classifying trend movements in the msci u.s.a. capital market index—a comparison of regression,...

Computers Ops Res. Vol. 23, No. 6, pp. 611-622, 1996 Copyright © 1996 Elsevier Science Ltd

Pergamon 030s-0s~gs)0o0r, s-s Printed in Great Britain. All rights reserved 0305-0548/96 $15.00 + 0.00

C L A S S I F Y I N G T R E N D M O V E M E N T S I N T H E M S C I U . S . A .

C A P I T A L M A R K E T I N D E X - - A C O M P A R I S O N O F

R E G R E S S I O N , A R I M A A N D N E U R A L N E T W O R K M E T H O D S

Douglas Wood1" and Bhaskar DasguptaJ~ Manchester Business School, Booth Street West, Manchester M15 6PB, England

SeOl)e ~ Pm'l)(me--This study explores a new tool for modeling financial markets; namely neural networks. Neural networks emerged from the artificial intelligence field, where attempts were made to simulate human b¢haviour. This has received considerable attention in the last few years, specially from the financial markets, who have found that this non-parametric non-linear tool can help in providing above average returns and provide a new methodology for modeling financial markets [Hutchinson, Lo and Poggio, MIT (1991)]. Our study applies neural networks in modeling the Morgan Stanley U.S.A. Capital Market Index, and evaluates the performance over two years. The performance of the model was judged in terms of the predictive power to forecast the one month ahead direction of the index as well as the percentage change. To provide a comparative basis, we also evaluate two alternative methodologies: multiple linear regression modeling and autoregressive integrated moving average modeling. We find that the neural network outperforms both the alternative models in terms of directional and percentage change prediction.

Al)ltraet--This paper describes our initial results in applying neural networks to forecast the MSCI U.S.A. Capital Market Index. The objective is to test the ability of an non-parametric learning network to provide valuable information to a global portfolio manager, who needs to assess investment opportunities in equity markets in order to shape a one month ahead asset allocation. Primarily, the objective is to test the directionai classification properties of the method with secondary objectives of higher magnitude prediction and lower RMS error. The system achieved fairly good results on the direetional classification criteria as well as the other criteria mentioned both in absolute terms and in comparison with multiple linear regression and two ARIMA models. Copyright © 1996 Elsevier Science Ltd

I. I N T R O D U C T I O N

Most portfolio and fund managers in the financial centers enjoy total cross border freedom and hence face a bewildering choice of assets, instruments and investment locations. The major component of almost all portfolios or funds is the equity market. In view of the prospective returns, the growth of cross-border capital movements and electronic banking, has created large flows of cross border funds [I]. A significant shift in the international component of U.S. Pension Funds has already occurred and the trend is predicted to continue [2]. The movement shown in Table I is equivalent to a new flow of $300 bn over the next five years as U.S. funds seek to exploit more attractive returns internationally. Since the amount currently invested outside the country by U.S. Pension Fund managers was over $150bn in 1992 and is projected to rise to $430bn in 1997, the direction of funds flow becomes a major issue for world markets. The primary reason for this investment shift is that, the foreign markets are performing well relative to the U.S. domestic market. An indication of the performance of domestic pension fund returns around the world is given in Fig. 1.

?Douglas Wood is National Westminster Bank Professor of Banking and Corporate Finance at Manchester Business School. His main interest has boen in financial economics and forecasting. He has authored 3 books and more than 75 journal articles. He is currently Director of the International Centre for Banking and Financial Services.

:~Bhatkar Dugupta is a doctoral researcher at Manchester Business School. His research and consultancy interests lie in the area of quantitative and math~natical modding of financial systems, applications of Artificial Intelligence to Finance and Decision Support Systems. He has published more than ten journal articles and five conference papers, His doctoral research is related to identification of non-linear anomalies in international financial and commodity markets.

611

612 Douglas Wood and Bhaskar Dasgupta

25 -

2O

DD Da.. 0

Mexico Phillipines Switzerland Canada HK Japan

UK Malaysia US NZ Australia

Market Source: Tke Watts Company

Fig. 1. Pension fund returns around the world, 1992.

