** **

**optimal
investment policies and oscilLators of stockmarket technical analysis. Application in the Impact of the war in
yugoslavia to the Greek stockmarket.**

*By*** ****Dr. Costas Kyritsis**

*Software Laboratory *

*of Athens*

** **

** **

**Abstract**

** **

In this paper
we analyze some price oscillators of Stockmarket technical analysis as linear
digital convolution filters. We review some of the theoretical published
research on optimal trading policies in stockmarkets and we give an application
of this approach in the negative shock impact of the war in Yugoslavia to the
Greek Stockmarket.

**Key words: Portfolio** Selection, technical analysis, stochastic optimal
control, digital linear filters, time series. JEL C3, C4, C5

** **

**1.Introduction**

In this paper we shall not investigate all the
negative impacts of the war and bombing in Yugoslavia, to the Greek Domestic
Economy, but we shall take the opportunity, to study only the shock-waves to
the Athens stock exchange market.

The
problem of profitable and optimal trading in Stockmarkets is the major choice
which investors are facing daily. On the other hand, such decisions are also
models of investment decision in other areas of Finance and Economics. The
solutions that are discovered for Stockmarkets are often formally the same with
solutions in the Microeconomics of firms or even of policies in Macroeconomics
of the Government. Academic research has considered the subject of much worth
for investigation and since 1967 till today, many papers, from famous
economists, has been published about it. It was soon understood that even
relatively simple and common problems that are decided empirically are far from
simple in their full theoretical formulation. Any one that has attempted to
apply any of the theoretical solutions in real situations of Stockmarkets has
realized that academic publications have very general and not easily applicable
abstractions. An example is the
utility function . At the same time they put very restrictive assumptions about
the movement of prices, that are rarely met with such simple symmetry, in
reality. A question is early put: “Is
academic research supporting the empirical trading rules of technical
analysis?” There are strong opponents of the concepts of the academic research,
among the technical analysts and investors, that would reverse the question to:
«Does reality of Stockmarkets supports the results of academic published
research?» It is to admit nevertheless that some of the results of academic
research make use of very sophisticated formulations, like the stochastic
differential equations, stochastic optimal control, Bellman’s maximum principle
etc., with which is familiar only a
small minority of the investors.The question of course remains if the above
results are enlightening and applicable even for the few that have familiarity
with all the involved formulations. The author’s experience is that the
academic publications are often much too complicated for direct applications.
They are nevertheless encouraging. Any serious application must involve
computer experiments and numerical specification of parameters for particular
Stockmarkets, that bridge the gap of theoretical concepts and practical
results. Academic research seems to be interested to describe the behavior of
investors and for this they introduce the utility functions of the profits,
which is not determined and is assumed to summarize the subjective or
irrational behavior of the investors. If this function is eliminated many of
the published results became trivial for trading applications. On the other
hand there are some important exceptions to this and furthermore the
theoretical formulation, even only with an abstract existence of solution,
offers much in the confidence of the investor for an appropriate trading system
that can bring to him profits with a consistent, best possible and rationally
explained method.

In
this paper we analyze explain popular price oscillators like the PrOsc
(5-70/50) and RSI, by abandoning the «null-oscillations» assumption, and by
considering oscillators as
discrete filters that extract
oscillations of the regression path . We give also a stochastic differential
equation formulation, in ITO’s calculus, of the concept of oscillations. We
give a formulation of the Eliot’s wave theory.
Finally we apply a discrete time version to study the impact of the war
in Yugoslavia to the Greek Stockmarket.
An almost clear oscillation appears and a lowering of the momentum of
the growth trend.

