estimating and predicting stock returns using artificial neural networks dissertation paper...
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Estimating and Predicting Estimating and Predicting Stock Returns Stock Returns
Using Artificial Neural Using Artificial Neural NetworksNetworks
Dissertation PaperDissertation Paper
BUCHAREST ACADEMY OF ECONOMIC STUDIESDOCTORAL SCHOOL OF FINANCE AND BANKING-DOFIN
Supervisor:Prof. Univ. Dr. Moisa Altar
MSc Student:
Catalin-Marius Untea
IntroductionIntroductionThis 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 2Slide 2
Theoretical background:Theoretical background: The Arbitrage Pricing Theory The Arbitrage Pricing Theory
(APT)(APT)
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.
Part IPart I
Slide 3Slide 3
The APT model can be described by two equations:the first equation expresses the stock returns based on the set of factors
it
k
jjtijitit FbRER
1
][ , t=1,...,T i=1,...,N j=1,...,k
iR represents the return on share i;
][ iRE where represents the expected return for share i;
jF represents the influence of factor j on stock return i;
ijb
the second equation is for the equilibrium expected return and expresses the no arbitrage opportunity:
k
jjtijtit bRE
10][
where 0 represents the free-risk rate return;
j represents the risk premiums corresponding to risk factor j;
represents the sensitivity of the return on asset i to the fluctuations of factor j;
Theoretical background:Theoretical background: The Arbitrage Pricing Theory The Arbitrage Pricing Theory
(APT)(APT)Part IIPart II
Slide 4Slide 4
Two-step cross-sectional regression Two-step cross-sectional regression procedure procedure
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:
itjtj
ijiit eFR
where itR represents portfolio return i;
jtF represents principal component j;
ij represents sensitivity coefficient for portfolio return i at factor fluctuations j.
Part IPart I
Slide 5Slide 5
Dependent Variable: RAND_PORT1Dependent Variable: RAND_PORT1
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 16:25Date: 06/24/07 Time: 16:25
Sample: 252 966Sample: 252 966
Included observations: 715Included observations: 715
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterationsConvergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. Std. ErrErroror
t-Statistict-Statistic Prob. Prob.
CC 0.0103700.010370 0.003510.0035133
2.9516642.951664 0.00330.0033
COMP_PRIN1COMP_PRIN1 0.0010690.001069 0.003130.0031399
0.3406120.340612 0.73350.7335
COMP_PRIN2COMP_PRIN2 0.0006210.000621 0.004110.0041111
0.1511060.151106 0.87990.8799
COMP_PRIN3COMP_PRIN3 -0.000596-0.000596 0.002940.0029433
-0.202683-0.202683 0.83940.8394
COMP_PRIN4COMP_PRIN4 0.0018720.001872 0.005890.0058999
0.3173850.317385 0.75100.7510
COMP_PRIN5COMP_PRIN5 -0.003121-0.003121 0.004430.0044333
-0.703955-0.703955 0.48170.4817
R-squaredR-squared 0.0015080.001508 Mean dependent varMean dependent var 0.010340.0103400
Adjusted R-Adjusted R-squaredsquared
-0.005533-0.005533 S.D. dependent varS.D. dependent var 0.074860.0748644
S.E. of regressionS.E. of regression 0.0750710.075071 Sum squared residSum squared resid 3.995633.9956300
Durbin-Watson Durbin-Watson statstat
1.7945841.794584 J-statisticJ-statistic 7.31E-337.31E-33
Dependent Variable: RAND_PORT1Dependent Variable: RAND_PORT1
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 16:27Date: 06/24/07 Time: 16:27
Sample: 1240 1714Sample: 1240 1714
Included observations: 475Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterationsConvergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. ErrorStd. Error t-Statistict-Statistic Prob. Prob.
CC 0.0002250.000225 0.0002220.000222 1.0121801.012180 0.31200.3120
COMP_PRIN1COMP_PRIN1 6.42E-066.42E-06 7.84E-067.84E-06 0.8193880.819388 0.41300.4130
COMP_PRIN2COMP_PRIN2 0.0001170.000117 0.0001190.000119 0.9878490.987849 0.32370.3237
COMP_PRIN3COMP_PRIN3 0.0001740.000174 0.0001760.000176 0.9891390.989139 0.32310.3231
COMP_PRIN4COMP_PRIN4 1.18E-061.18E-06 1.26E-051.26E-05 0.0936440.093644 0.92540.9254
COMP_PRIN5COMP_PRIN5 -2.48E-05-2.48E-05 2.74E-052.74E-05 -0.907004-0.907004 0.36490.3649
R-squaredR-squared 0.0024910.002491 Mean dependent varMean dependent var 0.000220.0002244
Adjusted R-Adjusted R-squaredsquared
-0.008143-0.008143 S.D. dependent varS.D. dependent var 0.004880.0048877
S.E. of regressionS.E. of regression 0.0049060.004906 Sum squared residSum squared resid 0.011290.0112900
Durbin-Watson Durbin-Watson statstat
2.0069552.006955 J-statisticJ-statistic 2.40E-282.40E-28
Estimation results for portfolio 1
2001 - 2003 2005 - 2006
Slide 6Slide 6
Dependent Variable: RAND_PORT2Dependent Variable: RAND_PORT2
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 16:22Date: 06/24/07 Time: 16:22
Sample: 252 966Sample: 252 966
Included observations: 715Included observations: 715
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterationsConvergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. ErrorStd. Error t-Statistict-Statistic Prob. Prob.
CC 0.0087330.008733 0.0030820.003082 2.8335672.833567 0.00470.0047
COMP_PRIN1COMP_PRIN1 -0.002338-0.002338 0.0039640.003964 -0.589960-0.589960 0.55540.5554
COMP_PRIN2COMP_PRIN2 -0.004659-0.004659 0.0029040.002904 -1.604640-1.604640 0.10900.1090
COMP_PRIN3COMP_PRIN3 -0.002547-0.002547 0.0028320.002832 -0.899318-0.899318 0.36880.3688
COMP_PRIN4COMP_PRIN4 -0.001160-0.001160 0.0044850.004485 -0.258574-0.258574 0.79600.7960
COMP_PRIN5COMP_PRIN5 -0.002081-0.002081 0.0024950.002495 -0.834141-0.834141 0.40450.4045
R-squaredR-squared 0.0117440.011744 Mean dependent varMean dependent var 0.008560.0085677
Adjusted R-Adjusted R-squaredsquared
0.0047740.004774 S.D. dependent varS.D. dependent var 0.056580.0565877
S.E. of regressionS.E. of regression 0.0564510.056451 Sum squared residSum squared resid 2.259402.2594044
Durbin-Watson Durbin-Watson statstat
1.4271651.427165 J-statisticJ-statistic 1.55E-311.55E-31
Dependent Variable: RAND_PORT2Dependent Variable: RAND_PORT2
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 16:23Date: 06/24/07 Time: 16:23
Sample: 1240 1714Sample: 1240 1714
Included observations: 475Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterationsConvergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. Std. ErrErroror
t-Statistict-Statistic Prob. Prob.
