1. Estimating and Predicting Stock Returns Using Artificial Neural Networks Dissertation Paper BUCHAREST ACADEMY OF ECONOMIC STUDIES DOCTORAL SCHOOL OF FINANCE AND BANKING-DOFIN Supervisor: Prof. Univ. Dr. Moisa Altar MSc Student: Catalin-Marius Untea

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 2

3. Theoretical background: The Arbitrage Pricing Theory (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 I Slide 3

4. The APT model can be described by two equations:  the first equation expresses the stock returns based on the set of factors, t=1,...,T i=1,...,N j=1,...,k represents the return on share i; where represents the expected return for share i; represents the influence of factor j on stock return i;  the second equation is for the equilibrium expected return and expresses the no arbitrage opportunity: where represents the free-risk rate return; 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: The Arbitrage Pricing Theory (APT) Part II Slide 4

5. Two-step cross-sectional regression 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: where represents portfolio return i; represents principal component j; represents sensitivity coefficient for portfolio return i at factor fluctuations j. Part I Slide 5

6. Dependent Variable: RAND_PORT1 Method: Generalized Method of Moments Date: 06/24/07 Time: 16:25 Sample: 252 966 Included observations: 715 Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0103700.0035132.9516640.0033 COMP_PRIN10.0010690.0031390.3406120.7335 COMP_PRIN20.0006210.0041110.1511060.8799 COMP_PRIN3-0.0005960.002943-0.2026830.8394 COMP_PRIN40.0018720.0058990.3173850.7510 COMP_PRIN5-0.0031210.004433-0.7039550.4817 R-squared0.001508 Mean dependent var Mean dependent var0.010340 Adjusted R-squared -0.005533 S.D. dependent var S.D. dependent var0.074864 S.E. of regression 0.075071 Sum squared resid Sum squared resid3.995630 Durbin-Watson stat 1.794584 J-statistic J-statistic7.31E-33 Dependent Variable: RAND_PORT1 Method: Generalized Method of Moments Date: 06/24/07 Time: 16:27 Sample: 1240 1714 Included observations: 475 Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0002250.0002221.0121800.3120 COMP_PRIN16.42E-067.84E-060.8193880.4130 COMP_PRIN20.0001170.0001190.9878490.3237 COMP_PRIN30.0001740.0001760.9891390.3231 COMP_PRIN41.18E-061.26E-050.0936440.9254 COMP_PRIN5-2.48E-052.74E-05-0.9070040.3649 R-squared0.002491 Mean dependent var Mean dependent var0.000224 Adjusted R-squared -0.008143 S.D. dependent var S.D. dependent var0.004887 S.E. of regression 0.004906 Sum squared resid Sum squared resid0.011290 Durbin-Watson stat 2.006955 J-statistic J-statistic2.40E-28 Estimation results for portfolio 1 2001 - 20032005 - 2006 Slide 6

7. Dependent Variable: RAND_PORT2 Method: Generalized Method of Moments Date: 06/24/07 Time: 16:22 Sample: 252 966 Included observations: 715 Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0087330.0030822.8335670.0047 COMP_PRIN1-0.0023380.003964-0.5899600.5554 COMP_PRIN2-0.0046590.002904-1.6046400.1090 COMP_PRIN3-0.0025470.002832-0.8993180.3688 COMP_PRIN4-0.0011600.004485-0.2585740.7960 COMP_PRIN5-0.0020810.002495-0.8341410.4045 R-squared0.011744 Mean dependent var Mean dependent var0.008567 Adjusted R-squared 0.004774 S.D. dependent var S.D. dependent var0.056587 S.E. of regression 0.056451 Sum squared resid Sum squared resid2.259404 Durbin-Watson stat 1.427165 J-statistic J-statistic1.55E-31 Dependent Variable: RAND_PORT2 Method: Generalized Method of Moments Date: 06/24/07 Time: 16:23 Sample: 1240 1714 Included observations: 475 Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C-2.12E-336.68E-19-3.17E-151.0000 COMP_PRIN13.08E-336.03E-195.11E-151.0000 COMP_PRIN2-1.00E-321.62E-18-6.20E-151.0000 COMP_PRIN34.62E-331.02E-184.53E-151.0000 COMP_PRIN41.39E-322.64E-185.26E-151.0000 COMP_PRIN51.16E-331.79E-186.45E-161.0000 Mean dependent var 0.000000 S.D. dependent var S.D. dependent var0.000000 S.E. of regression 1.74E-32 Sum squared resid Sum squared resid1.41E-61 Durbin-Watson stat 1.715121 J-statistic J-statistic0.049420 Estimation results for portfolio 2 2001 - 20032005 - 2006 Slide 7