** **

** **

**2. A short review of some of the
published academic research findings.**

The
first papers to formulate and solve the problem of optimal portfolio selection
and trading, were two papers by Samuelson and Merton in the same volume of Rev.
of Econ. Stat. in 1968 (see sAMUELSON
p.a. (1968), MERTON R.C (1968)) Samuelson solved it in discrete time and
Merton in continuous time. The later became known in the bibliography from then
on as the «Merton problem». In discrete time the solution was obtained using
dynamic programming of Bellman and in continuous time by using stochastic
differential equations and ITO’s calculus. Both authors, nevertheless, assumed
the possibility of costless transactions, thus portfolio adjustments could be
as often as one would like . Both authors solved it under the generality of a utility
function of profit and assuming that there was consumption at every time step
of a part of the investment, by the investor. The criterion of optimality was
the maximization of the total consumption in infinite horizon. Although the
costless transaction assumption was not realistic the reader got the idea of
the general form of an optimal trading tactic. The solution was among two
alternative assets, a risky stock and a riskless bank deposit or bond. The
optimal trading strategy that switches between these two alternatives was
synonymous with the problem of optimal portfolio selection, as the portfolio
was only these two alternatives. There was not given any solution for more than
two assets and the stock was assumed to have constant (exponential) trend. The
latter was an oversimplifying assumption, as we mentioned in the introduction,
from which nevertheless almost all-later advancements did not deviate. This
assumption that apart from the (exponential) trend the residual of the process
is white noise (or random walk for discrete time) is called in the bibliography
«the null» assumption. Almost all-later solutions, focused on infinite horizon,
were the existence of a stationary optimal policy was easy and possible to
obtain.

Many
years passed till the first publication of the solution of the same problem in
continuous time but with transaction costs (see Constantinides G.M. (1979)*,
*MAGILL m.j.p.- Constantinides G.M.
(1976)).

The papers by
Constantinides G.M. were pioneering and ahead of their time. They tried to
solve problems that required techniques of stochastic differential equations
that were known only to a very small minority of experts in the field and they
were not economists. Later an elegant and rigorous solution to the same problem
was published by Davis M.H. and Norman A. R. (see DAVIS M.H.-NORMAN A.R.
(1990)). The transaction costs are assumed to be a fixed percentage of the
amount in the transaction. The same problem was solved again under more general
forms of the transaction costs like convex and quadratic functions. The
solution was still for two only assets. For the first time appeared in the
publications the concept of «optimal buffer-region» or «optimal brake-region».
The optimal portfolio was not adjusted to changes of the stock prices, because
of the transaction costs until the first time deviation drives the previously
determined optimal portfolio outside an «optimal buffer-region» or «optimal
brake-region».

An
other author (see DUFFIE D. -SUN T. (1990)) solved the same problem but under
the additional assumption that each time a withdrawal (consumption) or
portfolio adjustment transaction is taking place there is, except of the
percentage transaction costs also a fixed cost as transaction cost or
management fees. This assumption, which is very realistic as each time a
trading offer is put in the computers a small constant cost occurs , turns
surprisingly the continuous time problem to a discrete one .It is proved that
the optimal policy requires transactions at constant time intervals! The
situation seems to be similar to the optimal time of ordering in inventory
control.

Other
authors (see DUMAS B. -LUCIANO EL. (1991), TASKAR M. -KLASS M.J.-ASSAF. D.
(1988)) solved the same problem as Davis and Norman, but without assuming
consumption. They required maximization is of the average value of the final
(in the infinite) utility of the wealth.

A
still later publication (see BROER D.P.-JANSEN W.J. (1998)) solved finally the
same problem but for more than two assets so that the term «optimal portfolio
selection policy» took its really literal meaning.

The
solution proved that the optimal portfolio was not at all in general
«mean-and-variance efficient»! This was surprising for many authors as almost
all of the bibliography in the theory of Portfolio Selection adapted the
approach of «mean-and-variance
efficient» portfolios (see ELTON E.J.-GRUBER M.J. (1991).

An
approach by far more risk averse and different from all the previous is to take
the optimality criterion to be not the maximization of the average value of a
utility but of the probability to succeed profits above a level, or of the
speed to succeed profits above some level. (See ROY S. (1995), HEATH D. -OREY S. -PESTIEN V. -SUDDERTH W.
(1987), PESTIEN V.C.-SUDDERTH W.D. (1985)* *SUDDERTH W.D.-WEERASINGHE AN (1989), BREIMAN L. (1961)) There have
been published results both in discrete and continuous time and finite and
infinite horizon. In continuous time and infinite horizon the results are
easier to obtain.

This
approach is related to the slogan «safety first». It is certainly by far more
valuable for applications as it avoids the abstract utility function for all
time steps. It may be relevant nevertheless to sequentially utility approaches
that have a separate utility function for each step (see PHELPS E. (1962)*,* SVENSSON L.E.O.* *(1989)*)*.

