estimating and predicting stock returns using artificial neural networks dissertation paper...

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


0.1511060.151106 0.87990.8799

COMP_PRIN3COMP_PRIN3 -0.000596-0.000596 0.002940.0029433

-0.202683-0.202683 0.83940.8394


0.3173850.317385 0.75100.7510


-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




Sample: 1240 1714Sample: 1240 1714








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_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







Estimation results for portfolio 1

2001 - 2003 2005 - 2006

Slide 6Slide 6













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







0.0047740.004774 S.D. dependent varS.D. dependent var 0.056580.0565877







Sample: 1240 1714Sample: 1240 1714










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




Mean dependent Mean dependent varvar


S.E. of regressionS.E. of regression 1.74E-321.74E-32 Sum squared residSum squared resid 1.41E-611.41E-61


1.7151211.715121 J-statisticJ-statistic 0.049420.0494200


2001 - 2003 2005 - 2006

Slide 7Slide 7













CC 0.0067100.006710 0.0024600.002460 2.7279742.727974 0.00650.0065















Sample: 1240 1714Sample: 1240 1714









CC 0.0028040.002804 0.0041180.004118 0.6808920.680892 0.49630.4963













2001 - 2003 2005 - 2006

Slide 8Slide 8








Convergence achieved after: 1 weight matrix, 2 total coef Convergence achieved after: 1 weight matrix, 2 total coef iterationsiterations






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


-1.221195-1.221195 0.22240.2224


-4.501326-4.501326 0.00000.0000


0.0576470.057647 0.95400.9540


0.8283800.828380 0.40770.4077

R-squaredR-squared 0.0356630.035663 Mean dependent Mean dependent varvar

0.014010.0140111









Sample: 1240 1714Sample: 1240 1714









CC -0.002295-0.002295 0.0047760.004776 -0.480618-0.480618 0.63100.6310






R-squaredR-squared 0.0351290.035129 Mean dependent varMean dependent var --0.00.001018787

33







2001 - 2003 2005 - 2006

Slide 9Slide 9














CC 0.0116290.011629 0.004880.0048888

2.3793642.379364 0.01760.0176


1.1880061.188006 0.23520.2352


-2.145833-2.145833 0.03220.0322


-3.048595-3.048595 0.00240.0024


-0.564091-0.564091 0.57290.5729


-0.214617-0.214617 0.83010.8301


0.011390.0113900









Sample: 1240 1714Sample: 1240 1714









CC 0.0048610.004861 0.0043640.004364 1.1140301.114030 0.26580.2658













2001 - 2003 2005 - 2006

Slide 10Slide 10











VariableVariable CoefficientCoefficient Std. Std. ErrErroror


CC 0.0188450.018845 0.004700.0047099

4.0022434.002243 0.00010.0001


0.8679550.867955 0.38570.3857


-2.213911-2.213911 0.02720.0272


-4.327917-4.327917 0.00000.0000


-1.159018-1.159018 0.24680.2468


0.5590610.559061 0.57630.5763


0.018970.0189744









Sample: 1240 1714Sample: 1240 1714







VariableVariable CoefficientCoefficient Std. ErrorStd. Error t-Statistict-Statistic Prob. Prob.