CC -2.12E-33-2.12E-33 6.68E-196.68E-19 -3.17E-15-3.17E-15 1.00001.0000
COMP_PRIN1COMP_PRIN1 3.08E-333.08E-33 6.03E-196.03E-19 5.11E-155.11E-15 1.00001.0000
COMP_PRIN2COMP_PRIN2 -1.00E-32-1.00E-32 1.62E-181.62E-18 -6.20E-15-6.20E-15 1.00001.0000
COMP_PRIN3COMP_PRIN3 4.62E-334.62E-33 1.02E-181.02E-18 4.53E-154.53E-15 1.00001.0000
COMP_PRIN4COMP_PRIN4 1.39E-321.39E-32 2.64E-182.64E-18 5.26E-155.26E-15 1.00001.0000
COMP_PRIN5COMP_PRIN5 1.16E-331.16E-33 1.79E-181.79E-18 6.45E-166.45E-16 1.00001.0000
Mean dependent Mean dependent varvar
0.0000000.000000 S.D. dependent varS.D. dependent var 0.000000.0000000
S.E. of regressionS.E. of regression 1.74E-321.74E-32 Sum squared residSum squared resid 1.41E-611.41E-61
Durbin-Watson Durbin-Watson statstat
1.7151211.715121 J-statisticJ-statistic 0.049420.0494200
Estimation results for portfolio 2
2001 - 2003 2005 - 2006
Slide 7Slide 7
Dependent Variable: RAND_PORT3Dependent Variable: RAND_PORT3
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 16:18Date: 06/24/07 Time: 16:18
Sample: 252 966Sample: 252 966
Included observations: 715Included observations: 715
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterationsConvergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. ErrorStd. Error t-Statistict-Statistic Prob. Prob.
CC 0.0067100.006710 0.0024600.002460 2.7279742.727974 0.00650.0065
COMP_PRIN1COMP_PRIN1 -0.000543-0.000543 0.0029280.002928 -0.185520-0.185520 0.85290.8529
COMP_PRIN2COMP_PRIN2 0.0033820.003382 0.0036590.003659 0.9244280.924428 0.35560.3556
COMP_PRIN3COMP_PRIN3 -0.004555-0.004555 0.0035140.003514 -1.296118-1.296118 0.19540.1954
COMP_PRIN4COMP_PRIN4 0.0020280.002028 0.0050670.005067 0.4002080.400208 0.68910.6891
COMP_PRIN5COMP_PRIN5 -0.002333-0.002333 0.0041690.004169 -0.559522-0.559522 0.57600.5760
R-squaredR-squared 0.0060250.006025 Mean dependent varMean dependent var 0.006930.0069377
Adjusted R-Adjusted R-squaredsquared
-0.000985-0.000985 S.D. dependent varS.D. dependent var 0.066360.0663688
S.E. of regressionS.E. of regression 0.0664010.066401 Sum squared residSum squared resid 3.126023.1260222
Durbin-Watson Durbin-Watson statstat
1.9757621.975762 J-statisticJ-statistic 1.10E-311.10E-31
Dependent Variable: RAND_PORT3Dependent Variable: RAND_PORT3
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 16:20Date: 06/24/07 Time: 16:20
Sample: 1240 1714Sample: 1240 1714
Included observations: 475Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterationsConvergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. ErrorStd. Error t-Statistict-Statistic Prob. Prob.
CC 0.0028040.002804 0.0041180.004118 0.6808920.680892 0.49630.4963
COMP_PRIN1COMP_PRIN1 0.0016720.001672 0.0010250.001025 1.6314851.631485 0.10350.1035
COMP_PRIN2COMP_PRIN2 -0.011449-0.011449 0.0038520.003852 -2.971858-2.971858 0.00310.0031
COMP_PRIN3COMP_PRIN3 -0.008726-0.008726 0.0033180.003318 -2.629798-2.629798 0.00880.0088
COMP_PRIN4COMP_PRIN4 0.0054820.005482 0.0035420.003542 1.5476951.547695 0.12240.1224
COMP_PRIN5COMP_PRIN5 0.0061240.006124 0.0049810.004981 1.2294811.229481 0.21950.2195
R-squaredR-squared 0.0347140.034714 Mean dependent varMean dependent var 0.003430.0034311
Adjusted R-Adjusted R-squaredsquared
0.0244230.024423 S.D. dependent varS.D. dependent var 0.098170.0981788
S.E. of regressionS.E. of regression 0.0969720.096972 Sum squared residSum squared resid 4.410234.4102300
Durbin-Watson Durbin-Watson statstat
2.0668032.066803 J-statisticJ-statistic 5.30E-315.30E-31
Estimation results for portfolio 3
2001 - 2003 2005 - 2006
Slide 8Slide 8
Dependent Variable: RAND_PORT4Dependent Variable: RAND_PORT4
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 16:13Date: 06/24/07 Time: 16:13
Sample: 252 966Sample: 252 966
Included observations: 715Included observations: 715
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef Convergence achieved after: 1 weight matrix, 2 total coef iterationsiterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. Std. ErrErroror
t-Statistict-Statistic Prob. Prob.
CC 0.0138910.013891 0.004260.0042600
3.2612723.261272 0.00120.0012
COMP_PRIN1COMP_PRIN1 -4.93E-05-4.93E-05 0.004570.0045733
-0.010774-0.010774 0.99140.9914
COMP_PRIN2COMP_PRIN2 -0.005801-0.005801 0.004750.0047500
-1.221195-1.221195 0.22240.2224
COMP_PRIN3COMP_PRIN3 -0.017844-0.017844 0.003960.0039644
-4.501326-4.501326 0.00000.0000
COMP_PRIN4COMP_PRIN4 0.0003190.000319 0.005530.0055388
0.0576470.057647 0.95400.9540
COMP_PRIN5COMP_PRIN5 0.0057380.005738 0.006920.0069277
0.8283800.828380 0.40770.4077
R-squaredR-squared 0.0356630.035663 Mean dependent Mean dependent varvar
0.014010.0140111
Adjusted R-Adjusted R-squaredsquared
0.0288630.028863 S.D. dependent varS.D. dependent var 0.102160.1021600
S.E. of regressionS.E. of regression 0.1006750.100675 Sum squared residSum squared resid 7.186017.1860133
Durbin-Watson Durbin-Watson statstat
1.8835551.883555 J-statisticJ-statistic 4.66E-314.66E-31
Dependent Variable: RAND_PORT4Dependent Variable: RAND_PORT4
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 16:15Date: 06/24/07 Time: 16:15
Sample: 1240 1714Sample: 1240 1714
Included observations: 475Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterationsConvergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. ErrorStd. Error t-Statistict-Statistic Prob. Prob.
CC -0.002295-0.002295 0.0047760.004776 -0.480618-0.480618 0.63100.6310
COMP_PRIN1COMP_PRIN1 -0.001392-0.001392 0.0015190.001519 -0.916460-0.916460 0.35990.3599
COMP_PRIN2COMP_PRIN2 -0.012073-0.012073 0.0039250.003925 -3.076084-3.076084 0.00220.0022
COMP_PRIN3COMP_PRIN3 -0.010983-0.010983 0.0039340.003934 -2.791577-2.791577 0.00550.0055
COMP_PRIN4COMP_PRIN4 0.0011900.001190 0.0046290.004629 0.2570300.257030 0.79730.7973
COMP_PRIN5COMP_PRIN5 0.0044510.004451 0.0025420.002542 1.7511891.751189 0.08060.0806
R-squaredR-squared 0.0351290.035129 Mean dependent varMean dependent var --0.00.001018787
33
Adjusted R-Adjusted R-squaredsquared
0.0248430.024843 S.D. dependent varS.D. dependent var 0.104250.1042599
S.E. of regressionS.E. of regression 0.1029560.102956 Sum squared residSum squared resid 4.971374.9713733
Durbin-Watson Durbin-Watson statstat
2.0685892.068589 J-statisticJ-statistic 7.70E-327.70E-32
Estimation results for portfolio 4
2001 - 2003 2005 - 2006
Slide 9Slide 9
Dependent Variable: RAND_PORT5Dependent Variable: RAND_PORT5
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 16:06Date: 06/24/07 Time: 16:06
Sample: 252 966Sample: 252 966
Included observations: 715Included observations: 715
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef Convergence achieved after: 1 weight matrix, 2 total coef iterationsiterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. Std. ErrErroror
t-Statistict-Statistic Prob. Prob.
