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International Conference On Applied Economics – ICOAE 2011 463 INFORMATION ENTROPY AND EFFICIENT MARKET HYPOTHESIS DANIEL TRAIAN PELE1,ANA-MARIA ȚEPUȘ2 Abstract This study aims to demonstrate that the extreme values of returns distribution are mostly associated with particular periods of stock markets inefficiency, when their level of uncertainty reaches a local minimum. We propose an estimator of uncertainty, through the entropy of probability density function of returns. The relationship between the level of uncertainty of a stock market and extreme values of returns distribution is illustrated trough a binary logistic regression model estimated for main indexes of four stock markets from Central and Eastern Europe. JEL codes: G01 - Financial Crises, G14 - Information and Market Efficiency; Event Studies, G15 - International Financial Markets. Keywords: entropy, entropy of a function, efficient market hypothesis. 1.Introduction Modeling the capital markets is closely linked to the efficient market hypothesis, a concept found in conjunction with the rationality of investor behavior. Since Bachelier's groundbreaking study(1900), most of the twentieth century academic research in finance has been formed around the paradigm of efficient markets. From the classical definition of Fama(1970), until more recent developments of Timmerman and Granger(2004), Efficient Market Hypothesis(EMH) is inseparable from the concept of information and the mechanism of incorporation of a certain set of information in the trading price of a financial asset.Numerous studies have investigated how the efficient market hypothesis, as a theoretical model, is observed at the level of empirical reality. Stiglitz(1981) shows that the hypothesis of an efficient market, where ‖prices fully reflect available information‖(Fama, 1970), is not consistent with the notion of Pareto optimum. Moreover, the information reflected in stock prices are just information that does not require a cost to obtain them. Summers (1986) argue that statistical tests commonly used for testing the Efficient Market Hypothesis have a relatively low power. Thus, a basic statistical law says that if we cannot reject the null hypothesis, we cannot automatically accept it as a valid hypothesis. From this point of view, are forms of inefficiency that can not be discriminated by the usual statistical tests, and we cannot deduce from these tests a conclusion about the validity of Efficient Market Hypothesis. Malkiel (2003) discusses the efficiency of capital markets through the criticisms that have been made over time. He clearly describes the random walk hypothesis: the logic of this hypothesis is that if the information is immediately reflected in stock prices, then tomorrow's price change will reflect only the tomorrow‘s information and will be independent of today‘s price change. As the information is unpredictable, then price changes must be unpredictable and random. The link between a certain measure of complexity and the Efficient Market Hypothesis is quite clear: if we assume an efficient market (in weak form), then the stock price follows a random walk model, i.e. the time series of returns is a white noise process. In terms of quantitative measures, such a white noise process has the highest level of complexity; on the contrary, if the Efficient Market Hypothesis is not met, then the price is no longer a random walk process and consequently, the level of complexity of the market will be lower. For instance, if the price is a purely deterministic process, completely predictable, then a minimum level of complexity is achieved; if the price is a purely random process, completely unpredictable, then we are dealing with the maximum level of complexity. Risso(2008) uses entropy as a measure of complexity to investigate the hypothesis that stock market crashes are most closely associated with periods of low entropy. The relationship between stock market crashes and informational efficiency is as follows: if the market is inefficient, so the information is not instantaneously reflected in prices, then local trends appears in the evolution of the price. But once the information is incorporated into stock price, the investor‘s reaction may lead to a significant collapse of stock price. Our study aims to demonstrate that both the decline, as well as the significant growth of the trading prices can be explained by lower levels of stock market complexity, in relation to the Efficient Market Hypothesis. The paper is organized as follows: in Section 1 we describe the concept of entropy as a measure of complexity and propose a method of estimating entropy using probability density function of returns; in Section 2 we describe the theoretical model used to investigate the relationship between the level of complexity of stock market and the occurrence of extreme values of returns; in Section 3 are presented the results of the estimated model for stock markets of Romania, Bulgaria, Hungary and the Czech Republic and the last section is for conclusions. 