8. Dependent Variable: RAND_PORT3 Method: Generalized Method of Moments Date: 06/24/07 Time: 16:18 Sample: 252 966 Included observations: 715 Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0067100.0024602.7279740.0065 COMP_PRIN1-0.0005430.002928-0.1855200.8529 COMP_PRIN20.0033820.0036590.9244280.3556 COMP_PRIN3-0.0045550.003514-1.2961180.1954 COMP_PRIN40.0020280.0050670.4002080.6891 COMP_PRIN5-0.0023330.004169-0.5595220.5760 R-squared0.006025 Mean dependent var Mean dependent var0.006937 Adjusted R-squared -0.000985 S.D. dependent var S.D. dependent var0.066368 S.E. of regression 0.066401 Sum squared resid Sum squared resid3.126022 Durbin-Watson stat 1.975762 J-statistic J-statistic1.10E-31 Dependent Variable: RAND_PORT3 Method: Generalized Method of Moments Date: 06/24/07 Time: 16:20 Sample: 1240 1714 Included observations: 475 Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0028040.0041180.6808920.4963 COMP_PRIN10.0016720.0010251.6314850.1035 COMP_PRIN2-0.0114490.003852-2.9718580.0031 COMP_PRIN3-0.0087260.003318-2.6297980.0088 COMP_PRIN40.0054820.0035421.5476950.1224 COMP_PRIN50.0061240.0049811.2294810.2195 R-squared0.034714 Mean dependent var Mean dependent var0.003431 Adjusted R-squared 0.024423 S.D. dependent var S.D. dependent var0.098178 S.E. of regression 0.096972 Sum squared resid Sum squared resid4.410230 Durbin-Watson stat 2.066803 J-statistic J-statistic5.30E-31 Estimation results for portfolio 3 2001 - 20032005 - 2006 Slide 8

9. Dependent Variable: RAND_PORT4 Method: Generalized Method of Moments Date: 06/24/07 Time: 16:13 Sample: 252 966 Included observations: 715 Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0138910.0042603.2612720.0012 COMP_PRIN1-4.93E-050.004573-0.0107740.9914 COMP_PRIN2-0.0058010.004750-1.2211950.2224 COMP_PRIN3-0.0178440.003964-4.5013260.0000 COMP_PRIN40.0003190.0055380.0576470.9540 COMP_PRIN50.0057380.0069270.8283800.4077 R-squared0.035663 Mean dependent var Mean dependent var0.014011 Adjusted R-squared 0.028863 S.D. dependent var S.D. dependent var0.102160 S.E. of regression 0.100675 Sum squared resid Sum squared resid7.186013 Durbin-Watson stat 1.883555 J-statistic J-statistic4.66E-31 Dependent Variable: RAND_PORT4 Method: Generalized Method of Moments Date: 06/24/07 Time: 16:15 Sample: 1240 1714 Included observations: 475 Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C-0.0022950.004776-0.4806180.6310 COMP_PRIN1-0.0013920.001519-0.9164600.3599 COMP_PRIN2-0.0120730.003925-3.0760840.0022 COMP_PRIN3-0.0109830.003934-2.7915770.0055 COMP_PRIN40.0011900.0046290.2570300.7973 COMP_PRIN50.0044510.0025421.7511890.0806 R-squared0.035129 Mean dependent var Mean dependent var-0.001873 Adjusted R-squared 0.024843 S.D. dependent var S.D. dependent var0.104259 S.E. of regression 0.102956 Sum squared resid Sum squared resid4.971373 Durbin-Watson stat 2.068589 J-statistic J-statistic7.70E-32 Estimation results for portfolio 4 2001 - 20032005 - 2006 Slide 9