CC 0.0116290.011629 0.004880.0048888
2.3793642.379364 0.01760.0176
COMP_PRIN1COMP_PRIN1 0.0062900.006290 0.005290.0052955
1.1880061.188006 0.23520.2352
COMP_PRIN2COMP_PRIN2 -0.010741-0.010741 0.005000.0050066
-2.145833-2.145833 0.03220.0322
COMP_PRIN3COMP_PRIN3 -0.012818-0.012818 0.004200.0042055
-3.048595-3.048595 0.00240.0024
COMP_PRIN4COMP_PRIN4 -0.003786-0.003786 0.006710.0067122
-0.564091-0.564091 0.57290.5729
COMP_PRIN5COMP_PRIN5 -0.001225-0.001225 0.005700.0057099
-0.214617-0.214617 0.83010.8301
R-squaredR-squared 0.0252550.025255 Mean dependent Mean dependent varvar
0.011390.0113900
Adjusted R-Adjusted R-squaredsquared
0.0183800.018380 S.D. dependent varS.D. dependent var 0.108750.1087599
S.E. of regressionS.E. of regression 0.1077540.107754 Sum squared residSum squared resid 8.232218.2322188
Durbin-Watson Durbin-Watson statstat
1.6886301.688630 J-statisticJ-statistic 4.69E-304.69E-30
Dependent Variable: RAND_PORT5Dependent Variable: RAND_PORT5
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 16:08Date: 06/24/07 Time: 16:08
Sample: 1240 1714Sample: 1240 1714
Included observations: 475Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterationsConvergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. ErrorStd. Error t-Statistict-Statistic Prob. Prob.
CC 0.0048610.004861 0.0043640.004364 1.1140301.114030 0.26580.2658
COMP_PRIN1COMP_PRIN1 -0.002308-0.002308 0.0014530.001453 -1.588477-1.588477 0.11290.1129
COMP_PRIN2COMP_PRIN2 -0.022491-0.022491 0.0051280.005128 -4.386342-4.386342 0.00000.0000
COMP_PRIN3COMP_PRIN3 -0.040446-0.040446 0.0070500.007050 -5.737089-5.737089 0.00000.0000
COMP_PRIN4COMP_PRIN4 -0.000495-0.000495 0.0045230.004523 -0.109453-0.109453 0.91290.9129
COMP_PRIN5COMP_PRIN5 -0.001721-0.001721 0.0048940.004894 -0.351652-0.351652 0.72530.7253
R-squaredR-squared 0.1791570.179157 Mean dependent varMean dependent var 0.004630.0046300
Adjusted R-Adjusted R-squaredsquared
0.1704060.170406 S.D. dependent varS.D. dependent var 0.127860.1278677
S.E. of regressionS.E. of regression 0.1164640.116464 Sum squared residSum squared resid 6.361406.3614033
Durbin-Watson Durbin-Watson statstat
2.3237142.323714 J-statisticJ-statistic 3.19E-313.19E-31
Estimation results for portfolio 5
2001 - 2003 2005 - 2006
Slide 10Slide 10
Dependent Variable: RAND_PORT6Dependent Variable: RAND_PORT6
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 16:00Date: 06/24/07 Time: 16:00
Sample: 252 966Sample: 252 966
Included observations: 715Included observations: 715
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef Convergence achieved after: 1 weight matrix, 2 total coef iterationsiterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficientCoefficient Std. Std. ErrErroror
t-Statistict-Statistic Prob. Prob.
CC 0.0188450.018845 0.004700.0047099
4.0022434.002243 0.00010.0001
COMP_PRIN1COMP_PRIN1 0.0057470.005747 0.006620.0066211
0.8679550.867955 0.38570.3857
COMP_PRIN2COMP_PRIN2 -0.013967-0.013967 0.006300.0063099
-2.213911-2.213911 0.02720.0272
COMP_PRIN3COMP_PRIN3 -0.028096-0.028096 0.006490.0064922
-4.327917-4.327917 0.00000.0000
COMP_PRIN4COMP_PRIN4 -0.009366-0.009366 0.008080.0080811
-1.159018-1.159018 0.24680.2468
COMP_PRIN5COMP_PRIN5 0.0046710.004671 0.008350.0083555
0.5590610.559061 0.57630.5763
R-squaredR-squared 0.0709400.070940 Mean dependent Mean dependent varvar
0.018970.0189744
Adjusted R-Adjusted R-squaredsquared
0.0643880.064388 S.D. dependent varS.D. dependent var 0.123350.1233544
S.E. of regressionS.E. of regression 0.1193170.119317 Sum squared residSum squared resid 10.093710.093722
Durbin-Watson Durbin-Watson statstat
1.8306131.830613 J-statisticJ-statistic 7.01E-317.01E-31
Dependent Variable: RAND_PORT6Dependent Variable: RAND_PORT6
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 16:04Date: 06/24/07 Time: 16:04
Sample: 1240 1714Sample: 1240 1714
Included observations: 475Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterationsConvergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficientCoefficient Std. ErrorStd. Error t-Statistict-Statistic Prob. Prob.
CC 0.0095140.009514 0.005310.0053111
1.7913331.791333 0.07390.0739
COMP_PRIN1COMP_PRIN1 0.0013400.001340 0.001360.0013699
0.9782500.978250 0.32850.3285
COMP_PRIN2COMP_PRIN2 -0.030951-0.030951 0.004950.0049500
-6.253226-6.253226 0.00000.0000
COMP_PRIN3COMP_PRIN3 -0.061180-0.061180 0.006440.0064433
-9.495897-9.495897 0.00000.0000
COMP_PRIN4COMP_PRIN4 0.0041930.004193 0.004270.0042799
0.9798560.979856 0.32770.3277
COMP_PRIN5COMP_PRIN5 0.0066950.006695 0.003990.0039977
1.6751081.675108 0.09460.0946
R-squaredR-squared 0.2955580.295558 Mean dependent Mean dependent varvar
0.009080.0090866
Adjusted R-Adjusted R-squaredsquared
0.2880480.288048 S.D. dependent varS.D. dependent var 0.146430.1464388
S.E. of regressionS.E. of regression 0.1235600.123560 Sum squared residSum squared resid 7.160257.1602566
Durbin-Watson Durbin-Watson statstat
2.0951712.095171 J-statisticJ-statistic 2.32E-322.32E-32
Estimation results for portfolio 6
2001 - 2003 2005 - 2006
Slide 11Slide 11
Dependent Variable: RAND_PORT7Dependent Variable: RAND_PORT7
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 15:50Date: 06/24/07 Time: 15:50
Sample: 252 966Sample: 252 966
Included observations: 715Included observations: 715
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterationsConvergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. ErrorStd. Error t-Statistict-Statistic Prob. Prob.