At
a first glance it seems that the continuous time assumption complicates the
process of solution .We know much more about time series than stochastic
differential equations. We know how to handle only a few small classes of
stochastic differential equations that are Gaussian processes. In addition the
formulation for infinite horizon may seem that complicates the solutions. But
both assumptions have as net result a crucial simplification of the optimal
solution! It is like the law of large numbers in statistical kinetic theory of
gasses that has in the average much simpler equations (the equations of fluid
dynamics) than the exact statistical equations. In particular for finite
horizon and discrete time, in many cases of utilities it would be not possible
to find any optimal solution, which is stationary. But even for infinite
horizon the discrete time counterpart of many of the solved cases in DAVIS
M.H.-NORMAN A.R. (1990) have not yet been solved in spite the attempts (see ROY
S. (1995) .In this paper only some qualitative results are obtained. e.g. the
existence of a stationary solution, while in continuous time the same problem
has been solved!).

No
doubt, the problem of optimal portfolio selection and optimal trading is
separate from the problem of (optimal) forecasting. Any model of price
movements could be used and in the above publications was taken the simplest
possible .In other words of constant exponential trend as in the Black-Scholes
model.

There
are also papers that try to avoid any stochastic assumption for the prices and
they simply define optimality of the trading tactic according to profits
maximization when there is a suggested trading position for the next day
according to neural networks techniques. (See GENCAY R. (1998)). Such papers
concentrate on the forecasting problem and try to approach it in a way as
realistic as possible.

Neural
network techniques give new algorithms of time series forecasting by letting
the neural network to learn from past successes or failures. Extended research
on data for 10 years in Stockmarkets as published in the latter paper, proves
that the universal limit of forecasting the next day (even with the best time
series forecasting algorithms as are considered the neural network techniques)
is 60%. That is 60% of the next day forecasting (for the prices going up or down) are successful!

There
are also other approaches to the forecasting problem based on wavelets and
non-linear models of time series (see ZUOHONG PAN-XIAODI WANG (1998)).

* *

**3.The price oscillators of technical
analysis as digital linear filters and their frequency response.**

The
main objective in this paragraph is to introduce the use of spectral analysis.
We may use it to pickup qualitatively or quantitatively the frequencies of the
oscillations of the prices and then apply the more familiar moving average
indicators at the right time scale. In order to understand better the concepts
and techniques appropriate books of Fourier and spectral analysis should be
read. We apply in this paragraph spectral analysis also so
as to analyse the standard indicators as filters that cut some frequencies and
let other frequencies pass. In this way we may understand what we may expect
from an indicator in advance, and we may compare them in a unified way. In
addition, the frequency response of an indicator , shows if this indicator
filters a particular frequency from the others, or in other words if its
designed to forecast waves or oscillations of a particular period.

** **

A
moving average oscillator, denoted usually by PrOsc (n-m/k) is defined as
follows:

a)
First we take the average value of the last m stock’s price values q (m)

b)
Then again the average value of the last n stock’s price values q (n)

c)
Then we take the average value q (k) of the last k values of q (m)-q (n)

The
oscillator is defined by the «crossing the lines rule» of the curves q (k) and
q (m)-q (n). The “crossing of lines rule”
sais that a buying signal is given each time the curve q (m)-q (n)
crosses the curve q (k) from lower to higher and a selling signal each time the
crossing is from higher to lower.

The
q (k) is a linear function (in fact a convex combination or weighted average)
of the stock’s last n+m+k prices.

q
(k)=p(n)*c(1)+...+p(n-(n+m+k))*c(n+m+k) (1)

with
c (1)+...+c(n+m+k)=1 (2)

Therefore
it is identified as linear (digital) convolution filter of the time series (see
Koopmans Lambert H. (1995) Chapter
4, 6). It has a finite number of
weights. Its frequency response is defined as the action of the filter on a
base of functions, which is the harmonic (trigonometric) oscillations. As by
Fourier analysis any finite sequence can be written as a linear combination of
the sine and cosine oscillations, and the filter is linear, knowing the filter
is equivalent to knowing how it filters the sine and cosine oscillations at all
frequencies .Let a linear filter denoted by L .It can be proved that the
filtering of any harmonic oscillation exp(iat) (written in complex exponential
form) is simply multiplication by a complex constant :

L(exp(iat))=B(a)
exp(iat) (3)

The
complex function B(a) is called the transfer function of the linear filter L.

