signif
A1
-0.000920055
0.000341529
-2.69393
0.00706148
A2
-1.323749029
1.238420607
-1.0689
0.28511428
A3
-0.00042569
0.000984323
-0.43247
0.66539995
A4
0
0
0
0
A5
0.000380733
0.00093773
0.40602
0.68473097
A6
0
0
0
0
A7
0.000179902
0.000974496
0.18461
0.8535346
A8
0
0
0
0
A9
-0.001476967
0.000953245
-1.54941
0.12128345
0
0
0
0
A10
A1,A3,A5,A7,A9 represent sensitivity coefficients
A2,A4,A6,A8,A10 represents risk premiums
Sum of Squared
Residuals
R-squared
Portfolio 1
0.2571438926
0.023201
Portfolio 2
0.2460795643
0.023142
Portfolio 3
4.9523677034
-0.001896
Portfolio 4
5.3289069842
0.001606
Portfolio 5
7.9490393901
0.003667
Portfolio 6
10.340686535
-0.000686
Portfolio 7
5.6034113653
0.002944
Portfolio 8
13.047265316
0.000374
Portfolio 9
13.628142703
0.000343
Slide 22
As can be seen from the IN-SAMPLE estimation results, in interval 2005 – 2006, the
model can not explain the variation in portfolio returns. As a consequence, the
system of non-linear equations is tested without Portfolio 1 and 2, which show
extremely small variations in portfolios returns.
Coefficient estimates for the system of
equations in interval 2005 – 2006
Variable
Coeff
Std Error
T-Stat
IN-SAMPLE statistics in interval
2005 – 2006
Signif
Sum of Squared
Residuals
R-squared
A1
-0.002013038
0.000809396
-2.48709
0.01287935
A2
-4.070064364
2.046949338
-1.98836
0.04677231
Portfolio 1
-
-
A3
-0.027149089
0.002332764
-11.63816
0
Portfolio 2
-
-
A4
0
0
0
0
Portfolio 3
5.6893382042
-0.150990
A5
-0.041252281
0.002222344
-18.56251
0
Portfolio 4
5.9391562210
-0.112727
A6
0
0
0
0
Portfolio 5
6.6432311301
0.167337
A7
-0.001289061
0.002309475
-0.55816
0.57673376
Portfolio 6
7.6205001613
0.262551
A8
0
0
0
0
Portfolio 7
4.3690014784
0.222591
A9
0.005912239
0.002259113
2.61706
0.00886902
Portfolio 8
8.2155321847
0.370561
0
0
0
0
Portfolio 9
8.2095642004
0.397809
A10
A1,A3,A5,A7,A9 represent sensitivity coefficients
A2,A4,A6,A8,A10 represents risk premiums
Slide 23
The prediction results in interval 2001 – 2003 (20 observations)
k
Rˆ it   bij [ jt  F jt ]  -0.003233423 * [-3.613009384  F1t ] - 0.004907741* [0  F2t ] 
j 1
- 0.012755416 * [0  F3t ]  0.000228760 * [0  F4t ] - 0.005935964 * [0  F5t ]
Root Mean
Squared Error
Statistic
Success ratio
sign
prediction
Portfolio 1
0.023711444
0.35
Portfolio 2
0.053907075
0.35
Portfolio 3
0.025671769
0.3
Portfolio 4
0.052131384
0.35
Portfolio 5
0.075277526
0.4
Portfolio 6
0.090334582
0.7
Portfolio 7
0.067906465
0.7
Portfolio 8
0.083651886
0.65
Portfolio 9
0.084227419
0.55
Slide 24
The prediction results in interval 2005 – 2006 (20 observations)
OUT-OF-SAMPLE statistics in interval
2005 – 2006
in the absence of Portfolio 1 and 2
OUT-OF-SAMPLE statistics in
interval 2005 - 2006
Root Mean
Squared
Error
Statistic
Success ratio
sign
prediction
Root Mean
Squared Error
Statistic
Success ratio
sign
prediction
Portfolio 1
-
-
Portfolio 1
0.034453639
0.6
Portfolio 2
-
-
Portfolio 2
0.034453639
0.6
Portfolio 3
0.067947663
0.4
Portfolio 3
0.058890502
0.45
Portfolio 4
0.07441322
0.55
Portfolio 4
0.0729732
0.55
Portfolio 5
0.146170625
0.65
Portfolio 5
0.143621321
0.35
Portfolio 6
0.127809348
0.75
Portfolio 6
0.147609619
0.55
Portfolio 7
0.152325492
0.6
Portfolio 7
0.155310445
0.9
Portfolio 8
0.102203122
0.75
Portfolio 8
0.117576968
0.65
Portfolio 9
0.076814254
0.7
Portfolio 9
0.098318052
0.5
Slide 25
Feedforward Artificial Neural Networks
Part I
•Network architecture. The feedforward neural network
has a hidden layer, fully connected. The number of neurons
at the input layer is 5 (corresponding to the 5 principal
components) and a neuron on the output layer
(representing the portfolio return). The number of neurons
in the hidden layer is 15, a number set as a result of many
tests with different number of neurons on the hidden layer.
•Gradient descent terms. The BFGS (Boyden-FletcherGoldfarb-Shanno) algorithm approximates
H n1 at step n based on the change in gradient J n  J n1
, relative to the change in the parameters n  n1
. The epoch is kept always equal to one, meaning that the
weights are updated after each presentation of a training
pattern. This is the “on-line” or “stochastic” version of the
BFGS algorithm, as opposed to the “batch” version where
the weights are updated after the gradients have
accumulated over the whole training set.
•Transfer function, cost function and initial conditions.
The transfer function is the logsigmoid function. The cost
function used is the sum of squared differences between
actual and estimated values. The initial conditions do not
change through the training and prediction process.
Slide 26
Feedforward Artificial Neural Networks
Part II
Mathematically the feedforward neural network can be described by the following
equations:
I