2. Information Entropy as a measure of complexity Entropy is both a measure of uncertainty and complexity of a system, with numerous applications in physics ( the second principle of thermodynamics), in information theory, in biology (DNA sequence complexity), medicine, economics (complexity of a system). 1 Ph.D., Lecturer, Department of Statistics and Econometrics, University of Economics, Bucharest, Email: danpele@ase.ro. 2 Ph.D. Candidate, Department of Money, University of Economics, Bucharest, Email: anamaria.tepus@yahoo.com . 464 International Conference On Applied Economics – ICOAE 2011 x1......xn p ...... p X : If X is a discrete random variable, with probability distribution 1 n p P( X xi ) , 0 pi 1 and , where i pi 1 i H ( X ) pi log 2 pi i , then Shannon Information Entropy is defined as follows: . [1] H ( X ) (1 / n) log 2 (1 / n) log 2 n For uniform distribution the Shannon Entropy reaches his maximum: i , while the minimum value is attained for a distribution like the following: x1......xn X : 1.........0 , H(X ) 0 . In other words, high levels of entropy are obtained for situations with high uncertainty and low levels of entropy are associated with situations with lower uncertainty. The relation between entropy and capital markets is straightforward(Risso, 2008). 1, rt 0 st Let t r p p t t 1 t log P log P t 1 the logreturn of an asset and let 0, rt 0 a random variable associated to ‖bull- bear‖ states . Then for a certain period of time, one can define the information entropy of the 0 and 1 sequence: H p log 2 p (1 p) log 2 (1 p) , where p P(st 1) 1 P(st 0) . In other words, we can define the entropy as a measure of the complexity of the stock market, by transforming the time series of logreturns into a sequence of 0 and 1.From our point of view, the methodology should be extended by taking into account the continuous nature of returns distribution and in the following we propose such an extension of this methodology. 2.1. Entropy of a function Unlike the case of a discrete random variable, the entropy of a continuous random variable is difficult to define. If X is a continuous random variable with probability density function f (x) , then we can define, by analogy with the Shannon information entropy, differential entropy: H ( f ) f ( x) log 2 f ( x)dx A , [2] where A is the support set of X. A naive estimator of differential entropy could be used to quantify the complexity level of a stock market, and the results for BET Index of Bucharest Stock Exchange(Pele, 2011) shows a significant correlation between the values of this estimator and probability of extreme negative values of daily logreturns distribution. Unfortunately, differential entropy does not have all the properties of Shannon entropy: it can take negative values and in addition is not invariant to linear transformations of variables. However, we can define the entropy of a function that satisfies certain properties, through a transformation called quantization. We present the essential elements of this transformation, as they appear in Lorentz (2009). Definition 1. Let f : I [a, b] R a continuous real-valued function, let n N and define * xi a (i 1 / 2)h , for i 0,.., n 1 , where h (b a) / n . The sampled function for f is n S ( f )(i) f ( xi ) for i 0,.., n 1 . Quantization process refers to creating a simple function that approximates the original function. Let q 0 a quantum. Then the Qq ( f )( x) (i 1 / 2)q following function defines a quantization of f: , if f ( x) [iq, (i 1)q) . Definition 2. Let f measurable and essentially bounded on the interval [a, b] and let q 0 . Let i I [iq, (i 1)q) and Bi f 1 ( I i ) . Then H q ( f ) ( Bi ) log 2 ( ( Bi )) the entropy of f at quantum q is i , where is the Lebesgue measure. Following these definitions, one can compute the entropy of any continuous functions defined on a compact interval. H ( f ) log n If f ( x) x on [0,1] , then for a fixed quantum q 1 / n , q 2 , the maximum value of the entropy. The following theorem (Lorentz, 2009) provides a conceptual framework for defining an estimator of entropy of a function. Theorem International Conference On Applied Economics – ICOAE 2011 465 Let f continuous on [a,b] and let 1/n sampling space. Let S n ( f ) the sampled function and Qq S n a quantization of S n with c (i) card {(i 1 / 2)q Qq Sn } (number of values (i 1 / 2)q in Qq S n ) and let pn (i) probability of quantum q 0 . Denote n c (i) c (i) pn (i) n n cn ( j) n j value i: . lim p (i) log h 0 i n 2 pn (i) H q ( f ) Then . [3] The above theorem assures us that regardless of quantization and sampling, we obtain a consistent estimator of the entropy of a function. 1.2. Entropy of probability density function of returns This conceptual framework can be used in order to define the entropy of probability density function(pdf) of returns. Let f a continuous real-valued function such as f ( x) 0 and f ( x)dx 1 . Then the hypothesis of Lorentz theorem are H q ( f ). fulfilled and we can compute In reality, the analytical expression of probability density function is unknown, so we can estimate the density using a nonparametric approach, such as Kernel Density Estimation (KDE). ^ 1 n x xi f ( x) K nh i1 h . Thus, KDE estimator of pdf is [4] K ( x)dx 1 K is a kernel function, with the following properties: K ( x) 0, x R , K ( x) K ( x), x R , R and xK ( x)dx 0. R The parameter h is a scale parameter, whose choice determines the quality of the estimate (is also known as smoothing parameter or bandwidth). Basically, our methodology involves the following steps to estimate the entropy of the probability density function of returns: Let rt the time series of logreturns for a time period T , and let f (x) the probability density function. If n 2 , we estimate pdf with Kernel Density Estimation, obtaining values for i 0,.., n 1 . k f ( xi ) S Les n sampled function as n S ( f )(i) f ( xi ) for i 0,.., n 1 . Let q 2 k a quantum; then define Qq Sn ( f )( j ) (i 1 / 2)q , if f ( x j ) [iq, (i 1)q) . c (i) c (i) card{ f ( x j ) [iq , (i 1)q)} pn (i) n n cn ( j) n j n Compute probabilities . H q ( f ) pn (i) log 2 pn (i) ˆ One can estimate the entropy of probability density function as i . [5] This estimator of entropy of the probability density function of returns will be called, in what follows, PDF Entropy. Notes xmax xmin h x xmin (i 1 / 2)h Actually, we have chosen n and i , for i 0,.., n 1 . For computational reasons, we have used n 2 2 128 , and KDE was done using a Gaussian kernel: k 7 K ( x) exp( x 2 / 2) / 2 . There are many distributions for which the probability density function is not necessarily bounded (Chi-square distribution is one such example); moreover, even pdf is bounded, his range is different from distribution to distribution. To ensure comparability between the results of the estimation in various markets, we proceeded to standardize the values estimated by KDE: f ( xi ) min f f ( xi ) max f min f , where max f and min f are the extreme estimated values of probability density function. 466 International Conference On Applied Economics – ICOAE 2011 One can define a normalized estimator of entropy of probability density function of returns (Normalized PDF Entropy): pn (i) log 2 pn (i) ˆ HN q ( f ) i log 2 n . [6] This will ensure comparability among different markets; in fact, this estimator of complexity will be used below to illustrate the relationship between the degree of complexity of the market and the likelihood of extreme events. 3. The theoretical model of entropy as a predictor of extreme values of returns distribution Entropy can be regarded as a measure of informational efficiency of a stock market; if the market is weak form efficient, then the price follows a random walk process, therefore the bull and bear market situations are likely probable. In terms of information entropy, market efficiency is equivalent to the situation of maximum entropy, maximum complexity or maximum uncertainty. Conversely, when the price exhibits a predominant trend(upwards or downwards), the level of certainty is high, and such periods are described by lower values of entropy. Our working hypothesis is that the likelihood of both tails of returns distribution could be explained by lower values of entropy. To verify this hypothesis we estimate the following logistic regression model: exp( 0 1 H t ) P(Yt* 1) 1 exp( 0 1 H t ) . [7] In the above equation, we have: - Yt* 1 , where Yt* {1 | rt rt , P(rt rt ) 0.01} {1 | rt rt , P(rt rt ) 0.01} (upper and lower tail). - Ht is market information entropy at time t, quantified by Normalized Shannon Entropy and by Normalized PDF Entropy. Normalized Shannon Entropy was estimated using the methodology from Risso(2008). 