For detailed definitions and proofs of the
properties of the transfer function see Koopmans
Lambert H. (1995) Chapter 4 pp
82-83. In particular the simple moving average filter of backward
horizon of k steps has frequency response:

B(a)=exp(-ia(k-1)/2)sin(ak/2)/k*sin(a/2), -π<a<π (4)

(see
again Koopmans Lambert H. (1995) Chapter
6 example 6.2 p 171.)

From
this it is not difficult to derive the frequency response of the PrOsc filter.
Subtracting two linear filters has as transfer function the subtraction of their transfer functions
while composition of them has as frequency response the multiplication of their
frequency response.

As
it is reported in DIMOPOULOS D (1998) ,
research on data of the Greek Stockmarket
proved that the most profitable results by trading with such price oscillators and the «crossing of
lines rule” are obtained with the PrOsc(5-70/50).

The
frequency response of this price oscillator
is :

B(a)=(exp(-ia2)(sin2.5a)/5sin(a/2)-exp(-i35a)sin(35a)/70sin(a/2))(1-exp(-25a)sin(25a)/50sin(a/2)) (5)

Its
absolute value multiplies determines
the effect of the filter on the amplitude of the oscillations and its argument
the shift on the phase .

The
next is a program in visual basic at excel that computes the norm (gain) of the transfer function of
PrOsc(5-70/50) with steps of 0.05 in the frequency domain. The unit of time is
one day.

Sub
ConvolutionFilterfrequencyresponse()

Dim
x As Single

Dim
y As Single

Dim
h As Integer

x
= 0

For
h = 1 To 90

x
= x + h * (0.05)

y
= Sqr((((Cos(2 * x) / 5 * Sin(x / 2)) - (Cos(35 * x) / 70 * Sin(x / 2))) ^ 2 +
(((Sin(35 * x)) ^ 2 / 70 * Sin(x / 2) - (Sin(2 * x) * Sin(2.5 * x) / 5 * Sin(x
/ 2))) ^ 2) * ((1 - Cos(25 * x) / 50 * Sin(x / 2)) ^ 2 + ((Sin(25 * x)) ^ 2 /
50 * Sin(x / 2)) ^ 2)))

Workbooks("prosctf.xls").Worksheets("sheet2").Cells(h,
1).Value = y

Next
h

End
Sub

The
results of the previous computations are shown in the next diagram.

**Figure 1 Frequency response of the
PrOsc(5-70/50).**

The x-axis is the frequency domain
with unit of time one day.

** **

** **

**4.The simple moving average
oscillators and their frequency response.**

Except
of the two-moving averages oscillator it is also used in technical analysis
the simple moving average oscillator .
We take again a moving average of k
last prices and then the difference of it from the actual price of that the
present day, in other words the residual of the filter. From the definition of the frequency
response (or transfer function) (see Koopmans
Lambert H. (1995) Chapter 6) we
obtain that the frequency response is :

B(a)=1-exp(-ia((k-1)/2)sin(ak/2)/(ksin(a/2)) (6)

In
–π<a<π

For
a moving average of 10 days we get the next diagram of the absolute value of
the frequency response:

We
use again the previous script in visual basic.

The
x-axis counts steps of length 0.05 in the frequency domain. The unit of time is
taken to be one day.

**5. The momentum oscillators of
technical analysis as linear filters and their frequency response.**

In
technical analysis momentum is defined as the :

D(k)=price-(price
before k days) (7)

In terms of the lag operator of time series
L(x(n))=x(n-1) we may rewrite the k-order momentum D(k) as

D(k)=1-L^{k } (8)^{ }

We
estimate the frequency response of this operator to be:

B(a)=
1-exp(-iak) -π<a<π (9)

The
frequency response of the Lag operator L is exp(-ia) (10)

In
the next figure we see the frequency response for the momentum of 10 days.

The x-axis is the frequency with time unit one day .

**6. The RSI oscillator and the coefficient
of variation (relative standard deviation)**

There
is in technical analysis an other price oscillator called relative strength
index (RSI) that was introduced by
J.W.Wilder and presented in 1978.(see MURPHY J.J. chapter 10 p 295).