10. Dependent Variable: RAND_PORT5 Method: Generalized Method of Moments Date: 06/24/07 Time: 16:06 Sample: 252 966 Included observations: 715 Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0116290.0048882.3793640.0176 COMP_PRIN10.0062900.0052951.1880060.2352 COMP_PRIN2-0.0107410.005006-2.1458330.0322 COMP_PRIN3-0.0128180.004205-3.0485950.0024 COMP_PRIN4-0.0037860.006712-0.5640910.5729 COMP_PRIN5-0.0012250.005709-0.2146170.8301 R-squared0.025255 Mean dependent var Mean dependent var0.011390 Adjusted R-squared 0.018380 S.D. dependent var S.D. dependent var0.108759 S.E. of regression 0.107754 Sum squared resid Sum squared resid8.232218 Durbin-Watson stat 1.688630 J-statistic J-statistic4.69E-30 Dependent Variable: RAND_PORT5 Method: Generalized Method of Moments Date: 06/24/07 Time: 16:08 Sample: 1240 1714 Included observations: 475 Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0048610.0043641.1140300.2658 COMP_PRIN1-0.0023080.001453-1.5884770.1129 COMP_PRIN2-0.0224910.005128-4.3863420.0000 COMP_PRIN3-0.0404460.007050-5.7370890.0000 COMP_PRIN4-0.0004950.004523-0.1094530.9129 COMP_PRIN5-0.0017210.004894-0.3516520.7253 R-squared0.179157 Mean dependent var Mean dependent var0.004630 Adjusted R-squared 0.170406 S.D. dependent var S.D. dependent var0.127867 S.E. of regression 0.116464 Sum squared resid Sum squared resid6.361403 Durbin-Watson stat 2.323714 J-statistic J-statistic3.19E-31 Estimation results for portfolio 5 2001 - 20032005 - 2006 Slide 10

11. Dependent Variable: RAND_PORT6 Method: Generalized Method of Moments Date: 06/24/07 Time: 16:00 Sample: 252 966 Included observations: 715 Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0188450.0047094.0022430.0001 COMP_PRIN10.0057470.0066210.8679550.3857 COMP_PRIN2-0.0139670.006309-2.2139110.0272 COMP_PRIN3-0.0280960.006492-4.3279170.0000 COMP_PRIN4-0.0093660.008081-1.1590180.2468 COMP_PRIN50.0046710.0083550.5590610.5763 R-squared0.070940 Mean dependent var Mean dependent var0.018974 Adjusted R-squared 0.064388 S.D. dependent var S.D. dependent var0.123354 S.E. of regression 0.119317 Sum squared resid Sum squared resid10.09372 Durbin-Watson stat 1.830613 J-statistic J-statistic7.01E-31 Dependent Variable: RAND_PORT6 Method: Generalized Method of Moments Date: 06/24/07 Time: 16:04 Sample: 1240 1714 Included observations: 475 Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0095140.0053111.7913330.0739 COMP_PRIN10.0013400.0013690.9782500.3285 COMP_PRIN2-0.0309510.004950-6.2532260.0000 COMP_PRIN3-0.0611800.006443-9.4958970.0000 COMP_PRIN40.0041930.0042790.9798560.3277 COMP_PRIN50.0066950.0039971.6751080.0946 R-squared0.295558 Mean dependent var Mean dependent var0.009086 Adjusted R-squared 0.288048 S.D. dependent var S.D. dependent var0.146438 S.E. of regression 0.123560 Sum squared resid Sum squared resid7.160256 Durbin-Watson stat 2.095171 J-statistic J-statistic2.32E-32 Estimation results for portfolio 6 2001 - 20032005 - 2006 Slide 11