1, rt 0 st Thus, for a certain time period T, let 0, rt 0 . p P(st 1) . Using a rolling window T , one can compute the probability Then the Normalized Shannon Entropy for the entire interval of length T is H [ p log 2 p (1 p ) log 2 (1 p )] / log 2 (n) . [8] Also, we have estimated Normalized PDF Entropy for several time intervals T; moreover, since the time series of daily returns is very noisy, the model was estimated using returns calculated on a local time window : t t r p p t . As we need to discriminate among several models, we should use a performance indicator of the logistic regression model. In general, such an indicator is defined by comparing the likelihood function of estimated model with the likelihood function of the model when the exogenous variable is removed. 2 One can define pseudo- R , as a measure of model‘s performance(Nagelkerke,1991): R 2 1 exp{2[log L(M ) log L(0)] / n} , where L(M ) and L(0) are likelihood function of the model, with and without the exogenous variable. log(1 R 2 ) 2[log L(M ) log L(0)] / n , this could be interpreted as the surplus of Rewriting the expression as 2 information due to explanatory variable. Unfortunately, R will never reach 1, not even for a perfect model, so the following adjustment is made(Nagelkerke,1991): Radj R 2 /[1 exp( 2 log L(0) / n)] 2 . We choose the model that best describes the correlation between reality and theoretical hypothesis using as criterion the 2 Radj maximization of . 4. Results We estimated the Normalized Shannon Entropy and the Normalized PDF Entropy using various rolling-windows and various time periods, for main indexes of stock markets from four countries of Central and Eastern Europe (Romania, Bulgaria, Hungary, and Czech Republic). There are several studies in the literature dealing with the market efficiency of those countries, and the results are quite contradictory. Emerson et al.(1997), found varying levels of efficiency and varying speeds of movement towards efficiency within a sample of four shares, selected from Sofia Stock Exchange. International Conference On Applied Economics – ICOAE 2011 467 Rockinger and Urga(2000), in a study covering the period 1993-1999, argue that Hungarian market always satisfies weak efficiency, while for the Czech market, they document convergence toward efficiency. Hájek(2007), on a study for the period 1995- 2005, concludes that the weak form of the EMH cannot be validated on the Czech stock market, since daily price changes of both individual stocks and indices are systematically linearly dependent. Also, the level of efficiency increases for weekly or monthly returns. Dragotă and Mitrică(2004) and Dragotă et. al.(2009) found different conclusions regarding the efficiency of Romanian stock market; while in the first paper, the Efficient Market Hypothesis is rejected, the second paper reveals a significant movement toward efficiency. Our analysis shows that are different degrees of efficiency among the four stock markets investigated and this result could be explained using information entropy as a measure of stock market uncertainty. 4.1. Estimation results for BET Index of Bucharest Stock Exchange To estimate the model, we have used daily logreturns of BET Index, for time period 19 September 1997-15 March 2011(3364 daily observations). 2 Table 1: Adjusted Pseudo - R for the two estimators of complexity for BET Index Normalized Shannon Entropy Normalized PDF Entropy 1 2 3 4 5 1 2 3 4 5 T=60 0.012 0.008 0.015 0.016 0.003 0.056 0.044 0.026 0.021 0.008 T=100 0.017 0.014 0.024 0.023 0.002 0.043 0.023 0.010 0.009 0.009 T=150 0.011 0.006 0.010 0.015 0.002 0.023 0.014 0.002 0.006 0.016 T=200 0.011 0.007 0.010 0.015 0.004 0.009 0.001 0.000 0.000 0.006 T=240 0.005 0.004 0.007 0.010 0.004 0.003 0.000 0.000 0.000 0.001 2 Radj 60 and As the values shows, the best results are obtained using Normalized PDF Entropy as explanatory variable, with T 1. The fact that the best results for BET Index were obtained for T 60 suggests that the Romanian stock market has no long temporal memory, the local temporal context being most relevant. In addition, market complexity estimation using entropy of probability density function of returns provides better results than the classical Shannon entropy. 