In allocating assets, portfolio managers rely to a significant extent on quantitative techniques. One of the common techniques is an index tracking mechanism in which the fund manager builds a portfolio with proxies that mirror the asset weightings in the target index [3]. This procedure may relate to investments in a single national index in which the fund manager is investing solely domestically, but the procedure can easily be extended to investments in some or all of the assets available internationally. When asset allocation involves cross border or world wide investments, it becomes necessary to track the performance of a number of foreign markets. This is usually done by forecasting the relevant index. Indexation, either domestically or internationally, has now become straightforward given the availability of traded index derivatives in most major markets. Trading index derivatives simplifies trading and increases the liquidity of the portfolio. But as cross border investment aims to secure risk return improvements, it also creates currency exposure in terms of the fund manager's liabilities. Optimization of the portfolio consequently requires a forecast of the portfolio's home currency version of the foreign index. In this paper, we develop a system based on neural networks, which forecasts the index, in this case the U.S.A. (U.S.$ denominated) Morgan Stanley Capital Index. The reason for selecting this index is that it is a component element in the Morgan Stanley Capital World Index, thereby allowing a fund manager to use the MSCI U.S.A. Index as a component of an indexed or speculative global equity portfolio.

2. S T O C K M A R K E T F O R E C A S T I N G

Forecasting stock market prices or indices, involves an assumption that past information about the prices/indices and other indicators have some predictive value in defining future prices. This runs

Table 1. Predicted change in projected U.S. pension fund asset mix

Type of investment 1992 1995

Major category Sub category Estimated (%)

Common stocks Active 30.000 30.500 Passive 13.900 14.800 Total 43.900 45.300

International Active stocks 4.000 6.000 Passive stocks 1.400 1.900 Bonds 1.300 1.800 Total 6.700 9.700

Bonds Active 26.800 25.100 Immunized or dedicated 2.500 2,400 Other passive 4.300 3.900 Total 33.600 31.460

Guaranteed investment contracts 4.000 3.000 Equity real estate 4.000 4.200 Cash and short.term securities 4.100 3.200 Other 3.700 3.300

Source: Greenwich Associates, U.S.A.

Modeling financial markets 613

counter to the general perception of efficiency of markets. As Fama [4] defines the concept, a market is considered to be efficient if current security prices fully reflect all available information, and if efficiency exists, then arbitrage opportunities through time series forecasts should not be attainable. However, this is an empirical issue and there is considerable evidence that markets are not fully efficient, or in other words it is possible to predict, the future performance of the markets better than randomly. Most of the results on which Fama's concept rests, rely on linear models, in which exhaustive inclusion of all plausible explanatory variables struggle to explain much more than 40% of market variance. The other 60% would then be described as noise, with market activity based on technical analysis or other pattern behaviour described as noise trading.

Generally forecasting or prediction concerns either the index itself, or its volatility (variance). In this paper we have concentrated on the index level although a parallel approach could be used to predict variance. The literature reflects a growing interest in the ability of non-linear models to explain residual variance and hence produce better predictions of price movements [5, 6]. In fact, Neural Networks (NN) are being used more and more in this fast growing area [7]. Interestingly, the literature in this area falls into two groups. The first group reports that NN are not useful and are unable to improve explanation of variances. The other group, mainly in the practitioner and general management press, reports that NN's provide excellent and amazing techniques for enhanced forecasting. The most likely explanation seems to be that while NN's are capable of excellent performance, due to commercial intelligence and competitive advantage factors, these successes are not reported in the detail required for academic journals. Effectively, the results, of course, cannot be validated or duplicated due to proprietary inputs, transformations, and/or techniques.

