Remarks on the time-scale invariance property on the financial markets.

Dima, Bogdan ; Preda, Ciprian Ion ; Pirtea, Gabriel Marilen 等

1. INTRODUCTION

The evidences from financial markets indicate that the data series display complex structures with time adapting hierarchies and sophisticated evolution laws. Zooming out, the data series seem to behave like random walks (no pattern can be summed up), but, once as the "resolution" is magnified (the data frequency is increased), a set of special properties ("fat-tails" effects, long range correlations in volatility, the presence of informational leverage etc.) shows up. Among these properties, it is worth noticing the possibility to describe the data as a collection of interrelated fractal families ("multifractal" objects). Also, the associated "free-scale" property is especially important for a more accurate description of a complex reality. The goal of this paper is to advance an empirical analysis of a key European market index in order to examine the invariant time-scale issue.

2. LITERATURE REVIEW

One of the critical issues in the analysis of the movements in financial assets' prices is generated by the fact that if prices' changes are independent, then it should not be any noticeable streaks in the data. Or, the empirical evidence shows that such increasing/decreasing streaks are highly frequent in a manner that is improbable under the classic Gaussian model. An alternative approach was proposed by Mandelbrot et al. (1997) with the so-called Multifractal Model of Asset Returns (MMAR) and largely discussed and developed in the literature (Mandelbrot & Hudson, 2004; Eisler & Kertesz, 2004; Lux, 2003; Lux & Kaizoji, 2004). The meta-assumption of this approach is that the dynamics of prices reflects a fractal property with the same characteristics as the initial data series. This property stays intact while shifting from low to high "resolution" (on the time-scale). Inserting a random component to this property guarantees a more accurate description of the prices' behavior. This implies that the economic subjects are neutral with respect to the informational leverage time scale (i.e. they are acting similarly no matter the frequency of the new information relevant for their decisions).

3. THE ANALYTICAL FRAMEWORK

Our starting point consists in the thesis that if the information is "imperfect" (i.e. it is incomplete, unequal distributed and there are costs of obtaining, updating and using it), a bounded rationality mechanism of prices formation could be described as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

In formula (1), [P.sub.t], is the price at time t, [[alpha].sub.0] reflects the price trend ("central tendency" which is obtained overall an anterior interval of time), [t-1.summmation over (i-t-k)] [[beta].sub.i] [P.sub.i] is a convex combination of the k lagged prices values (i.e. [[beta].sub.i] [member of] [01] and [t-1.summmation over (i-t-k)] [[beta].sub.i] = 1). In fact, [[beta].sub.i] represent the weights of past information (about the prices level) at the current prices level. Also [I.sub.t], is an "informational indicator" at time t, which captures the status of all the relevant information regarding other variables susceptible to influence prices' evolution. Moreover, [[epsilon].sup.2.sub.i] are the anticipation errors committed by the economic subjects in the previous periods. We used again a convex combination with the anticipation errors (i.e. [[theta].sub.i] [member of][01] and [t-1.summmation over (i-t-k)] [[theta].sub.i] = 1) to adjust the anticipated level of prices for the current [[zeta].sub.t] denotes a parameter reflecting the dominant relative risk aversion on the price formation market and [[alpha].sub.1t], [[alpha].sub.2t], [[alpha].sub.3t] are time-depending weights in price evolution of the corresponding variables.

The bounded rationality model implies that all the relevant information (about price formation) obtained at an "efficient level of implied costs" is provided by previous and current periods. The goal is to support the thesis that the economic subjects are always trying to adopt the "second best" decisions with "incomplete information".

The time-invariance property assumes that the properties of prices' mechanisms are conserved for all the time computational frequencies. More exactly, this means that the characteristics of the distribution, volatility and auto- regressive behavior are invariant with respect to the shifts from one time frequency to another. It also implies that the parameters of prices formation do not vary if the analysis frequency is changed--or that the spreads between coefficients can be described as pure random walk processes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Where [[epsilon].sub.t] ~ N(0, [[sigma].sup.2]) and t, [t.sup.*] are two distinct time frequencies.

A stronger version of the time-invariance is involved, if the markets display different degrees of informational efficiency and the prices' evolution itself is close to a pure random walk process (eventually with drift). Then [[alpha].sub.1] [approximately equal to] 1, [[beta].sub.i] [approximately equal to] 1, k [approximately equal to] 1, [[alpha].sub.2] [approximately equal to] 0 and the spread between two distinct time frequencies could be described as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Where [epsilon]'.sub.[tau]] ~ N (0,[[sigma].sup.2]).

Relation (3) implies that if the markets are efficient then the spread between two arbitrary frequencies is fairly described as a random (with possible a drift) random process.