CC 0.0159130.015913 0.0042370.004237 3.7559093.755909 0.00020.0002
COMP_PRIN1COMP_PRIN1 -0.001386-0.001386 0.0055200.005520 -0.251057-0.251057 0.80180.8018
COMP_PRIN2COMP_PRIN2 -0.013094-0.013094 0.0047410.004741 -2.761637-2.761637 0.00590.0059
COMP_PRIN3COMP_PRIN3 -0.025616-0.025616 0.0056110.005611 -4.565651-4.565651 0.00000.0000
COMP_PRIN4COMP_PRIN4 -0.002776-0.002776 0.0061310.006131 -0.452871-0.452871 0.65080.6508
COMP_PRIN5COMP_PRIN5 -0.009707-0.009707 0.0061740.006174 -1.572245-1.572245 0.11630.1163
R-squaredR-squared 0.0730870.073087 Mean dependent varMean dependent var 0.015800.0158066
Adjusted R-Adjusted R-squaredsquared
0.0665500.066550 S.D. dependent varS.D. dependent var 0.109230.1092344
S.E. of regressionS.E. of regression 0.1055370.105537 Sum squared residSum squared resid 7.896857.8968511
Durbin-Watson Durbin-Watson statstat
1.8450981.845098 J-statisticJ-statistic 2.88E-302.88E-30
Dependent Variable: RAND_PORT7Dependent Variable: RAND_PORT7
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 15:52Date: 06/24/07 Time: 15:52
Sample: 1240 1714Sample: 1240 1714
Included observations: 475Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterationsConvergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. ErrorStd. Error t-Statistict-Statistic Prob. Prob.
CC 0.0096330.009633 0.0041510.004151 2.3204282.320428 0.02070.0207
COMP_PRIN1COMP_PRIN1 -0.000167-0.000167 0.0010060.001006 -0.165679-0.165679 0.86850.8685
COMP_PRIN2COMP_PRIN2 -0.023277-0.023277 0.0044320.004432 -5.252230-5.252230 0.00000.0000
COMP_PRIN3COMP_PRIN3 -0.038118-0.038118 0.0053670.005367 -7.102547-7.102547 0.00000.0000
COMP_PRIN4COMP_PRIN4 0.0016510.001651 0.0032840.003284 0.5027870.502787 0.61530.6153
COMP_PRIN5COMP_PRIN5 -3.72E-05-3.72E-05 0.0036820.003682 -0.010109-0.010109 0.99190.9919
R-squaredR-squared 0.2383830.238383 Mean dependent varMean dependent var 0.009630.0096300
Adjusted R-Adjusted R-squaredsquared
0.2302640.230264 S.D. dependent varS.D. dependent var 0.106020.1060299
S.E. of regressionS.E. of regression 0.0930240.093024 Sum squared residSum squared resid 4.058464.0584600
Durbin-Watson Durbin-Watson statstat
2.0966402.096640 J-statisticJ-statistic 3.21E-313.21E-31
Estimation results for portfolio 7
2001 - 2003 2005 - 2006
Slide 12Slide 12
Dependent Variable: RAND_PORT8Dependent Variable: RAND_PORT8
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 15:19Date: 06/24/07 Time: 15:19
Sample: 252 966Sample: 252 966
Included observations: 715Included observations: 715
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef Convergence achieved after: 1 weight matrix, 2 total coef iterationsiterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. Std. ErrErroror
t-Statistict-Statistic Prob. Prob.
CC 0.0222620.022262 0.005420.0054299
4.1004484.100448 0.00000.0000
COMP_PRIN1COMP_PRIN1 -0.001124-0.001124 0.008180.0081800
-0.137423-0.137423 0.89070.8907
COMP_PRIN2COMP_PRIN2 -0.028633-0.028633 0.007730.0077322
-3.703067-3.703067 0.00020.0002
COMP_PRIN3COMP_PRIN3 -0.061396-0.061396 0.007950.0079522
-7.720936-7.720936 0.00000.0000
COMP_PRIN4COMP_PRIN4 -0.012017-0.012017 0.010590.0105988
-1.133904-1.133904 0.25720.2572
COMP_PRIN5COMP_PRIN5 0.0015340.001534 0.008000.0080022
0.1917270.191727 0.84800.8480
R-squaredR-squared 0.2076510.207651 Mean dependent Mean dependent varvar
0.022440.0224477
Adjusted R-Adjusted R-squaredsquared
0.2020630.202063 S.D. dependent varS.D. dependent var 0.153200.1532077
S.E. of regressionS.E. of regression 0.1368560.136856 Sum squared residSum squared resid 13.279213.279211
Durbin-Watson Durbin-Watson statstat
1.8644001.864400 J-statisticJ-statistic 3.95E-323.95E-32
Dependent Variable: RAND_PORT8Dependent Variable: RAND_PORT8
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 15:20Date: 06/24/07 Time: 15:20
Sample: 1240 1714Sample: 1240 1714
Included observations: 475Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterationsConvergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. ErrorStd. Error t-Statistict-Statistic Prob. Prob.
CC 0.0216350.021635 0.0053450.005345 4.0479774.047977 0.00010.0001
COMP_PRIN1COMP_PRIN1 -0.000399-0.000399 0.0012270.001227 -0.324855-0.324855 0.74540.7454
COMP_PRIN2COMP_PRIN2 -0.045252-0.045252 0.0048960.004896 -9.242509-9.242509 0.00000.0000
COMP_PRIN3COMP_PRIN3 -0.089708-0.089708 0.0073880.007388 -12.14312-12.14312 0.00000.0000
COMP_PRIN4COMP_PRIN4 -0.004756-0.004756 0.0038320.003832 -1.241197-1.241197 0.21520.2152
COMP_PRIN5COMP_PRIN5 0.0094630.009463 0.0039240.003924 2.4116392.411639 0.01630.0163
R-squaredR-squared 0.5065220.506522 Mean dependent varMean dependent var 0.020560.0205677
Adjusted R-Adjusted R-squaredsquared
0.5012610.501261 S.D. dependent varS.D. dependent var 0.163420.1634233
S.E. of regressionS.E. of regression 0.1154120.115412 Sum squared residSum squared resid 6.247046.2470433
Durbin-Watson Durbin-Watson statstat
2.0232082.023208 J-statisticJ-statistic 2.08E-312.08E-31
Estimation results for portfolio 8
2001 - 2003 2005 - 2006
Slide 13Slide 13
Dependent Variable: RAND_PORT9Dependent Variable: RAND_PORT9
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 15:18Date: 06/24/07 Time: 15:18
Sample: 252 966Sample: 252 966
Included observations: 715Included observations: 715
Kernel: Bartlett, Bandwidth: Fixed (6), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (6), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterationsConvergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. ErrorStd. Error t-Statistict-Statistic Prob. Prob.
CC 0.0126720.012672 0.0043240.004324 2.9307342.930734 0.00350.0035
COMP_PRIN1COMP_PRIN1 0.0090910.009091 0.0061600.006160 1.4759421.475942 0.14040.1404
COMP_PRIN2COMP_PRIN2 -0.028994-0.028994 0.0055710.005571 -5.204247-5.204247 0.00000.0000
COMP_PRIN3COMP_PRIN3 -0.055857-0.055857 0.0065060.006506 -8.585827-8.585827 0.00000.0000
COMP_PRIN4COMP_PRIN4 -0.010915-0.010915 0.0070770.007077 -1.542251-1.542251 0.12350.1235
COMP_PRIN5COMP_PRIN5 -0.010612-0.010612 0.0067690.006769 -1.567750-1.567750 0.11740.1174
R-squaredR-squared 0.2656820.265682 Mean dependent varMean dependent var 0.012540.0125400
Adjusted R-Adjusted R-squaredsquared
0.2605030.260503 S.D. dependent varS.D. dependent var 0.125230.1252311
S.E. of regressionS.E. of regression 0.1076910.107691 Sum squared residSum squared resid 8.222598.2225944
Durbin-Watson Durbin-Watson statstat
1.8746111.874611 J-statisticJ-statistic 2.93E-312.93E-31
Dependent Variable: RAND_PORT9Dependent Variable: RAND_PORT9
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 15:16Date: 06/24/07 Time: 15:16
Sample: 1240 1714Sample: 1240 1714
Included observations: 475Included observations: 475
Kernel: Bartlett, Bandwidth: Fixed (5), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (5), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterationsConvergence achieved after: 1 weight matrix, 2 total coef iterations
Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3
COMP_PRIN4 COMP_PRIN5COMP_PRIN4 COMP_PRIN5
VariableVariable CoefficienCoefficientt
Std. ErrorStd. Error t-Statistict-Statistic Prob. Prob.
