Serial correlation and heterogeneous volatility in financial markets: beyond the LeBaron effect

arXiv:0810.4912v1 [q-fin.ST] 27 Oct 2008 Serial correlation and heterogeneous volatility in financial markets: beyond the LeBaron effect Simone Bianco Department of Applied Science, College of William and Mary, Williamsburg, Va 23187-8795, USA Fulvio Corsi and Roberto Ren`o Dipartimento di Economia Politica, Universit`a di Siena, Piazza S.Francesco 7, 53100, Siena, Italy We study the relation between serial correlation of financial returns and volatility at intraday level for the S&P500 stock index. At daily and weekly level, serial correlation and volatility are known to be negatively correlated (LeBaron effect). While confirming that the LeBaron effect holds also at intraday level, we go beyond it and, complementing the efficient market hyphotesis (for returns) with the heterogenous market hyphotesis (for volatility), we test the impact of unexpected volatility, defined as the part of volatility which cannot be forecasted, on the presence of serial correlations in the time series. We show that unexpected volatility is instead positively correlated with intraday serial correlation. Serial correlation of asset prices is one of the most elusive quantity of financial economics. According to the theory of efficient markets [1, 2], it should not exist at all, and when it exists it represents an anomaly of financial markets. Many economists and physicist devoted themselves to the study of stock return predictability [3, 4]. Historical returns should prevent any forecasting technique, even if it has been shown, like in [5], that the random walk hypothesis holds only weakly. On the other hand the variance of financial returns on a fixed time interval, which is called volatility, is a highly predictable quantity [6, 7], with its probability distribution function showing fat tails [8]. The natural association of volatility to financial risk forecast and control makes its analysis paramount in Economics. To some extent, it seems obvious therefore to link volatility to returns serial correlation. If anything else, the link between volatility and serial correlations can reveal basic properties of the price formation mechanism. A notable stylized fact on serial correlation is the LeBaron effect [9], according to which volatility is negatively correlated to serial correlation, relation that has been empirically confirmed at daily level on several markets [10]. We exploit, as in [11, 12], high-frequency information (that is, intraday data) to test this effect more precisely than in previous studies. With respect to [11, 12], not only we extend considerably our data set, but also our work is set within the framework of the Heterogenous Market Hypothesis [13], which implies the modification of some basic assumptions on financial market dynamics due to empirical findings. The main feature of this model is the assumption that agents must be considered heterogenous, namely, they react differently to incoming news. This implies a completely new interpretation of the concept of time in the market, as now every agent has its own dealing time and frequency. More importantly, the transactions time, in the agent framework, has to be related to short-, medium-, and long-term decisions. Thus, volatility will be itself consistently composed by a cascade of several time components. In this work, the heterogeneous market hypothesis is the basis of our method of volatility estimation, introduced for the first time in [14]. A notable advantage in modeling volatility in the context of heterougeneous markets is that it allows to reconcile the measure with some empirical findings on volatility (e.g., long-range dependence and fat tails), a feature which simple autoregressive volatility forecasting models lacks. The purpose of this paper is then to exploit intraday measures of volatility and serial correlation to investigate the LeBaron effect at a deeper level with a very large and liquid dataset. We find that the LeBaron effect holds. However, we also refine the finding of LeBaron by showing that volatility can be splitted into two components: a predictable one and an unpredictable one. While the first is negatively correlated with serial correlation, the latter is instead positively correlated with serial correlation. I. METHODS AND DATA DESCRIPTION Let us assume to have r1, . . . , rn intraday logarithmic returns. To quantify volatility, we construct daily realized volatility measures defined as the cumulative sum of squared intraday five-minutes returns [15]: RVt = n X i=1 r2 i,t (1) where ri,t is the i-th return in day t. Since volatility has been proved to be log-Normal [16], we use the logarithm of RVt to obtain Normal distributions. 2 Measuring daily serial correlation is more subtle, and we borrow from [12] using a modified overlapped Variance Ratio. Define: ˆµ ≡1 n n X k=1 rk (2) ˆσ2 a ≡ 1 n −1 n X k=1 (rk −ˆµ)2 (3) ˆσ2 c ≡1 m n X k=q   k X j=k−q+1 rj −qˆµ   2 (4) where m = q(n −q + 1)

1 −q n ! . (5) We define the variance ratio as follows: V R(q) =  ˆσ2 c ˆσ

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