0.15 1 0.1 0.95 0.05 0.9 0 0.85 -0.05 0.8 -0.1 0.75 -0.15 0.7 12/15/1997 12/15/2002 12/15/2003 12/15/2004 12/15/2005 12/15/2006 12/15/2008 12/15/1998 12/15/1999 12/15/2000 12/15/2001 12/15/2007 12/15/2009 12/15/2010 12/15/1998 12/15/1999 12/15/2002 12/15/2003 12/15/2006 12/15/2007 12/15/2010 12/15/1997 12/15/2000 12/15/2001 12/15/2004 12/15/2005 12/15/2008 12/15/2009 Graph 1: Daily logreturns of BET Index Graph 2: Normalized PDF Entropy( T 60 , 1 ) The results of the optimum model, for Normalized PDF Entropy, T 60 and 1 , are presented below. Table 2: Estimation results of logistic regression for BET Index Parameter Estimate Standard Error Wald Chi-Square Pr > ChiSq 0 10.9884 2.2109 24.702 <0.0001 1 -15.7776 2.4034 43.0948 <0.0001 2 Radj Observations 3304 Pseudo- 0.0558 Chi-Square(Hosmer-Lemeshow -2 Log L 701.381 Goodness of Fit Test) 13.2112 2 Pseudo- R 0.0112 Pr > ChiSq 0.1048 468 International Conference On Applied Economics – ICOAE 2011 Analyzing the results of the estimation, we can see that the entropy adversely affect the likelihood of extreme values of daily returns. Thus, if entropy increases by 0.1 units (e.g. from 0.8 to 0.9), then the odds of occurrence of extreme values of BET returns drops by around 80%. Table 3: Normalized PDF Entropy of BET Index(tails and body of returns distribution) Descriptive N Mean Median Mode Std. Minimum Maximum Statistics Deviation Yt* 1 78 0.9167 0.9234 0.7417 0.0486 0.7417 78 Yt* 0 3226 0.9449 0.9528 0.9054 0.0355 0.7624 3226 Moreover, the average entropy is significantly lower in the days corresponding to the extreme values of returns distribution than the other days. 4.2. Estimation results for SOFIX Index of Sofia Stock Exchange To estimate the model, we have used daily logreturns of SOFIX Index, for time interval 26 november 2001 – 4 January 2011 (2239 daily observations). 2 Table 4: Adjusted Pseudo - R for the two estimators of complexity for SOFIX Index Normalized Shannon Entropy Normalized PDF Entropy 1 2 3 4 5 1 2 3 4 5 T=60 0.018 0.019 0.004 0.001 0.002 0.077 0.082 0.171 0.078 0.089 T=100 0.021 0.040 0.031 0.015 0.004 0.115 0.087 0.145 0.069 0.040 T=150 0.028 0.039 0.047 0.056 0.045 0.089 0.064 0.178 0.079 0.058 T=200 0.070 0.091 0.161 0.144 0.102 0.141 0.041 0.162 0.090 0.061 T=240 0.092 0.100 0.110 0.087 0.088 0.098 0.020 0.132 0.107 0.093 2 Radj As can be seen from the values of for estimated logistic regression models, the best performance offers Normalized Entropy Estimator for T 150 and 3 as well as Normalized PDF Entropy Estimator for T 60 and 3 . The results are substantially similar, but based on the results of Hosmer-Lemeshow test we chose the model with T 60 and 3 , since the other model presents a lack of fit. For SOFIX Index also, market complexity estimation using entropy of probability density function of returns provides better results than the classical Shannon entropy. 0.1 1 0.98 0.05 0.96 0.94 0 0.92 0.9 -0.05 0.88 0.86 -0.1 0.84 0.82 -0.15 0.8 2/27/2002 8/27/2002 2/27/2003 8/27/2003 2/27/2005 2/27/2006 8/27/2006 2/27/2007 8/27/2007 2/27/2009 2/27/2010 2/27/2004 8/27/2004 8/27/2005 2/27/2008 8/27/2008 8/27/2009 8/27/2010 7/15/2002 1/15/2003 1/15/2004 7/15/2004 7/15/2005 1/15/2006 7/15/2007 1/15/2008 1/15/2009 7/15/2010 7/15/2003 1/15/2005 7/15/2006 1/15/2007 7/15/2008 7/15/2009 1/15/2010 Graph 3: Daily logreturns of SOFIX Index Graph 4: Normalized PDF Entropy( T 60 , 3 ) The results of the optimum model, with Normalized PDF Entropy as exogenous, T 60 and 3 , are presented below. Table 5: Estimation results of logistic regression for SOFIX Index Estim Standard Wald Chi- Pr > Ch Parameter ate Error Square iSq 0 33 27.58 3.7150 55.1294 <0.0001 - 4.0649 68.9309 <0.0001 1 33.7489 International Conference On Applied Economics – ICOAE 2011 469 Observati 217 R2 0.1 ons 9 Pseudo- adj 710 349. Chi-Square(Hosmer- 8.5 -2 Log L 314 Lemeshow Goodness of Fit Test) 538 0.02 Pseudo- 0.3 96 Pr > ChiSq R2 813 Analyzing the results of the estimation, we can see that the entropy negatively affect the likelihood of extreme values of daily returns. Thus, if entropy increases by 0.1 units (e.g. from 0.8 to 0.9), then the odds of occurrence of extreme values of SOFIX returns drops by around 97%. Table 6: Normalized PDF Entropy of SOFIX Index(tails and body of returns distribution) Descript N Mea Medi Mod Std. Minim Maxim ive Statistics n an e Deviation um um 42 0.90 0.91 0.84 0.034 0.8423 0.9759 Yt* 1 95 32 23 1 0.95 0.95 0.89 0.028 0.8404 0.9954 2137 27 87 Yt* 0 48 4 Moreover, the average entropy is significantly lower for the days with extreme values of returns (0.90) than the other days (0.95). 