CC 0.0126200.012620 0.0044260.004426 2.8515112.851511 0.00450.0045
COMP_PRIN1COMP_PRIN1 -0.006055-0.006055 0.0059330.005933 -1.020475-1.020475 0.30800.3080
COMP_PRIN2COMP_PRIN2 -0.057778-0.057778 0.0083770.008377 -6.897398-6.897398 0.00000.0000
COMP_PRIN3COMP_PRIN3 -0.090673-0.090673 0.0056440.005644 -16.06432-16.06432 0.00000.0000
COMP_PRIN4COMP_PRIN4 -0.018203-0.018203 0.0155710.015571 -1.169005-1.169005 0.24300.2430
COMP_PRIN5COMP_PRIN5 0.0245970.024597 0.0165010.016501 1.4906651.490665 0.13670.1367
R-squaredR-squared 0.5902840.590284 Mean dependent varMean dependent var 0.011880.0118811
Adjusted R-Adjusted R-squaredsquared
0.5859160.585916 S.D. dependent varS.D. dependent var 0.168000.1680033
S.E. of regressionS.E. of regression 0.1081090.108109 Sum squared residSum squared resid 5.481455.4814533
Durbin-Watson Durbin-Watson statstat
2.0891962.089196 J-statisticJ-statistic 6.06E-316.06E-31
Estimation results for portfolio 9
2001 - 2003 2005 - 2006
Slide 14Slide 14
Two-step cross-sectional Two-step cross-sectional regression procedure regression procedure
Part IIPart 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 cross-section regression, represents an estimation for the risk premium associated to the exposure to unexpected variation in one of the factors.
u
returnportfoliomean
PRINCOMPPRINCOMP
PRINCOMPPRINCOMPPRINCOMPPRINCOMPPRINCOMPPRINCOMP
PRINCOMPPRINCOMP
5_5_
4_4_3_3_2_2_
1_1_1
ˆˆ
ˆˆˆˆˆˆ
ˆˆˆ__
Slide 15Slide 15
Dependent Variable: MEDII_PORT_01_03Dependent Variable: MEDII_PORT_01_03
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 18:17Date: 06/24/07 Time: 18:17
Sample: 1 9Sample: 1 9
Included observations: 9Included observations: 9
Kernel: Bartlett, Bandwidth: Fixed (2), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (2), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef iterationsConvergence 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_01_03 BETA_COMP_PRIN2_01_
03 BETA_COMP_PRIN3_01_03 BETA_COMP_PRIN4_01_0303 BETA_COMP_PRIN3_01_03 BETA_COMP_PRIN4_01_03
BETA_COMP_PRIN5_01_03BETA_COMP_PRIN5_01_03
VariableVariable CoefficientCoefficient Std. ErrorStd. Error t-Statistict-Statistic Prob. Prob.
CC 0.0105910.010591 0.0017420.001742 6.0811216.081121 0.00890.0089
BETA_COMP_PRINBETA_COMP_PRIN1_01_031_01_03
-0.359591-0.359591 0.2491040.249104 --1.41.4
4354353737
0.24460.2446
BETA_COMP_PRINBETA_COMP_PRIN2_01_032_01_03
-0.086729-0.086729 0.2276780.227678 --0.30.3
8098092626
0.72860.7286
BETA_COMP_PRINBETA_COMP_PRIN3_01_033_01_03
-0.064155-0.064155 0.0983610.098361 --0.60.6
5225223737
0.56080.5608
BETA_COMP_PRINBETA_COMP_PRIN4_01_034_01_03
-0.414972-0.414972 0.3956120.395612 --1.01.0
4894893636
0.37130.3713
BETA_COMP_PRINBETA_COMP_PRIN5_01_035_01_03
0.3152640.315264 0.1374190.137419 2.2941872.294187 0.10550.1055
R-squaredR-squared 0.7832100.783210 Mean dependent Mean dependent varvar
0.013440.0134466
Adjusted R-Adjusted R-squaredsquared
0.4218930.421893 S.D. dependent varS.D. dependent var 0.004970.0049799
S.E. of regressionS.E. of regression 0.0037860.003786 Sum squared residSum squared resid 4.30E-054.30E-05
Durbin-Watson Durbin-Watson statstat
1.4223911.422391 J-statisticJ-statistic 6.18E-296.18E-29
Dependent Variable: MEDII_PORT_05_06Dependent Variable: MEDII_PORT_05_06
Method: Generalized Method of MomentsMethod: Generalized Method of Moments
Date: 06/24/07 Time: 18:51Date: 06/24/07 Time: 18:51
Sample: 1 9Sample: 1 9
Included observations: 9Included observations: 9
Kernel: Bartlett, Bandwidth: Fixed (2), No prewhiteningKernel: Bartlett, Bandwidth: Fixed (2), No prewhitening
Simultaneous weighting matrix & coefficient iterationSimultaneous weighting matrix & coefficient iteration
Convergence achieved after: 1 weight matrix, 2 total coef Convergence achieved after: 1 weight matrix, 2 total coef iterationsiterations
Instrument list: BETA_COMP_PRIN1_05_06 BETA_COMP_PRIN2_05_Instrument list: BETA_COMP_PRIN1_05_06 BETA_COMP_PRIN2_05_
06 BETA_COMP_PRIN3_05_06 BETA_COMP_PRIN4_05_0606 BETA_COMP_PRIN3_05_06 BETA_COMP_PRIN4_05_06
BETA_COMP_PRIN5_05_06BETA_COMP_PRIN5_05_06
VariableVariable CoefficieCoefficientnt
Std. Std. ErrErroror
t-Statistict-Statistic Prob. Prob.