l



 k ,i  Ci ,t
k ,0
 k ,t
i 1

1

 Lk ,t 
1  exp( lk ,t )

K

Y



 k Lk ,t
0
 t
k 1
where we have I=5 input variables and K=15 neurons in the hidden layer.
Slide 27
Elman Recurrent Artificial Neural Networks
Elman Recurrent Neural Network
allow neurons in the hidden layer to
depend not only on independent
variables Ck at moment t, but also on
their own lags. A “memory” effect is
created in the neuron structure,
similar to the moving average (MA)
process in time-series analysis.
The mathematical representation of
the Elman Recurrent Network can be
illustrated as follows:
I
K

l





C

 k ,i i ,t  k  lk ,t 1
k ,0
 k ,t
i 1
k 1

1

Lk ,t 

1  exp( lk ,t )

K

Y



 k Lk ,t
t
0

k 1
Slide 28
IN-SAMPLE estimation results for the interval
2001 – 2003
Two-step cross sectional
regression procedure
Sum of
Squared
Residuals
R-squared
One-step system of
non-linear seemingly
unrelated equations
Sum of
Squared
Residuals
R-squared
Feedforward Artificial
Neural Networks
Sum of
Squared
Residuals
R-squared
Elman Recurrent
Neural Networks
Sum of
Squared
Residuals
R-squared
Portfolio 1
3.992550
0.001666
4.3905947134
-0.008163
3.7488
0.044070
3.757174
0.065192
Portfolio 2
2.261520
0.011703
2.3835046056
0.013809
2.16395
0.038402
2.04291
0.091478
Portfolio 3
3.149652
0.006129
3.3469906452
-0.010630
3.0061
0.038804
2.5355
0.18261
Portfolio 4
7.197417
0.035787
7.3023258879
0.050090
6.9744
0.060692
6.520185
0.121192
Portfolio 5
8.226738
0.025215
8.5221651357
0.031992
7.9631
0.056421
7.69197
0.095457
Portfolio 6
10.12331
0.070938
10.680690442
0.063338
9.7164
0.099906
9.0618
0.14375
Portfolio 7
7.918961
0.072261
8.1580819435
0.074072
7.4407
0.11369
7.28092
0.132126
Portfolio 8
13.29215
0.207786
15.654610604
0.089775
12.617
0.24891
12.158
0.26725
Portfolio 9
8.244830
0.264719
10.298273372
0.108219
7.8103
0.30498
7.2766
0.3405
Slide 29
OUT-OF-SAMPLE prediction results for the interval
2001 – 2003
Two-step cross sectional
regression procedure
Root Mean
Squared
Error
Statistic
Success
ratio sign
prediction
One-step system of
non-linear seemingly
unrelated equations
Root Mean
Squared
Error
Statistic
Feedforward Artificial
Neural Networks
Success
ratio sign
prediction
Root Mean
Squared
Error
Statistic
Success
ratio sign
prediction
Elman Artificial
Neural Networks
Root Mean
Squared
Error
Statistic
Success
ratio sign
prediction
Portfolio 1
0.023481364
0.4
0.023711444
0.35
0.02219102
0.35
0.02201845
0.35
Portfolio 2
0.055578377
0.25
0.053907075
0.35
0.05406676
0.4
0.05258817
0.4
Portfolio 3
0.022071454
0.35
0.025671769
0.3
0.02051426
0.3
0.01715430
0.3
Portfolio 4
0.051902829
0.4
0.052131384
0.35
0.04964692
0.35
0.04641826
0.65
Portfolio 5
0.074037862
0.3
0.075277526
0.4
0.0736478
0.25
0.06945371
0.5
Portfolio 6
0.086823189
0.7
0.090334582
0.7
0.08749857
0.7
0.08612201
0.75
Portfolio 7
0.062637405
0.65
0.067906465
0.7
0.0572132
0.65
0.060679652
0.65
Portfolio 8
0.080724635
0.65
0.083651886
0.65
0.08333967
0.7
0.080740324
0.75
Portfolio 9
0.074483077
0.55
0.084227419
0.55
0.0741114
0.6
0.069364256
0.65
Slide 30