4.3. Estimation results for BUX Index of Budapest Stock Exchange To estimate the logistic regression model, we have used daily logreturns of BUX Index, for time horizon 1 April 1997 – 4 January 2011 (3442 daily observations). 2 Table 7: Adjusted Pseudo - R for the two estimators of complexity for BUX Index Normalized Shannon Entropy Normalized PDF Entropy 1 2 3 4 5 1 2 3 4 5 T=60 0.001 0.001 0.001 0.001 0.019 0.169 0.167 0.127 0.187 0.193 T=100 0.000 0.000 0.002 0.005 0.024 0.183 0.185 0.156 0.205 0.190 T=150 0.000 0.001 0.000 0.003 0.018 0.183 0.189 0.190 0.237 0.227 T=200 0.005 0.004 0.006 0.000 0.004 0.138 0.142 0.156 0.198 0.184 T=240 0.016 0.016 0.018 0.003 0.000 0.167 0.154 0.137 0.163 0.172 2 Radj As can be seen from the values of for estimated logistic regression models, the best performance offers Normalized Entropy Estimator for T 150 and 4 . For BUX Index also, market complexity estimation using entropy of probability density function of returns provides better results than the classical Shannon entropy. 0.2 1 0.15 0.98 0.96 0.1 0.94 0.05 0.92 0 0.9 0.88 -0.05 0.86 -0.1 0.84 -0.15 0.82 0.8 -0.2 11/5/1999 11/5/2001 11/5/2002 11/5/2003 11/5/2004 11/5/2005 11/5/2010 11/5/1997 11/5/1998 11/5/2000 11/5/2006 11/5/2007 11/5/2008 11/5/2009 11/5/1997 11/5/1998 11/5/1999 11/5/2002 11/5/2003 11/5/2007 11/5/2000 11/5/2001 11/5/2004 11/5/2005 11/5/2006 11/5/2008 11/5/2009 11/5/2010 Graph 5: Daily logreturns of BUX Index Graph 6: Normalized PDF Entropy( T 150 , 4 ) The results of the optimum model, with Normalized PDF Entropy as exogenous, T 150 and 4 , are presented below. Table 8: Estimation results of logistic regression for BUX Index Estim Stand Wald Chi- Pr > Ch Parameter ate ard Error Square iSq 0 71 28.91 5 2.909 98.7797 <0.0001 - 3.271 120.0030 <0.0001 1 35.8376 5 470 International Conference On Applied Economics – ICOAE 2011 Observati 329 R2 0.17 ons 2 Pseudo- adj 10 492. Chi-Square(Hosmer-Lemeshow 11.1 -2 Log L 411 Goodness of Fit Test) 252 Pseudo- 0.04 0.19 R2 13 Pr > ChiSq 47 Analyzing the results of the estimation, we can see that the entropy negatively affect the likelihood of extreme values of daily returns of BUX Index. Thus, if entropy increases by 0.1 units (e.g. from 0.8 to 0.9), then the odds of occurrence of extreme values of BUX Index returns drops by around 98%. Table 9: Normalized PDF Entropy of BUX Index(tails and body of returns distribution) Descrip N Mea Medi Mod Std. Mini Maxi tive n an e Deviation mum mum Statistics 64 0.88 0.88 0.81 0.040 0.811 0.9668 Yt 1 * 46 2 14 5 4 3228 0.94 0.95 0.86 0.034 0.817 0.9906 Yt 0 * 46 05 32 4 2 For BUX Index, average entropy is significant higher in the body of returns distribution (0.94), than in the upper and lower tail(0.88). 4.4. Estimation results for PX Index of Prague Stock Exchange To estimate the binary logistic regression model, we have used daily logreturns of PX Index, for time interval 7 September 1993 – 5 January 2011 (4184 daily observations). 2 Table 10: Adjusted Pseudo - R for the two estimators of complexity for PX Index Normalized Shannon Entropy Normalized PDF Entropy 1 2 3 4 5 1 2 3 4 5 T=60 0.001 0.000 0.001 0.002 0.000 0.033 0.081 0.067 0.039 0.099 T=100 0.000 0.004 0.002 0.002 0.000 0.071 0.141 0.115 0.050 0.111 T=150 0.022 0.016 0.011 0.015 0.016 0.063 0.173 0.132 0.077 0.125 T=200 0.009 0.005 0.000 0.003 0.002 0.086 0.141 0.161 0.075 0.147 T=240 0.007 0.003 0.000 0.000 0.000 0.126 0.157 0.131 0.075 0.118 2 Radj As can be seen from the values of for estimated logistic regression models, the best performance offers Normalized Entropy Estimator for T 150 and 2 as well as Normalized PDF Entropy Estimator for T 200 and 3 . The results are substantially similar, but based on the results of Hosmer-Lemeshow test we chose the model with T 200 and 3 , since the other model presents a lack of fit. For PX Index also, market complexity estimation using entropy of probability density function of returns provides better results than the classical Shannon entropy. 