12. Dependent Variable: RAND_PORT7 Method: Generalized Method of Moments Date: 06/24/07 Time: 15:50 Sample: 252 966 Included observations: 715 Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0159130.0042373.7559090.0002 COMP_PRIN1-0.0013860.005520-0.2510570.8018 COMP_PRIN2-0.0130940.004741-2.7616370.0059 COMP_PRIN3-0.0256160.005611-4.5656510.0000 COMP_PRIN4-0.0027760.006131-0.4528710.6508 COMP_PRIN5-0.0097070.006174-1.5722450.1163 R-squared0.073087 Mean dependent var Mean dependent var0.015806 Adjusted R-squared 0.066550 S.D. dependent var S.D. dependent var0.109234 S.E. of regression 0.105537 Sum squared resid Sum squared resid7.896851 Durbin-Watson stat 1.845098 J-statistic J-statistic2.88E-30 Dependent Variable: RAND_PORT7 Method: Generalized Method of Moments Date: 06/24/07 Time: 15:52 Sample: 1240 1714 Included observations: 475 Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0096330.0041512.3204280.0207 COMP_PRIN1-0.0001670.001006-0.1656790.8685 COMP_PRIN2-0.0232770.004432-5.2522300.0000 COMP_PRIN3-0.0381180.005367-7.1025470.0000 COMP_PRIN40.0016510.0032840.5027870.6153 COMP_PRIN5-3.72E-050.003682-0.0101090.9919 R-squared0.238383 Mean dependent var Mean dependent var0.009630 Adjusted R-squared 0.230264 S.D. dependent var S.D. dependent var0.106029 S.E. of regression 0.093024 Sum squared resid Sum squared resid4.058460 Durbin-Watson stat 2.096640 J-statistic J-statistic3.21E-31 Estimation results for portfolio 7 2001 - 20032005 - 2006 Slide 12

13. Dependent Variable: RAND_PORT8 Method: Generalized Method of Moments Date: 06/24/07 Time: 15:19 Sample: 252 966 Included observations: 715 Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0222620.0054294.1004480.0000 COMP_PRIN1-0.0011240.008180-0.1374230.8907 COMP_PRIN2-0.0286330.007732-3.7030670.0002 COMP_PRIN3-0.0613960.007952-7.7209360.0000 COMP_PRIN4-0.0120170.010598-1.1339040.2572 COMP_PRIN50.0015340.0080020.1917270.8480 R-squared0.207651 Mean dependent var Mean dependent var0.022447 Adjusted R-squared 0.202063 S.D. dependent var S.D. dependent var0.153207 S.E. of regression 0.136856 Sum squared resid Sum squared resid13.27921 Durbin-Watson stat 1.864400 J-statistic J-statistic3.95E-32 Dependent Variable: RAND_PORT8 Method: Generalized Method of Moments Date: 06/24/07 Time: 15:20 Sample: 1240 1714 Included observations: 475 Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0216350.0053454.0479770.0001 COMP_PRIN1-0.0003990.001227-0.3248550.7454 COMP_PRIN2-0.0452520.004896-9.2425090.0000 COMP_PRIN3-0.0897080.007388-12.143120.0000 COMP_PRIN4-0.0047560.003832-1.2411970.2152 COMP_PRIN50.0094630.0039242.4116390.0163 R-squared0.506522 Mean dependent var Mean dependent var0.020567 Adjusted R-squared 0.501261 S.D. dependent var S.D. dependent var0.163423 S.E. of regression 0.115412 Sum squared resid Sum squared resid6.247043 Durbin-Watson stat 2.023208 J-statistic J-statistic2.08E-31 Estimation results for portfolio 8 2001 - 20032005 - 2006 Slide 13