Neural Networks or nonparametric learning networks are data driven methods generally built around non-linear activation functions (or non-linear transfer functions). These networks are allowed to determine the non-parametric dynamics of the indices with minimal assumptions on the indices. The inputs to these learning networks are mathematical transformations of the indices and we define the one period ahead prediction to be the output into which the learning network maps the inputs. The mapping is generally non-linear in nature although it is also capable of dealing with ARIMA like models [8], and give an excellent description of how these learning networks can be used to approximate and model ARIMA models. Non-parametric models offer some major advantages over traditional parametric methods. Firstly as they do not force restrictive parametric assumptions such as log-normality or sample-path continuity, they tend to be robust to the specification errors that plague parametric models. Second, they are adaptive and respond to structural changes in the data generating processes in ways that parametric models could only approximate with ad hoe procedures and finally they are flexible enough to implement and cater for a wide range of input vectors and high dimensionality. For example, Hutchinson et al. [9] report when modeling the Black Scholes model, the entire learning network becomes the model itself.

Generally, for prediction of time series, two functions are normally used, logistic activation giving rise to sigmoid transfer functions (a.k.a. multi layer perceptrons) and the other is Gaussian activation giving rise to radial basis transfer functions [7]. For sigmoid functions, let ~v denote the input vector including a bias bh into a sigmoid or logistic hidden unit h, so

d

~0 = ~ WhiXi "q- bh = wh " x + b h i=1

where xi stands for xt_i, the time series xt, the value of input i, and whi is the parameter/weight between input unit i and the hidden unit h. The contribution wh. x is the projection of the input vector x = (xl , x2, . . . , Xd) on the weight vector xh = (Xhl , Xh2, . . . , Xhd). The activation Sh of the hidden unit is then

Sh = S((h) = 1 + e-a~h = 1 + tan h (h

This sigmoid function performs a smooth mapping, (-e~, + ~ ) ---, (0,1). The slope of the sigmoid can be absorbed into the weights and biases without loss of generality and is set to one. The Radial Basis Function depends only on the distance ~ = [I x - #h II between the input x and the center of the function/~h also of input dimension d, so §(x) = §(11 x - • . II) = §(•) , choosingfto be Gaussian

614 Doughs Wood and Bhaskar Dasgupta

and II * II to be the Euclidean norm, the activation Gh of the hidden unit h is given by

= e x p ;=1

the standard deviation represents the width of the Oaussian, and the normalization of Oaussian radial basis function would be between zero and one. Maruyama [10] show that there is no difference between the radial basis functions and the multilayer perceptrons when normalized inputs are used in multilayer perceptrons. We are using normalized inputs and using the Nevprop Software (details are available from the authors). We use the multilayer perceptron derivative of these learning networks, using the feed forward back propagation architecture with a sigmoid transformation in the hidden layer (see Fig. 2 for a schematic diagram of the backpropagation- feedforward process).

3. T H E N E U R A L N E T W O R K M O D E L

NN models for forecasting stock prices or indices generally follow two approaches, univariate time series forecasting or multivariate time series forecasting. The univariate approach believes that the previous performance of the series itself and in various transformations would be sufficient to forecast the future performance, and since this is dealing solely in terms of the dependent variable, it is described as univariate time series forecasting. In multivariate analysis, the future price of a security is determined by analyzing other economic series such as T-Bills rate, base rates, balance of trade etc., reflecting an assumption that changes in one or more of these external macro economic series would have a lagged impact on the dependent variable. In the approach we follow, we have taken a hybrid approach including some external variables as well as transformations of past values of the dependent variable.

As far as neural networks go, their unique self adaptive capabilities and their facility for pattern recognition in very noisy environments [11] make them promising devices for locating patterns in the financial markets. In turn their patterns support forecasts which allow traders and investment managers to move assets into the most promising market or asset for the next period. For a portfolio manager though, attractiveness of markets involves risks as well as returns. This means that the relative performance of each country's stock price index would have to be judged within a mean/ variance efficiency framework [12, 13]. This requires a forecast for each matrix of assets of interest in consistent terms to provide updated expected returns and covariance. Since the specific indices for each country are to some extent idiosyncratic in their coverage and weighting, a standardized approach using MSCI indices for each stock market seems an appropriate target variable.

Fig. 2. The back propagation algorithm for neural networks.


