Self-similarity, Heavy Tails and Long-range
Dependence as Measures for Financial
Market Inefficiency - the Case of Bulgaria 1
Boyan Lomev and Ivan Ivanov
Department of Statistics and Econometrics,
Faculty of Economics and Business Administration,
Sofia University “ St. Kliment Ohridski”,
125 Tzarigradsko chaussee blvd., bl.3, Sofia 1113, Bulgaria,
e-mail: lomev@feb.uni-sofia.bg
e-mail: i_ivanov@feb.uni-sofia.bg
The notion for Market Efficiency – so called efficient market hypothesis implies that security
prices fully reflect all available information. The random walk process is a restrictive version
of the weak form of the efficient market hypothesis and assumes that successive returns are
independent and identically (normally) distributed.
In that work several deviations of log-returns of Bulgarian Stock Exchange index from normal
distribution are explored – namely:

Heavy Tails;

Long-range Dependence;

Possibility for Forecasting on the basis of historical information.
The data consists of daily quotes of Bulgarian Stock Exchange index SOFIX and six major
world indices: DAX, DJIA, Hang Seng, NASDAQ, FTSE100 and NIKKEI for the period
2002-2008.
The heavy tails of the indexes are studied through the following methods:

Histograms

Empirical Moments

Maximum/Sum Ratio

Mean Excess Function

GARCH modeling
Empirical histograms for SOFIX and DJIA are presented below on Figure 1 as an illustration
together with theoretical histogram for the normal distribution.
1
This paper was financially supported under the Sofia University “St. Kl. Ohridski” research project 2009.
Soffix
DJIA
450
160
400
140
350
120
300
100
250
80
200
60
150
100
40
50
20
0
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Figure 1.
It is seen, that SOFIX has much more clearly expressed leptokurtosis than DJIA.
All other approaches also clearly show that SOFIX differ substantially from the other indexes.
For instance GARCH Models can explain the heavy tails for all Indexes except SOFIX.
The correlation function for standard ARIMA(p,d,q) processes decays rapidly. There is a class
of processes with fractional integration order d and if 0≤d≤0.5 then we have a long-range
dependence [1].
The long range dependence was studied by several methods, based on time and frequency
analysis of the data. In Table 1 results obtained by Whittle method [2] are presented.
Table 1
SOFIX
DAX
DJIA
Hang Seng NASDAQ FTSE100 NIKKEI
d=0.0987 d=-0.0355 d=-0.0579 d=-0.017
d=-0.0613 d=-0.0824 d=-0.0096
The other methods give similar results, confirming that only SOFIX can be considered to
express long-range dependence.
Possibility for Forecasting is an important evidence for inefficiency. If the process is a
Random walk, then the best forecast for the return is zero (or the best forecast for the price is
previous value). At that stage we use methods from Chaos theory, based on the reconstruction
of the phase space and searching for similarities in the trajectory [3, 4]. Obtained results for
several local predicting functions are presented in Table 2.
Table 2
Forecasting Method
MSE
Closer Neighbor
0.0004131
Mean Neighbors Forecast
0.0003907
Linear Forecast
0.0003612
Quadratic Forecast
0.0012486
Statistical Mean Forecast
0.0004291
Zero Forecast
0.0004068
We see that forecasting with linear local approximation function gives the smallest mean
square error (MSE) and thus confirms the principle predictability of SOFIX quotes.
In conclusion we can say that our study clearly shows inefficiency of the Bulgarian Financial
Market which is not following a random walk and possess long-range memory.
[2] Hosking,J.R.M. Fractional differencing, Biometrika, 68,165-176, 1981.
[2] Taqqu,M., V.Teverovsky On Estimating the Intensity of Long-Range Dependance in
Finite and Infinite Variance Time Series,A practical Guide to Heavy Tails, ed. Adler,R.,
M.Taqqu, Birkhauser, Boston,177-217, 1998.
[3] Takens, F., Detecting strange attractors in turbulence,Warwick Lecture Notes in Math.,
vol 898, Springer pp.366-381, 1980.
[4] Farmer, J. и J.Sidorowich, Predicting chaotic time series, Physical review letters,1987