14. Dependent Variable: RAND_PORT9 Method: Generalized Method of Moments Date: 06/24/07 Time: 15:18 Sample: 252 966 Included observations: 715 Kernel: Bartlett, Bandwidth: Fixed (6), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0126720.0043242.9307340.0035 COMP_PRIN10.0090910.0061601.4759420.1404 COMP_PRIN2-0.0289940.005571-5.2042470.0000 COMP_PRIN3-0.0558570.006506-8.5858270.0000 COMP_PRIN4-0.0109150.007077-1.5422510.1235 COMP_PRIN5-0.0106120.006769-1.5677500.1174 R-squared0.265682 Mean dependent var Mean dependent var0.012540 Adjusted R-squared 0.260503 S.D. dependent var S.D. dependent var0.125231 S.E. of regression 0.107691 Sum squared resid Sum squared resid8.222594 Durbin-Watson stat 1.874611 J-statistic J-statistic2.93E-31 Dependent Variable: RAND_PORT9 Method: Generalized Method of Moments Date: 06/24/07 Time: 15:16 Sample: 1240 1714 Included observations: 475 Kernel: Bartlett, Bandwidth: Fixed (5), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: COMP_PRIN1 COMP_PRIN2 COMP_PRIN3 COMP_PRIN4 COMP_PRIN5 COMP_PRIN4 COMP_PRIN5 VariableCoefficient Std. Error t-StatisticProb. C0.0126200.0044262.8515110.0045 COMP_PRIN1-0.0060550.005933-1.0204750.3080 COMP_PRIN2-0.0577780.008377-6.8973980.0000 COMP_PRIN3-0.0906730.005644-16.064320.0000 COMP_PRIN4-0.0182030.015571-1.1690050.2430 COMP_PRIN50.0245970.0165011.4906650.1367 R-squared0.590284 Mean dependent var Mean dependent var0.011881 Adjusted R-squared 0.585916 S.D. dependent var S.D. dependent var0.168003 S.E. of regression 0.108109 Sum squared resid Sum squared resid5.481453 Durbin-Watson stat 2.089196 J-statistic J-statistic6.06E-31 Estimation results for portfolio 9 2001 - 20032005 - 2006 Slide 14

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 cross- section regression, represents an estimation for the risk premium associated to the exposure to unexpected variation in one of the factors. Slide 15

16. Dependent Variable: MEDII_PORT_01_03 Method: Generalized Method of Moments Date: 06/24/07 Time: 18:17 Sample: 1 9 Included observations: 9 Kernel: Bartlett, Bandwidth: Fixed (2), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: BETA_COMP_PRIN1_01_03 BETA_COMP_PRIN2_01_ 03 BETA_COMP_PRIN3_01_03 BETA_COMP_PRIN4_01_03 03 BETA_COMP_PRIN3_01_03 BETA_COMP_PRIN4_01_03 BETA_COMP_PRIN5_01_03 BETA_COMP_PRIN5_01_03 VariableCoefficient Std. Error t-StatisticProb. C0.0105910.0017426.0811210.0089 BETA_COMP_PRIN1 _01_03 -0.3595910.249104-1.4435370.2446 BETA_COMP_PRIN2 _01_03 -0.0867290.227678-0.3809260.7286 BETA_COMP_PRIN3 _01_03 -0.0641550.098361-0.6522370.5608 BETA_COMP_PRIN4 _01_03 -0.4149720.395612-1.0489360.3713 BETA_COMP_PRIN5 _01_03 0.3152640.1374192.2941870.1055 R-squared0.783210 Mean dependent var Mean dependent var0.013446 Adjusted R-squared 0.421893 S.D. dependent var S.D. dependent var0.004979 S.E. of regression 0.003786 Sum squared resid Sum squared resid4.30E-05 Durbin-Watson stat 1.422391 J-statistic J-statistic6.18E-29 Dependent Variable: MEDII_PORT_05_06 Method: Generalized Method of Moments Date: 06/24/07 Time: 18:51 Sample: 1 9 Included observations: 9 Kernel: Bartlett, Bandwidth: Fixed (2), No prewhitening Simultaneous weighting matrix & coefficient iteration Convergence achieved after: 1 weight matrix, 2 total coef iterations Instrument list: BETA_COMP_PRIN1_05_06 BETA_COMP_PRIN2_05_ 06 BETA_COMP_PRIN3_05_06 BETA_COMP_PRIN4_05_06 06 BETA_COMP_PRIN3_05_06 BETA_COMP_PRIN4_05_06 BETA_COMP_PRIN5_05_06 BETA_COMP_PRIN5_05_06 VariableCoefficient Std. Error t-StatisticProb. C0.0003090.0001841.6740790.1927 BETA_COMP_PRIN1 _05_06 4.2830850.20184721.219450.0002 BETA_COMP_PRIN2 _05_06 -0.8791580.071276-12.334560.0011 BETA_COMP_PRIN3 _05_06 0.1954350.0359025.4435650.0122 BETA_COMP_PRIN4 _05_06 -1.3917300.093100-14.948840.0006 BETA_COMP_PRIN5 _05_06 -0.8349890.047847-17.451220.0004 R-squared0.992682 Mean dependent var Mean dependent var0.006397 Adjusted R-squared 0.980485 S.D. dependent var S.D. dependent var0.007140 S.E. of regression 0.000997 Sum squared resid Sum squared resid2.98E-06 Durbin-Watson stat 2.211269 J-statistic J-statistic3.59E-25 Cross-section regression 2001 - 20032005 - 2006 Slide 16