We had to make a decision as to the time period for forecasting. We had the choice between intra-day, daily, weekly, monthly or quarterly. Long forecast periods would not provide enough data points to create a good, statistically reliable data set. If short time periods are used, more data is available, but the assumptions about transaction costs then become critical. As a compromise, it was decided to go for the monthly series, since it is based on a portfolio manager's decision on a medium term international weighting probability. The raw data series for U.S.A, denominated in U.S.$ is shown in Fig. 3.

The raw data consists of 142 data points, starting from February 1982 till November 1993. The data was divided into two segments, the training set of 119 points from February 1982 till December 1991 and the testing set of 23 points starting from January 1992 till November 1993. The inputs were as follows:

(l) Percentage change of the 3 month moving average of German Index (U.S.$) (2) Percentage change of the 3 month moving average of German Index (Din) (3) Percentage change of the 3 month moving average of Japanese Index (U.S.$) (4) Percentage change of the 3 month moving average of Japanese Index (Yen) (5) Percentage change of the 3 month moving average of U.K. Index (U.S.$) (6) Percentage change of the 3 month moving average of U.K. Index (£) (7) Percentage change of the 2 month moving average of U.S. Index (U.S.$) (8) Percentage change of the 3 month moving average of U.S. Index (U.S.$)

The dependent (output) variable was the percentage change of the 4 month moving average of U.S.A. (U.S.$). There are three hidden nodes in the one hidden layer. The architecture of the model is illustrated in Fig. 4. The model was set up using the normal backpropagation algorithm and the NevProp Software. The setup of the system is explained in Table 2.

400--

300 - -

2 0 0 -

100 -

I I I I I I I I t I I I I I I I I I I , I Feb-S2 Feb-S3 Feb-S4 Feb-85 Feb-86 Feb-S? Feb-S8 Feb-89 Feb-90 Feb-91

Aug-S2 Au|-S3 Au|-84 Au|-85 Aug-86 Aug-87 Aug-S8 Aug-89 Aug-90 Aug-91

Month/Year &m~c: Dataatvum

Fig. 3. Morgan Stanley U.S.A. (U.S.$) Index.

Table 2. Model parameters

WeightRange 0.1 HyperErr 0 SismoidPrimeOffset 0.1 El~ilon 0.1 SpliW.p,ilon 1 Momentum 0.1 WashtDecay -0.0001 ScoreThreshold O. 1 QPMaxFactor 1.75 QPModeSwitchThreshold 0


I

3 month rn.a. C-ermany(US$) / / ~ ~ Percentage change ~ ~

3 month rn.a Germany (Din) ( ~ < ! " ' ' ~ , ~ Percentege change ~ : ~ . ~ . . _ j ~ ,

Percentage change ~ ~ j \ I I

3 month m.a. Japan (Yen) : t, Percentage change , :

J , / ' 4 month m.a. USA f I Percentage change , , ~ (US $)

( / ~ _ ~ Percontage c h a n g e ' ' 3 month m.a UK (£) ', ',

2 month m.a. US(US$) Percentage c h a n g e ~ ~ _ ~ )

3 month m.a. US(US$) Percent change

INPUT HIDDEN OUTPUT LAYER LAYER LAYER

In the interests of clarity, all the connections have not been shown.

Fig. 4, The neural network architecture model.

The scaled data of the output is displayed in Fig. 5. Scaling reflected the requirements of the software for the output to be between +0.5 and -0 .5 , and furthermore, the scaling helps the network train better [14, 15]. Scaling is one way of reducing the impact of noise which confuses the system, and distracts the network from concentrating on the major patterns and information present in the data. Another element is the use of moving averages for promoting trend and other desirable characteristics of the data as well as to suppress noise. As can be seen in Fig. 5, the moving average percentage changes have patterns which are not immediately apparent from the much noisier raw data.

0.10 -

o

-0.05 -

-o.10 I I I I I I I I I 1 I i I I 1 I I I I 1 I I I May-S2 May-S3 May-S4 May-S5 May-S6 May-S7 May4S May-S9 May-90 May-91 May-92 May-93

Nov-S2 Nov-S3 Nov.84 Nov-aS Nov-S6 Nov-S7 Nov-U Nov-S9 Nov-SM) Nov-91 Nov-92

Month/Year

Fig. 5. 4-period moving average percentage change.