It
is defined by the equation:

RSI
in%=1-(100/(1+(sum of daily price units gained
only in the upward days during the last k days )/( sum of daily price
units lost only in the downward days during the last k days))) (11)

Let
us denote by D(x(n))=x(n)-x(n-1) then as the “sum of daily price units
gained only in the upward days during
the last k days”=sum of D(x(n))
for n=n-k to n, we rewrite it by simplifying the formula and we get an equivalent form. We put
(x(n)-abs(x(n)))/2 and
(x(n)+abs(x(n)))/2 for the price points
gained in the nth down or up days
respectively .Then with simplification
on the quotients we get the next

(12)

We
denote by x with bar the average of the signed price points gained in k-days
(as are the smoothing days of the RSI) and by absolute value of x with a bar the average price points in absolute
value gained in k days.

For
normal random variables it holds that

(13)

thus
the RSI becomes a simple formula of the coefficient of variation .

(14)

From
this we deduce that this oscillator and its success is not accidental but is related to a well known
and very useful coefficient in
statistics.

**7. Eliot’s wave theory .**

There
is a very attractive old and classical theory of the movement of prices in
stockmarkets, known as «Elliot’s wave theory».
A concise description of the theory can be found in Murphy J.J. chapter 13 pp. 371-413 (see
also ELLIOT,R.N. (1980) and FROST ,A.J.-PRECHTER R.R.(1978)). This theory has
come out from long experience and is mainly an informal and empirical theory.
We assess that this theory supports our approach that there are 1st moment
oscillations in the time series. We shall give a formulation to some of the ideas of the theory with well-known
wave equations that are partial differential equations . The basic tenets of
the Elliot Wave theory are three and in the next order of importance : a)
pattern b) ratio c)time . The main pattern according to Elliot’s theory is the
wave pattern. The basic ratio of importance is the retracement ratio. In other
words after an upward movement and a reversal of the trend ,what percentage of
it is the downward movement .Finally timing according to Elliot is described
very often with numerical relations that come from the Fibonacci numbers
(x(n+1)=x(n)+x(n-1), x(1)=1=x(2)).

A
summary of the Elliot’s approach is contained in the next statements:

*0) A wave is a monotonic movements
after and before two other movements in the opposite direction.(we notice that
a more correct term would be monotonic growth impulse rather that wave. Then we
could define as wave or oscillation two successive monotonic growth impulses
that alternate .)*

* *

*1) A trend is divided into three waves three longer in the direction of the trend and two shorter in the reverse direction ,in total five waves. Opposite
direction waves alternate making a zigzag .*

* *

*2) A trend correction or
retracement is divided into three waves
. Two longer in the reverse direction of the trend and no shorter in the direction of the trend . Opposite direction
waves alternate. (Elliot classifies many variations of the main correction
pattern calling them zigzags, flats, triangles, and double and triple threes.)*

* *

*3) An upward market cycle is composed from eight waves ,five up waves making an upward trend then
three down waves .Thus it is composed from an upward trend followed by a
downward trend correction.*

*Similarly is defined a downward
market cycle .*

*The next figure shows the basic
pattern*:

**Figure 4** The market cycle wave pattern of Elliot.

Wave5
wave8

Wave2

Wave1

*4)Waves can be expanded into waves
and subdivided into shorter waves.(This principle seems to be the forerunner of
the later theory of fractals of Mandelbrot)*

* *

*5) The number of waves in trends and
trend corrections follows the Fibonacci sequence. (We should not fail to notice
that the simplicity of the Elliot’s approach that combines waves and simple
numbers ,met even in patterns of flower shapes ,reminds of Pythagorean ideas of
patterns for understanding the world)*

* *

*6) Fibonacci numbers are used to
estimate the retracement ratios .The most common retracements are 62% ,50% and
38% .*

* *

*7) The theory was originally applied
to Stockmarket averages not on individual stocks and it works better in those markets that the largest public is
involved and the laws of large numbers hold better.*

If
we would like to translate Elliot’s concepts of wave ,trend and market cycle in
to our approach with time series we would make the next correspondence:

trend=
the superposition of a trend in the form of monotonic non-periodic regression
path (1st moment )of a time series and an oscillation of the regression
path during its period plus of course
some random innovation.

market
cycle=the superposition of an non-periodic trend part of the regression path
and two oscillations of the 1st moment
at different periods ,plus again a random innovation