0.15 1 0.1 0.95 0.05 0 0.9 -0.05 0.85 -0.1 0.8 -0.15 -0.2 0.75 30/01/1995 20/06/1996 19/02/1997 7/6/2000 5/2/2001 4/6/2002 26/09/2003 28/05/2004 26/01/2005 20/09/2005 19/05/2006 8/9/2009 14/10/1997 17/06/1998 3/2/2003 17/01/2007 13/09/2007 15/05/2008 13/01/2009 6/10/1995 12/2/1999 7/10/1999 3/10/2001 10/5/2010 17/06/1998 19/05/2006 17/01/2007 13/09/2007 15/05/2008 13/01/2009 30/01/1995 20/06/1996 19/02/1997 14/10/1997 26/09/2003 28/05/2004 26/01/2005 20/09/2005 3/2/2003 7/6/2000 5/2/2001 4/6/2002 8/9/2009 6/10/1995 12/2/1999 7/10/1999 3/10/2001 10/5/2010 International Conference On Applied Economics – ICOAE 2011 471 Graph 7: Daily logreturns of PX Index Graph 8: Normalized PDF Entropy( T 200 , 3 ) The results of the optimum model, with Normalized PDF Entropy as explanatory variable, T 200 and 3 , are presented below. Table 11: Estimation results of logistic regression for PX Index Standard Parameter Estimate Error Wald Chi-Square Pr > ChiSq 0 18.3822 1.8960 93.9932 <0.0001 1 -24.6228 2.1505 131.0977 <0.0001 R2 Observations 3983 Pseudo- adj 0.1611 Chi-Square(Hosmer-Lemeshow Goodness -2 Log L 653.879 of Fit Test) 11.3840 2 Pseudo- R 0.0283 Pr > ChiSq 0.1809 Analyzing the results of the estimation, we can see a negative correlation between entropy and the likelihood of extreme values of daily returns of PX Index. Thus, if entropy increases by 0.1 units (e.g. from 0.8 to 0.9), then the odds of occurrence of e xtreme values for PX Index returns drops by around 92%. Table 12: Normalized PDF Entropy of PX Index(tails and body of returns distribution) Descriptive N Mean Median Mode Std. Minimum Maximum Statistics Deviation Yt* 1 78 0.9060 0.9037 0.8161 0.0455 0.8161 0.9833 Yt* 0 3905 0.9352 0.9402 0.8937 0.0320 0.8048 0.9899 For PX Index, average entropy is significant higher in the body of returns distribution (0.93), than in the upper and lower tail(0.90). 5.Conclusions Informational efficiency of the stock markets is an intensely debated topic in recent years, especially in the context of current economic and financial crisis. From an information theory point of view, capital market efficiency may be associated with a high degree of uncertainty. Indeed, if a stock market is efficient, meaning that information is transmitted instantly and is completely incorporated in trading prices, it is virtually impossible to anticipate future evolutions in prices or returns. This translates into a high degree of uncertainty, specific to a random walk behavior of stock prices. As random walk process is the most complex in terms of predictability, it is natural to use entropy as a measure of uncertainty, in relationship to market efficiency. The main conclusion of the study is that periods characterized by a sharp drop in entropy, when the market complexity level reaches a local minimum, and uncertainty is low, are associated with the occurrence of extreme values of returns. In this study we analyzed the relationship between complexity and predictability of stock market crashes stock using entropy of probability density function of returns as a measure of complexity. Results of the estimated models show that this complexity estimator produces better results than classical Shannon entropy. The entropy of probability density function of returns can be used in two ways: as a tool to measure the degree of market efficiency, as well as to compare, in terms of efficiency, two or more stock markets. As a tool for measuring the degree of stock market efficiency, we proposed an indicator who can detect, at a time, the level of efficiency, taking into account the local temporal context. Unfortunately, this indicator has a number of disadvantages, among which the difficulty to deduce some theoretical properties, since the probability density function does not take values in a standardized interval, for any distribution. Another major disadvantage is that it depends very much on the local temporal context, which can be improved by estimating the indicator using intraday data.On the other hand, as a tool for comparison, PDF Normalized Entropy may be useful in evaluating the relative efficiency of two markets by comparing the sensitivity of the chance of occurrence of extreme values of the returns distribution to a change in entropy. According to Efficient Market Hypothesis, a growing level of market uncertainty should be reflected in an adverse change in the chances of occurrence of extreme values of returns. Lower the magnitude of expected odds change as a result of one unit increase of entropy, lower the degree of market efficiency. From this point of view, the Romanian stock market is the least efficient in relation to capital markets analyzed in this study. According to the estimated model, if entropy increases by 0.1 units (e.g. from 0.8 to 0.9), then the chance of occurrence of extreme values of BET return drops to around 80%. 