CC 0.0003090.000309 0.000180.0001844
1.6740791.674079 0.19270.1927
BETA_COMP_PRINBETA_COMP_PRIN1_05_061_05_06
4.2830854.283085 0.201840.2018477
21.2194521.21945 0.00020.0002
BETA_COMP_PRINBETA_COMP_PRIN2_05_062_05_06
--0.80.8
7917915858
0.071270.0712766
-12.33456-12.33456 0.00110.0011
BETA_COMP_PRINBETA_COMP_PRIN3_05_063_05_06
0.1954350.195435 0.035900.0359022
5.4435655.443565 0.01220.0122
BETA_COMP_PRINBETA_COMP_PRIN4_05_064_05_06
--1.31.3
9179173030
0.093100.0931000
-14.94884-14.94884 0.00060.0006
BETA_COMP_PRINBETA_COMP_PRIN5_05_065_05_06
--0.80.8
3493498989
0.047840.0478477
-17.45122-17.45122 0.00040.0004
R-squaredR-squared 0.9926820.992682 Mean dependent Mean dependent varvar
0.006390.0063977
Adjusted R-Adjusted R-squaredsquared
0.9804850.980485 S.D. dependent varS.D. dependent var 0.007140.0071400
S.E. of regressionS.E. of regression 0.0009970.000997 Sum squared residSum squared resid 2.98E-062.98E-06
Durbin-Watson Durbin-Watson statstat
2.2112692.211269 J-statisticJ-statistic 3.59E-253.59E-25
Cross-section regression 2001 - 2003 2005 - 2006
Slide 16Slide 16
Risk premiums associated with factors Risk premiums associated with factors
CCOMP_PRIN1γ̂COMP_PRIN2γ̂COMP_PRIN3γ̂COMP_PRIN4γ̂COMP_PRIN5γ̂
Interval 2001 – 2003Interval 2001 – 2003 0.0105910.010591 -0.359591-0.359591 -0.086729-0.086729 -0.064155-0.064155 -0.414972-0.414972 0.3152640.315264
Interval 2005 – 2006Interval 2005 – 2006 0.0003090.000309 4.2830854.283085 -0.879158-0.879158 0.1954350.195435 -1.391730-1.391730 -0.834989-0.834989
COMP_PRIN1γ̂ COMP_PRIN2γ̂ COMP_PRIN3γ̂C COMP_PRIN4γ̂ COMP_PRIN5γ̂
Slide 17Slide 17
Predictions using sensitivity coefficients estimated through Two-step cross sectional regression procedure
Root Mean Root Mean Squared Error Squared Error
StatisticStatistic
Success ratio Success ratio sign sign
predictionprediction
Portfolio 1Portfolio 1 0.0228274490.022827449 0.350.35
Portfolio 2Portfolio 2 0.054693490.05469349 0.40.4
Portfolio 3Portfolio 3 0.021521430.02152143 0.30.3
Portfolio 4Portfolio 4 0.0516321610.051632161 0.40.4
Portfolio 5Portfolio 5 0.0754669390.075466939 0.40.4
Portfolio 6Portfolio 6 0.085837620.08583762 0.750.75
Portfolio 7Portfolio 7 0.0627300210.062730021 0.650.65
Portfolio 8Portfolio 8 0.0810786540.081078654 0.70.7
Portfolio 9Portfolio 9 0.0756745440.075674544 0.650.65
Root Mean Root Mean Squared Error Squared Error
StatisticStatistic
Success ratio Success ratio sign sign
predictionprediction
Portfolio 1Portfolio 1 0.000285150.00028515 00
Portfolio 2Portfolio 2 8.81318E-338.81318E-33 00
Portfolio 3Portfolio 3 0.0560572930.056057293 0.20.2
Portfolio 4Portfolio 4 0.080002030.08000203 0.60.6
Portfolio 5Portfolio 5 0.1367080460.136708046 0.60.6
Portfolio 6Portfolio 6 0.1142733760.114273376 0.650.65
Portfolio 7Portfolio 7 0.1453403560.145340356 0.50.5
Portfolio 8Portfolio 8 0.0791471860.079147186 0.70.7
Portfolio 9Portfolio 9 0.0681766120.068176612 0.70.7
2001 - 2003 2005 - 2006
Slide 18Slide 18
One-step system of non-linear One-step system of non-linear seemingly unrelated equationsseemingly 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:
it
k
jjtij
k
jjtijtit FbbR
110
or by passing to the left side the risk-free rate of return, the last equation becomes:
it
k
jjtij
k
jjtijtit FbbR
110
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 19Slide 19
VariableVariable CoeffCoeff Std ErrorStd Error T-StatT-Stat SignifSignif
A1A1 -0.003233423-0.003233423 0.0016933660.001693366 -1.90946-1.90946 0.056202130.05620213
A2A2 -3.613009384-3.613009384 1.941765531.94176553 -1.86068-1.86068 0.062789010.06278901
A3A3 -0.004907741-0.004907741 0.0017100680.001710068 -2.86991-2.86991 0.004105890.00410589
A4A4 00 00 00 00
A5A5 -0.012755416-0.012755416 0.0014263020.001426302 -8.943-8.943 00
A6A6 00 00 00 00
A7A7 0.000228760.00022876 0.0025081740.002508174 0.091210.09121 0.927329060.92732906
A8A8 00 00 00 00
A9A9 -0.005935964-0.005935964 0.0026164180.002616418 -2.26874-2.26874 0.023284350.02328435
A10A10 00 00 00 00
Sum of SquaredSum of SquaredResidualsResiduals
R-squaredR-squared
Portfolio 1Portfolio 1 4.39059471344.3905947134 -0.008163-0.008163
Portfolio 2Portfolio 2 2.38350460562.3835046056 0.0138090.013809
Portfolio 3Portfolio 3 3.34699064523.3469906452 -0.010630-0.010630
Portfolio 4Portfolio 4 7.30232588797.3023258879 0.0500900.050090
Portfolio 5Portfolio 5 8.52216513578.5221651357 0.0319920.031992
Portfolio 6Portfolio 6 10.68069044210.680690442 0.0633380.063338
Portfolio 7Portfolio 7 8.15808194358.1580819435 0.0740720.074072
Portfolio 8Portfolio 8 15.65461060415.654610604 0.0897750.089775
Portfolio 9Portfolio 9 10.29827337210.298273372 0.1082190.108219
Coefficient estimates for the system of equations in interval 2001 – 2003
IN-SAMPLE statistics in interval 2001 – 2003
A1,A3,A5,A7,A9 represent sensitivity coefficientsA2,A4,A6,A8,A10 represents risk premiums
Slide 20Slide 20
VariableVariable CoeffCoeff Std ErrorStd Error T-StatT-Stat SignifSignif
A1A1 -0.000920055-0.000920055 0.0003415290.000341529 -2.69393-2.69393 0.007061480.00706148
A2A2 -1.323749029-1.323749029 1.2384206071.238420607 -1.0689-1.0689 0.285114280.28511428
A3A3 -0.00042569-0.00042569 0.0009843230.000984323 -0.43247-0.43247 0.665399950.66539995
A4A4 00 00 00 00
A5A5 0.0003807330.000380733 0.000937730.00093773 0.406020.40602 0.684730970.68473097
A6A6 00 00 00 00
A7A7 0.0001799020.000179902 0.0009744960.000974496 0.184610.18461 0.85353460.8535346
A8A8 00 00 00 00
A9A9 -0.001476967-0.001476967 0.0009532450.000953245 -1.54941-1.54941 0.121283450.12128345
A10A10 00 00 00 00
Sum of Squared Sum of Squared ResidualsResiduals
R-squaredR-squared
Portfolio 1Portfolio 1 0.25714389260.2571438926 0.0232010.023201
Portfolio 2Portfolio 2 0.24607956430.2460795643 0.0231420.023142
Portfolio 3Portfolio 3 4.95236770344.9523677034 -0.001896-0.001896
Portfolio 4Portfolio 4 5.32890698425.3289069842 0.0016060.001606
Portfolio 5Portfolio 5 7.94903939017.9490393901 0.0036670.003667
Portfolio 6Portfolio 6 10.34068653510.340686535 -0.000686-0.000686
Portfolio 7Portfolio 7 5.60341136535.6034113653 0.0029440.002944
Portfolio 8Portfolio 8 13.04726531613.047265316 0.0003740.000374
Portfolio 9Portfolio 9 13.62814270313.628142703 0.0003430.000343
Coefficient estimates for the system of equations in interval 2005 – 2006
IN-SAMPLE statistics in interval 2005 – 2006
A1,A3,A5,A7,A9 represent sensitivity coefficientsA2,A4,A6,A8,A10 represents risk premiums
Slide 21Slide 21
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.