0.10 -

O.OS

0.06

0.04

0.02

0 3 m German (US$) 3 m UK (US$) 3 m US (US$) 3 m UK (£) m UK (US$

m Japan (US$) 3 m Japan (Yea) 3 m UK (US$) 3 m USA (US$) 3 m UK (US$)

Connections

I Note: Only the Nodes have been mentioned; months moving [ input m: average

Fig. 6. 10 most significant connections by network weight (absolute).

The number of epochs for the training set was 1000 with the initial random number for initialization of the weights set to 0, to minimize the chance of the network being "swamped". The 10 highest absolute weights obtained in transformation are given in Fig. 6. As is evidenced by the figure, the percentage change of the 3 month moving average (MA) of Germany (U.S.$) input seems to be the most important, in terms of the highest weight. The other connections that seem to be important are the percent changes in 3 month MA of Japan (U,S.$), U.K. Market (U.S.$), Japan (Yen) and U.K. (U.K.£). In the post processing stage, the forecast of the moving average was re-transformed to the actual index. This step was carried out for better evaluability, statistical analysis and display options.

The only major information that can be drawn from this weight structure is that the most important input node seems to be the 3-period moving average of the German market, expressed in U.S.S. This is quite surprising, since the top four weights are all non U.S. market nodes. Even when the U.S. market is important, it is not the 3 month percent change of the moving average which is more important, but the 2 month percent change of the moving average. The 3 month percent change of the MA is important, but on the lower end of the 10 highest weights. It would not be appropriate to judge the weight structure or even try to explain the reasons behind it, because of the non-linear structure of the model, therefore the normal linear procedure of explanatory statements as "since x is highest in some criteria while determining/explaining y, x is the reason or some explanatory variable of y", cannot be used while using a non-linear procedure.

The in-sample forecasts of the NN system in Fig. 7 are quite good. The generally accepted meaning of the accuracy and efficiency of a model is to take a look at the correlation coefficients between the actual and the predicted series as well as the root mean square error. As mentioned earfier, the 4-period moving average is the series being predicted, but for comparison's sake, the statistical analysis has been conducted with respect to the decomposed series. The statistical analysis of the prediction process is given in Table 3, with the out-of-sample performance in Fig. 8.

To evaluate the NN forecasts for portfolio investment, two criteria were used. The first one is how well does it classify the future direction (trend) and secondly how well does it forecast the absolute

Table 3. Statistical analysis of the actual vs predicted values (NN model)

Variable Number of pairs Correlation 2-tail significance Mean Standard deviation RMS error

Actual, in-utmple 115.000 0.964 0.000 233.385 73.697 20.264 Predicted, in-ample 233.618 75.340

Actual, out-of-uunple 23.000 0.866 0.000 404.312 17.303 10.380 Predicted, out-of-rumple 406.904 19.067


4 0 0 - -

"i 290

190 Q Actual

¢ Predicted

o I I I I I I 1 I I I I I I I t I I I 1 I Jun-82 lan-83 Jan-84 Jua-8$ Jun-86 Iun-87 Jun-88 lun-89 Jan-90 |un-91

Dec-82 Dec-83 Dec-84 Dec-85 Dec-86 Dec-S7 Dec-88 Dec-89 Dec-90 Dec -91

Month/Year

Fig. 7. In-sample forecasts of the N N model.

4 4 0 - -

420 --

490

380 - - _ O Predicted

3e0 I I I I I I I I I I I I I I I I I I I I I I I Jan-92 Mar-92 May-92 Jul-92 Sup-92 Nov-92 Jan-93 Mar-93 May-93 Jul-93 Sep-93 Nov-93

Feb-92 Apt-92 Jun-92 Aug-92 Oct-92 Dec-92 Feb-93 Apr-93 Jun-93 Aug-93 Oct-93

Month/Year

Fig. 8. Out-of-sample forecasts of the N N model.

next per iod value, or the change in m a g n i t u d e [16]. As such, for each pair o f predicted and actual values, we can determine the two types o f errors. For Type A representing the trend predict ion, and Type B represent ing the abso lu te value predict ion, the ca lculat ions are made as fol lows:

Type A Parameter Successful Predict ion {P( t ) - A ( t - 1)} • { a ( t ) - A( t - 1)} > 0

Type B Parameter Successful Predict ion {P( t ) - P ( t - 1 ) } , {A( t ) - A ( t - 1)} > 0

Table 4. Percentage of Type A and B errors in NN performance

Set Type A (Trend) Type B (Absolute)

In-sample forecast 53.91% 58.77% Out-of-sample forecast 65.22% 47.83%


4 6 0 - -

440

420

4OO

380

o Actual

360 0 Predicted

340 I I I I I I I I I I I I I I 1 I I I I I I I Jan-92 Mar-92 May-92 Jul-92 Sep-92 Nov-92 Jaa-93 Mar-93 May-93 !ul-93 Sep-93 Nov-93

Feb-92 Apr-92 Jan-92 Aug-92 Oct-92 Dec-92 Feb-93 Apt-93 Jan-93 Aug-93 O¢t-93

Month /Year

Fig. 9. Out-of-sample forecasts of the regression model.

where P is the predicted price and A is the actual price, t is current period and t - 1 is current period less one. Given these parameters, the performance of the neural network model is as in Table 4.

4. T H E S T A T I S T I C A L R E G R E S S I O N M O D E L

The multiple regression model was constructed using the same parameters as the neural network model, with the same inputs and data horizon. The regression model was based on the stepwise entry method, with the final selection of variables comprising the model, the % change in the 3- month MA of U.S. (U.S.$), UK (£), Germany (Din) and Japan (Yen). The out-of-sample forecasts are illustrated in Fig. 9. The 4-period moving average series is the series being predicted, but the statistical analysis was carried out on the decomposed series. The related statistical results are as in Table 5. The related Type A and Type B error statistics are shown in Table 6. The regression model performs poorly when compared with the NN model, with decrease in performance as compared to the correlation, RMS, distribution and the Type A and B errors. This suggests that the non-linear component of the Japanese series is relatively larger, consequently the better performance of the NN model.

5. T H E S T A T I S T I C A L A R I M A M O D E L

For further analysis and comparison, it was decided to execute an ARIMA model for evaluating the performance of the NN model. For the sake of greater evaluability, it was decided to develop

Table 5. Statistical analysis of the actual vs predicted values (regression model)

Variable Number of pairs Correlation 2-tail significance Mean Standard deviation RMS error

Actual, in-sample 114.000 0.985 0.000 232.335 74.238 13.163 Predicted, in-sample 234.520 74.274

Actual, out-of-sample 23.000 0.825 0.000 404.312 17.303 15.526 Predicted, out-of-sample 407.280 25.048

Table 6. Percentage of Type A and B errors in regression performance

Set Type A (Trend) Type B (Absolute)



460 - -

44O

420

4OO

380

Q Actual

360 <> Moving average

A Index

34o I I I I I I I I I l I I I I I I I I I I I I I ran-92 Mar-92 May-92 Jnl-92 Sep-92 Nov-92 Jan-93 Mar-93 May-93 Iul-93 $ep-93 Nov-93

Feb-92 Apr-92 Jan-92 Aug-92 Oct-92 Dec-92 Feb-93 Apr-93 Jun-93 Aug-93 Oct-93

Month/Year

Fig. 10. Out-of-sample forecasts of the ARIMA models.

ARIMA models for the 4-period moving average as well as the actual index. The autocorrelations of the two variables is given in Table 7. Referring to the autocorrelations, the first order differencing is the most important, hence, for the MA model, the form would be (p, q, r) = (1,0, 1), where p is the auto regressive order, q is the order of differencing and r is the moving average order and for the index model, the form would be (0, 1,0). The error statistics for the models follow in Table 8, while the Type A and Type B errors are shown in Table 9. The performance is clearly around the level established by the NN and multiple linear regression model in the out of sample case. The out of sample forecasts are illustrated in Fig. 10.