17. Risk premiums associated with factors Interval 2001 – 2003 0.010591-0.359591-0.086729-0.064155-0.4149720.315264 Interval 2005 – 2006 0.0003094.283085-0.8791580.195435-1.391730-0.834989 Slide 17

18. Predictions using sensitivity coefficients estimated through Two-step cross sectional regression procedure Root Mean Squared Error Statistic Success ratio sign prediction Portfolio 1 0.0228274490.35 Portfolio 2 0.054693490.4 Portfolio 3 0.021521430.3 Portfolio 4 0.0516321610.4 Portfolio 5 0.0754669390.4 Portfolio 6 0.085837620.75 Portfolio 7 0.0627300210.65 Portfolio 8 0.0810786540.7 Portfolio 9 0.0756745440.65 Root Mean Squared Error Statistic Success ratio sign prediction Portfolio 1 0.000285150 Portfolio 2 8.81318E-330 Portfolio 3 0.0560572930.2 Portfolio 4 0.080002030.6 Portfolio 5 0.1367080460.6 Portfolio 6 0.1142733760.65 Portfolio 7 0.1453403560.5 Portfolio 8 0.0791471860.7 Portfolio 9 0.0681766120.7 2001 - 20032005 - 2006 Slide 18

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: or by passing to the left side the risk-free rate of return, the last equation becomes: 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 19

20. VariableCoeff Std Error T-StatSignif A1 -0.0032334230.001693366-1.909460.05620213 A2 -3.6130093841.94176553-1.860680.06278901 A3 -0.0049077410.001710068-2.869910.00410589 A4 0000 A5 -0.0127554160.001426302-8.9430 A6 0000 A7 0.000228760.0025081740.091210.92732906 A8 0000 A9 -0.0059359640.002616418-2.268740.02328435 A10 0000 Sum of Squared ResidualsR-squared Portfolio 1 4.3905947134-0.008163 Portfolio 2 2.38350460560.013809 Portfolio 3 3.3469906452-0.010630 Portfolio 4 7.30232588790.050090 Portfolio 5 8.52216513570.031992 Portfolio 6 10.6806904420.063338 Portfolio 7 8.15808194350.074072 Portfolio 8 15.6546106040.089775 Portfolio 9 10.2982733720.108219 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 coefficients A2,A4,A6,A8,A10 represents risk premiums Slide 20

21. VariableCoeff Std Error T-StatSignif A1 -0.0009200550.000341529-2.693930.00706148 A2 -1.3237490291.238420607-1.06890.28511428 A3 -0.000425690.000984323-0.432470.66539995 A4 0000 A5 0.0003807330.000937730.406020.68473097 A6 0000 A7 0.0001799020.0009744960.184610.8535346 A8 0000 A9 -0.0014769670.000953245-1.549410.12128345 A10 0000 Sum of Squared Residuals R-squared Portfolio 1 0.25714389260.023201 Portfolio 2 0.24607956430.023142 Portfolio 3 4.9523677034-0.001896 Portfolio 4 5.32890698420.001606 Portfolio 5 7.94903939010.003667 Portfolio 6 10.340686535-0.000686 Portfolio 7 5.60341136530.002944 Portfolio 8 13.0472653160.000374 Portfolio 9 13.6281427030.000343 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 coefficients A2,A4,A6,A8,A10 represents risk premiums Slide 21