IN-SAMPLE estimation results for the interval
2005 – 2006
Two-step cross
sectional regression
procedure
One-step system of
non-linear seemingly
unrelated equations
Sum of
Squared
Residuals
R-squared
Sum of
Squared
Residuals
Portfolio 1
0.012044
0.002491
0.2571438926
0.023201
Portfolio 2
3.36E-61
-
0.2460795643
Portfolio 3
4.410894
0.034472
Portfolio 4
4.984877
Portfolio 5
One-step system of
non-linear seemingly
unrelated equations (*)
Elman Artificial
Neural Networks
R-squared
Sum of
Squared
Residuals
R-squared
-
-
0.01134
0.003317
0.011336
0.0047311
0.023142
-
-
-
-
-
-
4.9523677034
-0.001896
5.6893382042
-0.150990
4.3565
0.038584
4.02997497
0.093225
0.034641
5.3289069842
0.001606
5.9391562210
-0.112727
4.764977
0.062526
4.1105
0.14498
6.361233
0.179266
7.9490393901
0.003667
6.6432311301
0.167337
6.1208
0.19856
5.8633
0.21055
Portfolio 6
7.153975
0.295882
10.340686535
-0.000686
7.6205001613
0.262551
6.83826
0.29476
6.3947
0.34006
Portfolio 7
4.081556
0.237497
5.6034113653
0.002944
4.3690014784
0.222591
3.7551
0.26623
3.61821376
0.302257
Portfolio 8
6.245082
0.506626
13.047265316
0.000374
8.2155321847
0.370561
5.9631
0.55452
5.606
0.57002
Portfolio 9
5.487160
0.590297
13.628142703
0.000343
8.2095642004
0.397809
2.93941
0.76771
4.82396474
0.633803
R-squared
Sum of
Squared
Residuals
Feedforward
Artificial Neural
Networks
Sum of
Squared
Residuals
(*)Estimation results in interval 2005 – 2006 using One-step System of Non-linear Seemingly Unrelated Equations
in the absence of Portfolios 1 and 2
R-squared
Slide 31
OUT-OF-SAMPLE prediction results for the
interval 2005 – 2006
Two-step cross
sectional regression
procedure
One-step system of
non-linear
seemingly unrelated
equations
One-step system of
non-linear seemingly
unrelated equations
(*)
Root Mean
Squared
Error
Statistic
Root Mean
Squared
Error
Statistic
Root Mean
Squared
Error
Statistic
Success
ratio sign
prediction
Success
ratio
sign
predictio
n
Feedforward
Artificial Neural
Networks
Success
ratio sign
prediction
Root Mean
Squared
Error
Statistic
Elman Artificial
Neural Networks
Success
ratio
sign
predictio
n
Root Mean
Squared
Error
Statistic
Success
ratio sign
prediction
Portfolio 1
0.000294522
0
0.034453639
0.6
-
-
0
0
0.00077611
0
Portfolio 2
1.24522E-32
0
0.034453639
0.6
-
-
-
-
-
-
Portfolio 3
0.055832265
0.15
0.058890502
0.45
0.067947663
0.4
0.05757300
0.2
0.05274362
0.3
Portfolio 4
0.079836677
0.5
0.0729732
0.55
0.07441322
0.55
0.08316120
0.45
0.08024026
0.55
Portfolio 5
0.136755982
0.6
0.143621321
0.35
0.146170625
0.65
0.13940947
0.65
0.13969073
0.5
Portfolio 6
0.113841636
0.35
0.147609619
0.55
0.127809348
0.75
0.11709767
0.65
0.11818629
0.55
Portfolio 7
0.145922651
0.45
0.155310445
0.9
0.152325492
0.6
0.14058805
0.6
0.14406041
0.65
Portfolio 8
0.079475156
0.7
0.117576968
0.65
0.102203122
0.75
0.07137226
0.7
0.08297590
0.7
Portfolio 9
0.06808954
0.7
0.098318052
0.5
0.076814254
0.7
0.05801578
0.8
0.07260961
0.8
(*) Prediction results in interval 2005 – 2006 using One-step System of Non-linear Seemingly Unrelated Equations
in the absence of Portfolios 1 and 2