472 International Conference On Applied Economics – ICOAE 2011 For the Bulgarian market this expected change is approximately 97%, for the Hungarian market is about 98%, and for the Prague Stock Exchange Index the expected change is about 92%. The explanations for these differences between the countries surveyed can be found in the existing inequalities in terms of economic and capital market developments. Acknowledgments This paper was cofinanced from the European Social Fund, through the Human Resources Development Operational Sectorial Program 2007-2013, project number POSDRU/89/1.5/S/59184, „Performance and excelence in the post-doctoral economic research in Romania‖. 6.References Bachelier, L.(1900), ‖Théorie de la Spéculation‖ (first two pages of English translation), Annales Scientifiques de l'Ecole Normale Superieure, I I I -17, 21-86. (English Translation;- Cootner (ed.), (1964) Random Character of Stock Market Prices, Massachusetts Institute of Technology pp17-78). Bouzebda, S., Elhattab, I.(2009), ‖Uniform in bandwidth consistency of the kernel-type estimator of the Shannon's entropy‖, Comptes Rendus Mathematique, Volume 348, Issues 5-6, March 2010, Pages 317-321, ISSN 1631-073X. Dragotă, V., Stoian, A., Pele, D. T., Mitrică, E., Bensafta, M.(2009), „The Development of the Romanian Capital Market: Evidences on Information Efficiency‖, Romanian Journal of Economic Forecasting, vol. 10 (2), 2009, pp. 147-160. Dragotă, V., Mitrică, E.(2004), ―Emergent capital market‘s efficiency: The case of Romania‖, European Journal of Operational Research, 155, 2004, pp. 353-360. Emerson, R., Stephen, H., Zalewska-Mitura, A.(1997), ‖Evolving Market Efficiency with an Application to Some Bulgarian Shares‖, Economics of Planning, Volume 30, Issue 2, Pages 75-90. Fama, E.(1970), ‖Efficient Capital Markets: A Review of Theory and Empirical Work‖, Journal of Finance, 25(2), 383–417. Hájek, J.(2007),"Czech Capital Market Weak-Form Efficiency, Selected Issues", Prague Economic Papers, University of Economics, Prague, vol. 2007(4), pages 303-318. Lorentz, R. (2009), ‖On the entropy of a function‖. J. Approx. Theory 158, 2 (June 2009), 145-150. Malkiel, B.(2003), ―The Efficient Market Hypothesis and Its Critics‖, Journal of Economic Perspectives, Volume 17, Number 1, Winter 2003, pp. 59 - 82. Mateev, M.(2004), ‖CAPM Anomalies and the Efficiency of Stock Markets in Transition: Evidence from Bulgaria‖, South- Eastern Europe Journal of Economics, 1 (2004) 35-58. Nagelkerke, N. J. D.(1991), ‖A note on a general definition of the coefficient of determination‖, Biometrika (1991) 78(3): 691- 692. Parzen, E.(1962), ‖On The Estimation Of Probability Density Function And Mode‖, The Annals Of Mathematical Statistics, 33, 1065-1076. Pele, D.T.(2011), ‖ Information entropy and occurrence of extreme negative returns‖, Journal of Applied Quantitative Methods, Forthcoming, 2011 Risso, A. (2008), ―The Informational Efficiency and the Financial Crashes‖, Research in International Business and Finance, Vol. 22, pp. 396-408. Rockinger, M. , Urga, G.(2000), ‖The Evolution of Stock Markets in Transition Economies‖, Journal of Comparative Economics, Volume 28, Issue 3, September 2000, Pages 456-472. Sain, S.(1994), Adaptive Kernel Density Estimation, Ph.D. Thesis, Houston, Texas. Saporta, G.(1990), Probabilité, Analyse des Données et Statistique , Ed.Technip. Scott, D. W.(1992), Multivariate Density Estimation: Theory, Practice And Visualisation, New York, John Wiley. Silverman, B. W.(1986), Density Estimation For Statistics And Data Analysis, London, Chapman and Hall. Stiglitz, J.(1981), ―The Allocation Role of the Stock Market: Pareto Optimality and Competition‖, The Journal of Finance, Vol. 36, No. 2, Papers and Proceedings of the Thirty Ninth Annual Meeting American Finance Association, Denver, September 5-7, 1980. (May, 1981), pp. 235-251. Summers, L.(1986), ‖Does the Stock Market Rationally Reflect Fundamental Values?‖, The Journal of Finance, 41(3), 591–601. Timmermann, A., Granger, C.(2004), ‖Efficient market hypothesis and forecasting‖, International Journal of Forecasting, 20(1), 15–27. *** Website of Sofia Stock Exchange: www.bse-sofia.bg *** Website of Bucharest Stock Exchange: www.bvb.ro *** Website of Budapest Stock Exchange: www.bse.hu *** Website of Prague Stock Exchange: www.pse.cz

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