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. VariableCoeff Std Error T-StatSignif A1 -0.0020130380.000809396-2.487090.01287935 A2 -4.0700643642.046949338-1.988360.04677231 A3 -0.0271490890.002332764-11.638160 A4 0000 A5 -0.0412522810.002222344-18.562510 A6 0000 A7 -0.0012890610.002309475-0.558160.57673376 A8 0000 A9 0.0059122390.0022591132.617060.00886902 A10 0000 Sum of Squared ResidualsR-squared Portfolio 1 -- Portfolio 2 -- Portfolio 3 5.6893382042-0.150990 Portfolio 4 5.9391562210-0.112727 Portfolio 5 6.64323113010.167337 Portfolio 6 7.62050016130.262551 Portfolio 7 4.36900147840.222591 Portfolio 8 8.21553218470.370561 Portfolio 9 8.20956420040.397809 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 coefficients A2,A4,A6,A8,A10 represents risk premiums Slide 22

23. Root Mean Squared Error Statistic Success ratio sign prediction Portfolio 1 0.0237114440.35 Portfolio 2 0.0539070750.35 Portfolio 3 0.0256717690.3 Portfolio 4 0.0521313840.35 Portfolio 5 0.0752775260.4 Portfolio 6 0.0903345820.7 Portfolio 7 0.0679064650.7 Portfolio 8 0.0836518860.65 Portfolio 9 0.0842274190.55 The prediction results in interval 2001 – 2003 (20 observations) Slide 23

24. Root Mean Squared Error Statistic Success ratio sign prediction Portfolio 1 0.0344536390.6 Portfolio 2 0.0344536390.6 Portfolio 3 0.0588905020.45 Portfolio 4 0.07297320.55 Portfolio 5 0.1436213210.35 Portfolio 6 0.1476096190.55 Portfolio 7 0.1553104450.9 Portfolio 8 0.1175769680.65 Portfolio 9 0.0983180520.5 OUT-OF-SAMPLE statistics in interval 2005 - 2006 The prediction results in interval 2005 – 2006 (20 observations) Root Mean Squared Error Statistic Success ratio sign prediction Portfolio 1 -- Portfolio 2 -- Portfolio 3 0.0679476630.4 Portfolio 4 0.074413220.55 Portfolio 5 0.1461706250.65 Portfolio 6 0.1278093480.75 Portfolio 7 0.1523254920.6 Portfolio 8 0.1022031220.75 Portfolio 9 0.0768142540.7 OUT-OF-SAMPLE statistics in interval 2005 – 2006 in the absence of Portfolio 1 and 2 Slide 24

25. Feedforward Artificial Neural Networks 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 at step n based on the change in gradient, relative to the change in the parameters. 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 I Slide 25

26. Feedforward Artificial Neural Networks Part II Mathematically the feedforward neural network can be described by the following equations: where we have I=5 input variables and K=15 neurons in the hidden layer. Slide 26

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: Slide 27

28. IN-SAMPLE estimation results for the interval 2001 – 2003 Two-step cross sectional regression procedure One-step system of non-linear seemingly unrelated equations unrelated equations Feedforward Artificial Neural Networks Neural Networks Elman Recurrent Neural Networks Sum of Squared SquaredResidualsR-squared Sum of SquaredResidualsR-squared Squared SquaredResidualsR-squared Sum of SquaredResidualsR-squared Portfolio 1 3.9925500.0016664.3905947134-0.0081633.74880.0440703.7571740.065192 Portfolio 2 2.2615200.0117032.38350460560.0138092.163950.0384022.042910.091478 Portfolio 3 3.1496520.0061293.3469906452-0.0106303.00610.0388042.53550.18261 Portfolio 4 7.1974170.0357877.30232588790.0500906.97440.0606926.5201850.121192 Portfolio 5 8.2267380.0252158.52216513570.0319927.96310.0564217.691970.095457 Portfolio 6 10.123310.07093810.6806904420.0633389.71640.0999069.06180.14375 Portfolio 7 7.9189610.0722618.15808194350.0740727.44070.113697.280920.132126 Portfolio 8 13.292150.20778615.6546106040.08977512.6170.2489112.1580.26725 Portfolio 9 8.2448300.26471910.2982733720.1082197.81030.304987.27660.3405 Slide 28