The
concept of Fibonacci numbers is very close also to the concept of recursive and
autoregressive relations ,*except that it
is not applied only in the time domain but also in the frequency domain*.
The latter is an approach that only relatively recently has been implemented
for forecasting in time series through the neural networks and multi-resolution
training of them .(see SCHALKOFF R.J .(1997) chapter 6 p146)

** **

Strictly speaking a wave in physics is not only an oscillation in time
but oscillations distributed in space also such that their phase difference
makes at any instant a waveform as the
inter-temporal waveform of an oscillator . In Stockmarket prices it is not
obvious what magnitude would play the role of space location. A suggestion was
given in the paper KYRITSIS C (1999) of the author, that we may define waves of
demand and supply that result to oscillations of prices and volume, but this
would be a totally new and different study that we not pursuit in this paper

**9. The ITO stochastic
differential equation of linear harmonic oscillator .**

It is worth formulating price oscillations with real stochastic
differential equations .

(21)

The
discrete time formulation of this model is:

p(n)=(bsin(ωn+c)+r+ε(n))p(n-1)

With b we denote the amplitude of the price rate r, per time unit , with p the price at time t, and B is a Brownian motion .The constant α gives the frequency of the oscillation ,
the constant c the initial phase of the oscillation and the σ the volatility or standard deviation of the rate. This ITO process is a
lognormal process. If b=0 then the solution of this equation is

p(t)=Pexp((r-0.5σ^2)t+B(t))

An other choice would be to postulate an explicit equation like the
next:

p(t)=Pexp((rsin(at+c)-0.5σ^2)t+B(t)) (22)

In the next paragraph we apply and estimate a normal rather than
lognormal discrete time model on real
data of the Greek Stockmarket . The
innovation and oscillation is on the price and not on the rate .The model has
equation

p(n)=a*n+b+ c*sin(ω*n+φ)+ε(n)

That is a superposition of a linear trend with an oscillation trend.

**10. An application to the impact of
the war in Yugoslavia in the Athens Stockmarket.**

Application
to the data of the last 40 stockmarket days of the general index of the Athens
Stockmarket till the end of April with straight-line trend and no retracement
phase shift, give for the model the
estimated parameters (we apply here the model not of the oscillating rate but
of the oscillating prices ).The computer solves such models with algorithms of
non-linear optimization. As the relevant theorems are only necessary conditions
and not sufficient and necessary ,the computed optimal fit depends much on the
initial values put by the computer .After experimentation by giving “best fit”
initially from graphical observation we estimated the parameters as shown below
.The amplitude and period was determined mainly graphically. The average path
has equation:

**p(n)=( -0.598550)n+( 3577.854)+
300*sin((6.28/14)*n+(198534.7)) **(23)

The day parameter n takes value from 80 to 120
and is till the end of April .

The reader should be warned nevertheless, that a high
goodness of fit of a forecasting model, for a particular short time interval,
as the above, is not adequate for a repetitive, trading based on it and for a long time (years). For a model to
be used for repetitive trading and for a long time (years), it should be tested
that for the goodness of fit at repetitive forecasting does remains high for long
times intervals, that must me at least 2 to 5 years, but even better 20-25
years.

The next
figure and tables describe the results.