CC 0.0095140.009514 0.005310.0053111

1.7913331.791333 0.07390.0739


0.9782500.978250 0.32850.3285


-6.253226-6.253226 0.00000.0000


-9.495897-9.495897 0.00000.0000


0.9798560.979856 0.32770.3277


1.6751081.675108 0.09460.0946


0.009080.0090866







2001 - 2003 2005 - 2006

Slide 11Slide 11













CC 0.0159130.015913 0.0042370.004237 3.7559093.755909 0.00020.0002















Sample: 1240 1714Sample: 1240 1714









CC 0.0096330.009633 0.0041510.004151 2.3204282.320428 0.02070.0207





COMP_PRIN5COMP_PRIN5 -3.72E-05-3.72E-05 0.0036820.003682 -0.010109-0.010109 0.99190.9919








2001 - 2003 2005 - 2006

Slide 12Slide 12














CC 0.0222620.022262 0.005420.0054299

4.1004484.100448 0.00000.0000


-0.137423-0.137423 0.89070.8907


-3.703067-3.703067 0.00020.0002


-7.720936-7.720936 0.00000.0000


-1.133904-1.133904 0.25720.2572


0.1917270.191727 0.84800.8480


0.022440.0224477









Sample: 1240 1714Sample: 1240 1714









CC 0.0216350.021635 0.0053450.005345 4.0479774.047977 0.00010.0001













2001 - 2003 2005 - 2006

Slide 13Slide 13













CC 0.0126720.012672 0.0043240.004324 2.9307342.930734 0.00350.0035















Sample: 1240 1714Sample: 1240 1714









CC 0.0126200.012620 0.0044260.004426 2.8515112.851511 0.00450.0045













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








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


-0.086729-0.086729 0.2276780.227678 --0.30.3

8098092626

0.72860.7286


-0.064155-0.064155 0.0983610.098361 --0.60.6

5225223737

0.56080.5608


-0.414972-0.414972 0.3956120.395612 --1.01.0

4894893636

0.37130.3713


0.3152640.315264 0.1374190.137419 2.2941872.294187 0.10550.1055


0.013440.0134466



S.E. of regressionS.E. of regression 0.0037860.003786 Sum squared residSum squared resid 4.30E-054.30E-05



Dependent Variable: MEDII_PORT_05_06Dependent Variable: MEDII_PORT_05_06








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



CC 0.0003090.000309 0.000180.0001844

1.6740791.674079 0.19270.1927


4.2830854.283085 0.201840.2018477

21.2194521.21945 0.00020.0002


--0.80.8

7917915858

0.071270.0712766

-12.33456-12.33456 0.00110.0011


0.1954350.195435 0.035900.0359022

5.4435655.443565 0.01220.0122


--1.31.3

9179173030

0.093100.0931000

-14.94884-14.94884 0.00060.0006


--0.80.8

3493498989

0.047840.0478477

-17.45122-17.45122 0.00040.0004


0.006390.0063977



S.E. of regressionS.E. of regression 0.0009970.000997 Sum squared residSum squared resid 2.98E-062.98E-06



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









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








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









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


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













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.


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 -- --












Slide 22Slide 22

Root Mean Root Mean Squared Squared

Error Error StatisticStatistic

Success Success ratio sign ratio sign predictionprediction










]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













OUT-OF-SAMPLE statistics in interval 2005 - 2006

The prediction results in interval 2005 – 2006 (20 observations)













OUT-OF-SAMPLE statistics in interval 2005 – 2006

in the absence of Portfolio 1 and 2

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

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







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


SuccessSuccess ratio sign ratio sign predictionprediction



SuccessSuccess ratio sign ratio sign predictionprediction
















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

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 -- -- -- -- -- --


-0.001896-0.001896 5.68933820425.6893382042 -0.150990-0.150990 4.35654.3565 0.0385840.038584 4.029974974.02997497 0.0932250.093225


0.0016060.001606 5.93915622105.9391562210 -0.112727-0.112727 4.7649774.764977 0.0625260.062526 4.11054.1105 0.144980.14498


0.0036670.003667 6.64323113016.6432311301 0.1673370.167337 6.12086.1208 0.198560.19856 5.86335.8633 0.210550.21055


-0.000686-0.000686 7.62050016137.6205001613 0.2625510.262551 6.838266.83826 0.294760.29476 6.39476.3947 0.340060.34006


0.0029440.002944 4.36900147844.3690014784 0.2225910.222591 3.75513.7551 0.266230.26623 3.618213763.61821376 0.3022570.302257


0.0003740.000374 8.21553218478.2155321847 0.3705610.370561 5.96315.9631 0.554520.55452 5.6065.606 0.570020.57002


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


Root Mean Squared

Error Statistic

Success ratio sign

prediction

Root Mean Squared

Error Statistic


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

estimating and predicting stock returns using artificial neural networks dissertation paper...

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