29. OUT-OF-SAMPLE prediction results for the interval 2001 – 2003 Two-step cross sectional regression procedure regression procedure One-step system of non-linear seemingly unrelated equations unrelated equations Feedforward Artificial Neural Networks Neural Networks Elman Artificial Neural Networks Root Mean Squared SquaredErrorStatisticSuccess ratio sign ratio signprediction Root Mean Squared SquaredErrorStatisticSuccess ratio sign ratio signprediction Root Mean Squared SquaredErrorStatisticSuccess ratio sign prediction Root Mean SquaredErrorStatisticSuccess ratio sign prediction Portfolio 1 0.0234813640.40.0237114440.350.022191020.350.022018450.35 Portfolio 2 0.0555783770.250.0539070750.350.054066760.40.052588170.4 Portfolio 3 0.0220714540.350.0256717690.30.020514260.30.017154300.3 Portfolio 4 0.0519028290.40.0521313840.350.049646920.350.046418260.65 Portfolio 5 0.0740378620.30.0752775260.40.07364780.250.069453710.5 Portfolio 6 0.0868231890.70.0903345820.70.087498570.70.086122010.75 Portfolio 7 0.0626374050.650.0679064650.70.05721320.650.0606796520.65 Portfolio 8 0.0807246350.650.0836518860.650.083339670.70.0807403240.75 Portfolio 9 0.0744830770.550.0842274190.550.07411140.60.0693642560.65 Slide 29

30. IN-SAMPLE estimation results for the interval 2005 – 2006 Two-step cross sectional regression sectional regression procedure procedure One-step system of non-linear seemingly non-linear seemingly unrelated equations unrelated equations One-step system of non-linear seemingly unrelated equations (*) unrelated equations (*)Feedforward Artificial Neural Artificial Neural Networks Networks Elman Artificial Neural Networks Neural Networks Sum of Squared SquaredResidualsR-squared Sum of Squared SquaredResidualsR-squared Sum of Squared SquaredResidualsR-squared Sum of Squared SquaredResidualsR-squared Sum of SquaredResidualsR-squared Portfolio 1 0.0120440.0024910.25714389260.023201--0.011340.0033170.0113360.0047311 Portfolio 2 3.36E-61-0.24607956430.023142------ Portfolio 3 4.4108940.0344724.9523677034-0.0018965.6893382042-0.1509904.35650.0385844.029974970.093225 Portfolio 4 4.9848770.0346415.32890698420.0016065.9391562210-0.1127274.7649770.0625264.11050.14498 Portfolio 5 6.3612330.1792667.94903939010.0036676.64323113010.1673376.12080.198565.86330.21055 Portfolio 6 7.1539750.29588210.340686535-0.0006867.62050016130.2625516.838260.294766.39470.34006 Portfolio 7 4.0815560.2374975.60341136530.0029444.36900147840.2225913.75510.266233.618213760.302257 Portfolio 8 6.2450820.50662613.0472653160.0003748.21553218470.3705615.96310.554525.6060.57002 Portfolio 9 5.4871600.59029713.6281427030.0003438.20956420040.3978092.939410.767714.823964740.633803 (*)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 30

31. OUT-OF-SAMPLE prediction results for the 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 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 predictio n Root Mean Squared Error Statistic Success ratio sign prediction Root Mean Squared Error Statistic Success ratio sign predictio n Root Mean Squared Error Statistic Success ratio sign prediction Portfolio 1 0.00029452200.0344536390.6--000.000776110 Portfolio 2 1.24522E-3200.0344536390.6------ Portfolio 3 0.0558322650.150.0588905020.450.0679476630.40.057573000.20.052743620.3 Portfolio 4 0.0798366770.50.07297320.550.074413220.550.083161200.450.080240260.55 Portfolio 5 0.1367559820.60.1436213210.350.1461706250.650.139409470.650.139690730.5 Portfolio 6 0.1138416360.350.1476096190.550.1278093480.750.117097670.650.118186290.55 Portfolio 7 0.1459226510.450.1553104450.90.1523254920.60.140588050.60.144060410.65 Portfolio 8 0.0794751560.70.1175769680.650.1022031220.750.071372260.70.082975900.7 Portfolio 9 0.068089540.70.0983180520.50.0768142540.70.058015780.80.072609610.8