**Figure 5**

** **

** **

** **

** Convergence
achieved after 8 iterations **

** **

** **

** Coefficient Std. Error t-Statistic Prob.
**

** **

**C(1) -0.598550 1.833884 -0.326384 0.7460**

**C(2)
3577.854 185.3935 19.29871 0.0000**

**C(5)
198534.7 10.17225 19517.28 0.0000**

** **

**R-squared 0.480133
Mean dependent var 3514.002**

**Adjusted R-squared 0.452032 S.D. dependent var 174.9727**

**S.E. of regression 129.5233 Akaike info criterion 9.799759**

**Sum squared resid 620722.1 Schwarz criterion 9.926425**

**Log likelihood -249.7527 F-statistic 17.08603**

**Durbin-Watson stat 0.640855 Prob(F-statistic) 0.000006**

** **

**Covariance matrix**

** C(1) C(2) C(5)**

**C(1)
3.363131 -337.9093 4.723391**

**C(2) -337.9093 34370.74 -472.8303**

**C(5)
4.723391 -472.8303 103.4747**

** 3334.86 3427.10 -92.2389 **

** 3444.09 3314.31 129.781 **

** 3467.28 3243.95 223.327 **

** 3471.98 3229.84 242.144 **

** 3537.90 3274.63 263.270 **

** 3625.20 3369.36 255.845 **

** 3598.09 3495.15 102.940 **

** 3630.82 3627.01 3.81423 **

** 3660.26 3738.71 -78.4539 **

** 3774.29 3808.05 -33.7632 **

** 3759.45 3821.19 -61.7353 **

** 3637.80 3775.39 -137.594 **

** 3471.48 3679.62 -208.140 **

** 3548.46 3552.70 -4.23638 **

** 3496.31 3419.62 76.6919 **

** 3511.02 3306.60 204.422 **

** 3376.37 3235.88 140.489 **

** 3121.39 3221.34 -99.9503 **

** 3303.49 3265.73 37.7551 **

** 3218.06 3360.16 -142.102 **

** 3300.20 3485.82 -185.620 **

** 3535.66 3617.73 -82.0667 **

** 3621.53 3729.66 -108.134 **

** 3720.16 3799.36 -79.2048 **

** 3732.64 3812.92 -80.2797 **

** 3640.94 3767.53 -126.588 **

** 3559.32 3672.05 -112.733 **

** 3386.09 3545.27 -159.177 **

** 3373.62 3412.14 -38.5183 **

** 3350.56 3298.89 51.6701 **

** 3281.96 3227.81 54.1487 **

** 3312.88 3212.85 100.032 **

** 3252.09 3256.84 -4.75230**

** 3431.31 3350.97 80.3392 **

** 3549.70 3476.49 73.2097 **

** 3523.54 3608.45 -84.9066 **

** 3617.42 3720.61 -103.192 **

** 3806.33 3790.67 15.6565 **

** 3776.18 3804.65 -28.4710 **

** 3799.36 3759.66 39.7000 **

From these
results we see that the impact of the war of Yugoslavia is an almost perfect
oscillation of period 14 days and amplitude 300 units of the general index .

Also if we
re-estimate the model in earlier days we find that the slope of the trend becomes
less as a result of the war. This analysis of the impact of the war is
obviously only in short terms and only based on the behavior of the investors
in the Stockmarket. A more complete analysis of the economic impact of the war
in Yugoslavia should include the effects on the Greek tourism, the chances of
cooperation of the Greek industry with Yugoslavia , the changes in the Labor
wages in the Greek industry because of emigrant workers ,in general changes in
the cost of labor in the Greek industry
(see also KIOCHOS P.(1993)) and effects on changes of the cost Greek military
resources. As any war it has negative impact on capital markets and the
Domestic Economy in Greece.

** **

**11. Conclusions**

We
summarize our conclusions:

1)
The assumption that
after subtracting a constant rate trend from the stock prices the residual is a
stationary time series and any oscillations are therefore of the variance
(Box-Jenkins and spectral analysis approach or null oscillations assumption) is
not seem to be supported from the statistical
data and traditional trading techniques of the investors.

2)
It seems more probable
that except of non-periodic part of the 1^{st} moment (regression path)
there is also a periodic or waving
part. Upon this it based the trading techniques with oscillators. There can be
performed of course statistical tests that they do not reject the
stationarity hypothesis but also analysis of variance that does not reject the
non-stationarity hypothesis (see also LIMA P.J.F. (1998). The choice of the
size of the horizon is crucial. After subtracting also the 1^{st}
moment oscillational part of the time series ,there might remain in the
residual a 2^{nd} moment (variance) oscillational part. Its
significance nevertheless for trading based on criteria of average value of profit
is by far less important .

3)
Econometrics of
stockmarkets seem to omit the analysis
of price time series with digital filters that extract oscillations of the
first moment . A reason is that digital filters were designed and computed
initially for signal theory. Nevertheless their mathematics are part of
statistics and time series and are very close relatives to the empirical
oscillators of technical analysis of stocks and commodities. The effect of the
empirical filters of technical analysis depends completely on the frequency
response of them.

4)
The impact of the war in
Yugoslavia to the Athens Stockmarket during March and April 1999 is best
approximated with an oscillation of the general index of amplitude about 300
units and period almost 14 days, plus a weakening of the ascending trend. It
has of course as any war a negative shock impact on the capital markets.
Further analysis, would show of course deeply negative effects of the war for
the Greek Economy, not necessarily related to the stock exchange market, but to
the Domestic Economy in General.

** **

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