VariableVariable CoeffCoeff Std ErrorStd Error T-StatT-Stat SignifSignif
A1A1 -0.002013038-0.002013038 0.0008093960.000809396 -2.48709-2.48709 0.012879350.01287935
A2A2 -4.070064364-4.070064364 2.0469493382.046949338 -1.98836-1.98836 0.046772310.04677231
A3A3 -0.027149089-0.027149089 0.0023327640.002332764 -11.63816-11.63816 00
A4A4 00 00 00 00
A5A5 -0.041252281-0.041252281 0.0022223440.002222344 -18.56251-18.56251 00
A6A6 00 00 00 00
A7A7 -0.001289061-0.001289061 0.0023094750.002309475 -0.55816-0.55816 0.576733760.57673376
A8A8 00 00 00 00
A9A9 0.0059122390.005912239 0.0022591130.002259113 2.617062.61706 0.008869020.00886902
A10A10 00 00 00 00
Sum of Squared Sum of Squared ResidualsResiduals
R-squaredR-squared
Portfolio 1Portfolio 1 -- --
Portfolio 2Portfolio 2 -- --
Portfolio 3Portfolio 3 5.68933820425.6893382042 -0.150990-0.150990
Portfolio 4Portfolio 4 5.93915622105.9391562210 -0.112727-0.112727
Portfolio 5Portfolio 5 6.64323113016.6432311301 0.1673370.167337
Portfolio 6Portfolio 6 7.62050016137.6205001613 0.2625510.262551
Portfolio 7Portfolio 7 4.36900147844.3690014784 0.2225910.222591
Portfolio 8Portfolio 8 8.21553218478.2155321847 0.3705610.370561
Portfolio 9Portfolio 9 8.20956420048.2095642004 0.3978090.397809
Coefficient estimates for the system of equations in interval 2005 – 2006
IN-SAMPLE statistics in interval 2005 – 2006
A1,A3,A5,A7,A9 represent sensitivity coefficientsA2,A4,A6,A8,A10 represents risk premiums
Slide 22Slide 22
Root Mean Root Mean Squared Squared
Error Error StatisticStatistic
Success Success ratio sign ratio sign predictionprediction
Portfolio 1Portfolio 1 0.0237114440.023711444 0.350.35
Portfolio 2Portfolio 2 0.0539070750.053907075 0.350.35
Portfolio 3Portfolio 3 0.0256717690.025671769 0.30.3
Portfolio 4Portfolio 4 0.0521313840.052131384 0.350.35
Portfolio 5Portfolio 5 0.0752775260.075277526 0.40.4
Portfolio 6Portfolio 6 0.0903345820.090334582 0.70.7
Portfolio 7Portfolio 7 0.0679064650.067906465 0.70.7
Portfolio 8Portfolio 8 0.0836518860.083651886 0.650.65
Portfolio 9Portfolio 9 0.0842274190.084227419 0.550.55
]0[*40.00593596-]0[*00.00022876]0[*60.01275541-
]0[*10.00490774-]84-3.6130093[*23-0.0032334][ˆ
543
211
ttt
ttjt
k
jjtijit
FFF
FFFbR
The prediction results in interval 2001 – 2003 (20 observations)
Slide 23Slide 23
Root Mean Root Mean Squared Squared
Error Error StatisticStatistic
Success Success ratio sign ratio sign predictionprediction
Portfolio 1Portfolio 1 0.0344536390.034453639 0.60.6
Portfolio 2Portfolio 2 0.0344536390.034453639 0.60.6
Portfolio 3Portfolio 3 0.0588905020.058890502 0.450.45
Portfolio 4Portfolio 4 0.07297320.0729732 0.550.55
Portfolio 5Portfolio 5 0.1436213210.143621321 0.350.35
Portfolio 6Portfolio 6 0.1476096190.147609619 0.550.55
Portfolio 7Portfolio 7 0.1553104450.155310445 0.90.9
Portfolio 8Portfolio 8 0.1175769680.117576968 0.650.65
Portfolio 9Portfolio 9 0.0983180520.098318052 0.50.5
OUT-OF-SAMPLE statistics in interval 2005 - 2006
The prediction results in interval 2005 – 2006 (20 observations)
Root Mean Root Mean Squared Squared
Error Error StatisticStatistic
Success Success ratio sign ratio sign predictionprediction
Portfolio 1Portfolio 1 -- --
Portfolio 2Portfolio 2 -- --
Portfolio 3Portfolio 3 0.0679476630.067947663 0.40.4
Portfolio 4Portfolio 4 0.074413220.07441322 0.550.55
Portfolio 5Portfolio 5 0.1461706250.146170625 0.650.65
Portfolio 6Portfolio 6 0.1278093480.127809348 0.750.75
Portfolio 7Portfolio 7 0.1523254920.152325492 0.60.6
Portfolio 8Portfolio 8 0.1022031220.102203122 0.750.75
Portfolio 9Portfolio 9 0.0768142540.076814254 0.70.7
OUT-OF-SAMPLE statistics in interval 2005 – 2006
in the absence of Portfolio 1 and 2
Slide 24Slide 24
Feedforward Artificial Neural Feedforward Artificial Neural NetworksNetworks
•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-Fletcher-Goldfarb-Shanno) algorithm approximates
1nH at step n based on the change in gradient 1 nn JJ
, relative to the change in the parameters 1 nn
. 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.
Part IPart I
Slide 25Slide 25
Feedforward Artificial Neural Feedforward Artificial Neural Networks Networks
Part IIPart II
Mathematically the feedforward neural network can be described by the following equations:
K
ktkkt
tktk
I
itiikktk
LY
lL
Cl
1,0
,,
1,,0,,
)exp(1
1
where we have I=5 input variables and K=15 neurons in the hidden layer.
Slide 26Slide 26
Elman Recurrent Artificial Neural Elman Recurrent Artificial Neural Networks Networks
K
ktkkt
tktk
I
i
K
ktkktiikktk
LY
lL
lCl
1,0
,,
1 11,,,0,,
)exp(1
1
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:
Slide 27Slide 27
IN-SAMPLE estimation results for the interval IN-SAMPLE estimation results for the interval 2001 – 20032001 – 2003
Two-step cross sectionalTwo-step cross sectionalregression procedureregression procedure
One-step system of One-step system of non-linear seeminglynon-linear seemingly unrelated equationsunrelated equations
Feedforward ArtificialFeedforward Artificial Neural NetworksNeural Networks
Elman Recurrent Elman Recurrent Neural NetworksNeural Networks
Sum ofSum of Squared Squared ResidualsResiduals
R-squaredR-squared Sum of Sum of Squared Squared ResidualsResiduals
R-squaredR-squared Sum ofSum of Squared Squared ResidualsResiduals
R-squaredR-squared Sum of Sum of Squared Squared ResidualsResiduals
R-squaredR-squared
Portfolio 1Portfolio 1 3.9925503.992550 0.0016660.001666 4.39059471344.3905947134 -0.008163-0.008163 3.74883.7488 0.0440700.044070 3.7571743.757174 0.0651920.065192
Portfolio 2Portfolio 2 2.2615202.261520 0.0117030.011703 2.38350460562.3835046056 0.0138090.013809 2.163952.16395 0.0384020.038402 2.042912.04291 0.0914780.091478
Portfolio 3Portfolio 3 3.1496523.149652 0.0061290.006129 3.34699064523.3469906452 -0.010630-0.010630 3.00613.0061 0.0388040.038804 2.53552.5355 0.182610.18261
Portfolio 4Portfolio 4 7.1974177.197417 0.0357870.035787 7.30232588797.3023258879 0.0500900.050090 6.97446.9744 0.0606920.060692 6.5201856.520185 0.1211920.121192
Portfolio 5Portfolio 5 8.2267388.226738 0.0252150.025215 8.52216513578.5221651357 0.0319920.031992 7.96317.9631 0.0564210.056421 7.691977.69197 0.0954570.095457
Portfolio 6Portfolio 6 10.1233110.12331 0.0709380.070938 10.68069044210.680690442 0.0633380.063338 9.71649.7164 0.0999060.099906 9.06189.0618 0.143750.14375