6. A D I S C U S S I O N O F T H E R E S U L T S

A summary of the statistical results for the out-of-sample case is given in Table 10. The performance of all models in the in-sample part of the series is quite good. The real test of a model's performance is in the out-of-sample case. As can be seen in Table 10, the models have been evaluated in terms of correlation (movements of one variable against other), RMS error (the error between the actual and predicted), mean and standard deviation (for measuring the distribution of the series), and the two Type A and B errors for prediction of the direction and absolute price prediction. According to the practitioner literature, the RMS error and Type A and B errors are the ones which count when evaluating a model. So, on those counts, the NN model performs better than

Table 7. Autocorrelation studies

% Change in 4-period moving average of index Actual index

LAG (Month) Autocorrelation statistic

1.000 0.711 0.960 2.000 0.411 0.926 3.000 0.110 0.891 4.000 -0.144 0.857 5.000 -0.106 0.823 6.000 -0.120 0.790 7.000 -0.100 0.758 8.000 -0.115 0.724 9.000 -0.162 0.693

10.000 -0.099 0.669 11.000 -0.102 0.641 12.000 -0.095 0.617


Table 8. Statistical analysis of the actual vs predicted values (ARIMA model)

Variable 4-period moving average Number of 2-tail Standard RMS of the index pairs Correlation significance Mean deviation error



Number of 2-tail Standard RMS Variable actual index pairs Correlation significance Mean deviation error



the other models, except for the absolute price prediction. Furthermore, the statistical distribution of the out of sample predicted series (NN) is also closely matching to the actual series, with the closest mean and standard deviation.

Another area of interest is the weight structure which comes out in the case of the N N model as well as the multiple regression. As has been mentioned, the regression model was developed on the stepwise method, and the major variables detected were the % change in the 3-month MA of U.S. (U.S.$), U.K. (£), Germany (Din) and Japan (Yen). This is logical since from a linear viewpoint, the most important variable is the % change of the 3-month MA of U.S. for forecasting the % change of the 4-month MA. Another surprising result was that the other variables are all important variables, thus giving rise to the explanation that the movements in the U.S. Index are driven to a large extent by itself, but even so, other markets are also important in explanation of the U.S. Index. This still leaves the point that the other indices are expressed in their local currencies and not in terms of the U.S. Dollar. One of the likely explanations for this result is that the currency movements of the local currencies: the British Pound, Japanese Yen and German D-Mark against the U.S. Dollar are not that important. In other words, the movements in U.S. Index and its relationship with the U.K., Japan and Germany does not involve the currency movements, but purely the movements of the indices themselves.

On the other hand, the N N model comes up with different variables which are important. The percent changes in 3-month MA of Germany (U.S.$), Japan (U.S.$), U.K. Market (U.S.$), Japan (Yen) and U.K. (U.K.£) are the important variables, surprisingly, in this case, all of the markets are denominated in U.S. Dollars, with Japan also given in Yen. One of the primary explanations of this

Table 9. Percentage of Type A and B errors in ARIMA performance

4-period MA of Index Type A (Trend) Type B (Absolute)


Actual index Type A (Trend) Type B (Absolute)

In-sample forecast 62.18% 49.58 % Out-of-sample forecast 60.87% 56.52%

Table I0. Summary of statistical analysis of the actual vs predicted values (out-of-sample)

RMS Standard Trend Price Variable Correlation error Mean deviation succass % success %

Actual 404.312 17.303 Predicted, MA, NN 0.866 10.380 406.904 19.067 65.22% 47.83% Predicted, MA, Resrexion 0.825 15.526 407.280 25.048 56.52% 39.13% Predicted, MA, ARIMA 0.878 12.887 412.526 19.130 60.87% 60.87% Predicted, Index, ARIMA 0.844 15.086 414.549 18.727 60.87% 56.52%


phenomena is that there is an impact of foreign exchange movements. This in itself a very significant result. Presently, the movements of the other indices are studied by the portfolio and fund managers, but this study shows that movements of other indices are important when studying the U.S. market. We conclude that it is also important to consider the movements of the exchange rate. This is proved by the fact that the NN model based on non-linearities is performing better than the linear regression model, which is catching only the index movements in the local currency and is performing worse. Another interesting finding is that the ARIMA model performed quite well. This means that there is sufficient information embedded in the index itself, which is sufficient to forecast quite adequately in the out-of-sample period. Furthermore, the ARIMA model which was built on the % change in 3-period MA is performing better than the ARIMA model build on the index itself, proof that the process of cleaning up and preprocessing the data does help in building better models.