4. THE EMPIRICALL FRAMEWORK

The Austrian Traded Index (ATX) is the Austrian market main index and comprises the blue chips of Wiener Borse from 1991. It was designed as an underlying reference for futures and options financial assets and it contains the most actively traded and highly liquid stocks on the prime market segment. In order to verify the preservation of the time-scale invariance, a time-scale variation indicator can be used:

[??]Indicator = Indicator {n}--Indicator {n * z} (4)

Where z is the number of the main observation period' sub-periods. Assuming the robustness of the market index about the time scale change, the time-scale variation indicator should be close to 0. By taking the same analysis period and considering hourly data (i.e. 9 hours of trade) over the ATX market, the time-scale variation indicator (computed on 9*5= 45 data) looks like in Tab. 1.

The general statistic properties of the time-scale variation indicator [??] display the same non-Gaussian distribution. Also, the mean varies over the data subsets. A critical aspect is that the time-scale variation indicator can not be described as a random walk as in Tab.2.

The so-called aggregational Gaussianity is present, which means that the price distribution converges to a Gaussian one (accordingly to the Central Limit Theorem), if the time horizon [DELTA]t increases slowly. Thus, it can be presumed that the time-scale variation indicator exhibits some stationarity properties. Or, the evidence suggests that the null hypothesis of the no unit roots could be accepted for this indicator. Thus, Tab.3 shows that a more detailed analysis is required.

5. COMMENTS AND FURTHER RESEARCH

Based on the proposed analysis, it can be concluded that the market does not fully exhibit the time-invariance property. The evidences appear to be mixed especially in terms of the time- scale variation properties. Despite some important caveats, we consider that the advanced analysis can provide a better explanation of the investors' behavior in the portfolio management processes under shifting time frames decisional frameworks. In our view, this extension requires minimally to: 1) adopt a more complex analytical framework (perhaps based on the computation of the Hurst exponent for the time spread indicator); 2) further analyze the time spread stationarity / distribution issues. Our objective is to go further in the amazing microcosm of data in order to admire the "simple beauty of the complexity".

6. REFERENCES

Calvet, L.; Mandelbrot, B. & Fisher, A. (1997), A multifractal model of asset returns. Cowles Foundation Discussion Paper, September 15

Eisler, Z., Kertesz, J. (2004), Multifractal model of asset returns with leverage effect. arXiv:cond-mat/0403767 v2, Accessed on: 2004-05-11

Lux, T. (2003), The Multi-Fractal Model of Asset Returns: Its Estimation via GMM and Its Use for Volatility Forecasting. University of Kiel, Working Paper

Lux, T., Kaizoji, T. (2004) Forecasting Volatility and Volume in the Tokyo Stock Market: The Advantage of Long Memory Models. University of Kiel, Working Paper

Mandelbrot, B., Hudson, R. (2004), The (Mis)behavior of Markets: A Fractal View of Risk, Ruin and Reward. New York: Basic Books, ISBN 0465043577

Tab. 1. The general statistic properties of the time-scale
variation indicator

Category Statistics
 Std. Err.
Indicator Count Mean Std. Dev. of Mean

[-20, -10) 7 -12.69143 2.079098 0.785825
[-10, 0) 17 -4.715882 2.556167 0.619961
[0, 10) 11 3.411818 2.323466 0.700551
[10, 20) 13 15.24000 2.735605 0.758720
[20, 30) 3 25.88000 4.120400 2.378914
All 51 2.829020 11.45631 1.604204

Tab. 2. The random walk test for time-scale variation indicator

 Final Root MSE z- Prob.
 State Statistic

[[epsilon].sub.t] 21.98000 20.64191 1.064824 0.2870

Log likelihood -229.2206 Akaike info criterion 9.208825

Parameters 1 Schwarz info criterion 9.247066

Diffuse priors 1 Hannan-Quinn info
 criterion 9.223388

Tab. 3. The stationarity test for time-scale variation indicator

Null Hypothesis: SPREAD has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 0 (Automatic based on Modified
HQ, MAXLAG=10)

 t-Statistic Prob. *

Augmented Dickey-Fuller -4.877778 0.0013
test statistic

Test critical 1% level -4.152511
values:
 5% level -3.502373

 10% level -3.180699

* MacKinnon (1996) one-sided p-values.

Null Hypothesis: SPREAD is stationary

Exogenous: Constant, Linear Trend

Bandwidth: 0 (Newey-West using Bartlett kernel)

Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.062163

Asymptotic 1% level 0.216000
critical
values *: 5% level 0.146000

 10% level 0.119000

* Kwiatkowski-Phillips-Schmidt-Shin (1992, Table 1)

Residual variance (no correction) 128.0817

HAC corrected variance (Bartlett kernel) 128.0817