Portfolio 7Portfolio 7 7.9189617.918961 0.0722610.072261 8.15808194358.1580819435 0.0740720.074072 7.44077.4407 0.113690.11369 7.280927.28092 0.1321260.132126
Portfolio 8Portfolio 8 13.2921513.29215 0.2077860.207786 15.65461060415.654610604 0.0897750.089775 12.61712.617 0.248910.24891 12.15812.158 0.267250.26725
Portfolio 9Portfolio 9 8.2448308.244830 0.2647190.264719 10.29827337210.298273372 0.1082190.108219 7.81037.8103 0.304980.30498 7.27667.2766 0.34050.3405
Slide 28Slide 28
OUT-OF-SAMPLE prediction results for the OUT-OF-SAMPLE prediction results for the interval 2001 – 2003interval 2001 – 2003
Two-step cross sectionalTwo-step cross sectional regression procedureregression procedure
One-step system of One-step system of non-linear seeminglynon-linear seemingly unrelated equationsunrelated equations
Feedforward ArtificialFeedforward Artificial Neural NetworksNeural Networks
Elman Artificial Elman Artificial Neural NetworksNeural Networks
Root MeanRoot Mean Squared Squared
Error Error StatisticStatistic
SuccessSuccess ratio sign ratio sign predictionprediction
Root MeanRoot Mean Squared Squared
Error Error StatisticStatistic
SuccessSuccess ratio sign ratio sign predictionprediction
Root MeanRoot Mean Squared Squared
Error Error StatisticStatistic
Success Success ratio sign ratio sign predictionprediction
Root Mean Root Mean Squared Squared
Error Error StatisticStatistic
Success Success ratio sign ratio sign predictionprediction
Portfolio 1Portfolio 1 0.0234813640.023481364 0.40.4 0.0237114440.023711444 0.350.35 0.022191020.02219102 0.350.35 0.022018450.02201845 0.350.35
Portfolio 2Portfolio 2 0.0555783770.055578377 0.250.25 0.0539070750.053907075 0.350.35 0.054066760.05406676 0.40.4 0.052588170.05258817 0.40.4
Portfolio 3Portfolio 3 0.0220714540.022071454 0.350.35 0.0256717690.025671769 0.30.3 0.020514260.02051426 0.30.3 0.017154300.01715430 0.30.3
Portfolio 4Portfolio 4 0.0519028290.051902829 0.40.4 0.0521313840.052131384 0.350.35 0.049646920.04964692 0.350.35 0.046418260.04641826 0.650.65
Portfolio 5Portfolio 5 0.0740378620.074037862 0.30.3 0.0752775260.075277526 0.40.4 0.07364780.0736478 0.250.25 0.069453710.06945371 0.50.5
Portfolio 6Portfolio 6 0.0868231890.086823189 0.70.7 0.0903345820.090334582 0.70.7 0.087498570.08749857 0.70.7 0.086122010.08612201 0.750.75
Portfolio 7Portfolio 7 0.0626374050.062637405 0.650.65 0.0679064650.067906465 0.70.7 0.05721320.0572132 0.650.65 0.0606796520.060679652 0.650.65
Portfolio 8Portfolio 8 0.0807246350.080724635 0.650.65 0.0836518860.083651886 0.650.65 0.083339670.08333967 0.70.7 0.0807403240.080740324 0.750.75
Portfolio 9Portfolio 9 0.0744830770.074483077 0.550.55 0.0842274190.084227419 0.550.55 0.07411140.0741114 0.60.6 0.0693642560.069364256 0.650.65
Slide 29Slide 29
IN-SAMPLE estimation results for the interval IN-SAMPLE estimation results for the interval 2005 – 20062005 – 2006
Two-step crossTwo-step cross sectional regressionsectional regression
procedureprocedure
One-step system ofOne-step system of non-linear seeminglynon-linear seemingly unrelated equationsunrelated equations
One-step system of One-step system of non-linear seeminglynon-linear seemingly
unrelated equations (*)unrelated equations (*)
FeedforwardFeedforward Artificial NeuralArtificial Neural
NetworksNetworks
Elman ArtificialElman Artificial Neural NetworksNeural Networks
Sum ofSum of Squared Squared ResidualsResiduals
R-squaredR-squared Sum ofSum of Squared Squared ResidualsResiduals
R-squaredR-squared Sum ofSum of Squared Squared ResidualsResiduals
R-squaredR-squared Sum ofSum of Squared Squared ResidualsResiduals
R-squaredR-squared Sum of Sum of Squared Squared ResidualsResiduals
R-squaredR-squared
Portfolio 1Portfolio 1 0.0120440.012044 0.0024910.002491 0.2571438920.25714389266
0.0232010.023201 -- -- 0.011340.01134 0.0033170.003317 0.0113360.011336 0.00473110.0047311
Portfolio 2Portfolio 2 3.36E-613.36E-61 -- 0.2460795640.24607956433
0.0231420.023142 -- -- -- -- -- --
Portfolio 3Portfolio 3 4.4108944.410894 0.0344720.034472 4.9523677034.95236770344
-0.001896-0.001896 5.68933820425.6893382042 -0.150990-0.150990 4.35654.3565 0.0385840.038584 4.029974974.02997497 0.0932250.093225
Portfolio 4Portfolio 4 4.9848774.984877 0.0346410.034641 5.3289069845.32890698422
0.0016060.001606 5.93915622105.9391562210 -0.112727-0.112727 4.7649774.764977 0.0625260.062526 4.11054.1105 0.144980.14498
Portfolio 5Portfolio 5 6.3612336.361233 0.1792660.179266 7.9490393907.94903939011
0.0036670.003667 6.64323113016.6432311301 0.1673370.167337 6.12086.1208 0.198560.19856 5.86335.8633 0.210550.21055
Portfolio 6Portfolio 6 7.1539757.153975 0.2958820.295882 10.3406865310.3406865355
-0.000686-0.000686 7.62050016137.6205001613 0.2625510.262551 6.838266.83826 0.294760.29476 6.39476.3947 0.340060.34006
Portfolio 7Portfolio 7 4.0815564.081556 0.2374970.237497 5.6034113655.60341136533
0.0029440.002944 4.36900147844.3690014784 0.2225910.222591 3.75513.7551 0.266230.26623 3.618213763.61821376 0.3022570.302257
Portfolio 8Portfolio 8 6.2450826.245082 0.5066260.506626 13.0472653113.0472653166
0.0003740.000374 8.21553218478.2155321847 0.3705610.370561 5.96315.9631 0.554520.55452 5.6065.606 0.570020.57002
Portfolio 9Portfolio 9 5.4871605.487160 0.5902970.590297 13.6281427013.6281427033
0.0003430.000343 8.20956420048.2095642004 0.3978090.397809 2.939412.93941 0.767710.76771 4.823964744.82396474 0.6338030.633803(*)Estimation results in interval 2005 – 2006 using One-step System of Non-linear Seemingly Unrelated Equations in the absence of Portfolios 1 and 2
Slide 30Slide 30
OUT-OF-SAMPLE prediction results for OUT-OF-SAMPLE prediction results for the interval 2005 – 2006the interval 2005 – 2006
(*) Prediction results in interval 2005 – 2006 using One-step System of Non-linear Seemingly Unrelated Equations in the absence of Portfolios 1 and 2
Slide 31Slide 31
Two-step cross sectional regression
procedure
One-step system of non-linear
seemingly unrelated equations
One-step system of non-linear seemingly unrelated equations
(*)
Feedforward Artificial Neural
Networks
Elman Artificial Neural Networks
Root Mean Squared
Error Statistic
Success ratio sign prediction
Root Mean Squared
Error Statistic
Success ratio sign
prediction
Root Mean Squared
Error Statistic
Success ratio sign prediction
Root Mean Squared
Error Statistic
Success ratio sign
prediction
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
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