7. CONCLUSIONS

This study gives rise to two conclusions. One is that the neural network model is better or comparable with the commonly used forecasting techniques such as regression analysis and the ARIMA models. The performance of the NN, is better than the 50% expected for a random walk, and has some advantages over the traditional statistical techniques such as self learning, tolerance of noisy data and identification and utilization of chaotic and non-linear data. Secondly, we would conclude that the American market is not efficient. If N N can comfortably improve on the 50% benchmark accuracy level, then there is prima facie' evidence that an efficiency conclusion is a product of poor forecasting techniques. It might be interesting to undertake further research on this aspect of the applications of neural networks. Although the literature is inconsistent on whether NN's can provide consistent above average returns, it has been mentioned that neural networks have still not created systems that can systematically generate profits or else at least be as good as a human trader [17]. The present study suggests that on a consistent basis, this N N model can make money, even when transaction costs are included. As Penrose [17] mentions, a very good human trader can get it right about 55% of the time, while the study that we conducted gives a performance of predicting the trend 65% of the time and the absolute price about 48% of the time. Further research is of course, required to further elucidate this position, but on the whole, it seems that the application of neural networks in the field of forecasting is showing great promise.

REFERENCES

1. B. Riley, Pension funds investment. Financial Times 6 May, Section III (1993). 2. P. Harverson, Eyes are switching to overseas markets. Financial Times Pension Fund Management 6 May, Section HI (1993). 3. N. Cohen, Less passivity, more activity. Financial Times Pension Funds Investment, Financial Times Survey, 6 May,

Section V (1993). 4. E. F. Fama, Efficient capital markets: II. J. Finance 46, 1575-1617 (1991). 5. T.C. Mills, Non=linear time series models in economics. J. Econ. Surveys 5, 215-241 (1990). 6. M. B. Priestley, Non-linear andNon-stationary Tm~e Series Analysis. Academic Press, London (1988). 7. A. S. Weigend, D. E. Rumelhart and B. A. Huberman, Back-propagation, weight.elimination and time series prediction.

Prec. 1990 Connectioniat Models Summer Schools 1, 105-116 (1990). 8. A. B. Bulsari and H. Saxen, A recurrent neural network for time-series modeling. Prec. Int. Congr. Neural Networks

1, pp. 285-291 (1993). 9. J. M. Hutchinson, A. W. Lo and T. Poggio, A nonparametric approach to pricing and hedging derivative securities via

learning networks. J. Finance XLIX, 3, 851-889 (1994). 10. M. Maruyama, F. Girosi and T. Poggio, A connection between GRBF and RBF. Araficial Intelligence Memo 1291,

Massachusetts Institute of Technology, AI Laboratory (1991). 11. H. White, Economic prediction using neural networks: the case of IBM daily stock returns. Prec. Int. Congr. Neural

Networks ti, pp. 451-458 (1987). 12. H. Markowitz, Portfolio selection. J. Finance, March, 77-91 (1952). 13. F. Black and R. Litterman, Global portfolio optimization. Financial Analysts J. Sel)tember-October, 28-43 (1992). 14. M. Jurik, Going fishing with a neural network. Futures Se~ember, 38-42 (1992a). 15. M. Jurik, Trading techniques: the case and feeding of a neural network. Futures October, 40-44 (1992b). 16. G. Papadourakis and G. Spanoudakis, Application of neural networks in short term stock price forecasting. Prec. First

Int. Workshop on Neural Networks in the Capital Markets 1, pp. 200-205, (1993). 17. P. Penrose, Star dealer who works in the dark. The Times 26 February, 28 (1993).

Readers may address inquiries to Professor Wood at: [email protected]

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