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Intraday LeBaron effects

Edited by H. Eugene Stanley, Boston University, Boston, MA, and approved April 28, 2009 (received for review February 6, 2009)
Abstract
We study the relation at intraday level between serial correlation and volatility of the Standard and Poor (S&P) 500 stock index futures returns. At daily and weekly levels, serial correlation and volatility forecasts have been found to be negatively correlated (LeBaron effect). After finding a significant attenuation of the original effect over time, we show that a similar but more pronounced effect holds by using intraday measures, by such as realized volatility and variance ratio. We also test the impact of unexpected volatility, defined as the part of volatility which cannot be forecasted, on the presence of intraday serial correlation in the time series by employing a model for realized volatility based on the heterogeneous market hypothesis. We find that intraday serial correlation is negatively correlated to volatility forecasts, whereas it is positively correlated to unexpected volatility.
Serial correlation of asset prices is one of the most elusive quantities 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 physicists devoted themselves to the study of stock return predictability (3, 4). Historical returns should prevent any forecasting technique, even if it has been shown, as in ref. 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 more usually called volatility, is a highly predictable quantity (6, 7), with its probability distribution function showing fat tails (8–10). 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 (or trading volume, as in ref. 11) to returns serial correlation. If anything else, the link between volatility and serial correlations can reveal basic properties of the priceformation mechanism.
A notable stylized fact on serial correlation is the LeBaron effect (12), according to which volatility forecasts are negatively correlated to serial correlation. In this work, we find milder evidence of such effect in the dataset we analyze which, being more recent than that used by LeBaron, suggests that market efficiency has increased. Most importantly, we improve on the existing literature by using measures of both volatility and serial correlation which are based on 5minute returns. The forecasting model we use is directly based on realized volatility measures, and it is set in the framework of what can be termed the heterogeneous market hypothesis (13). In this model, volatility is consistently composed by a cascade of several time components (14). The model is particularly successful in recovering the volatility dynamics and mimicking the longrange dependence and fat tails which are observed in the realized volatility time series. To quantify serial correlation, we use instead a modification of the variance ratio statistics with overlapping observations.
Usage of suitable intraday measures allows us to test an intraday version of the LeBaron effect with a very large and liquid dataset. We provide evidence of a negative relation between volatility forecasts and intraday serial correlation. Moreover, we also refine this finding by showing that volatility can be split into 2 components: a predictable one and an unpredictable one, with the latter being positively correlated with serial correlation.
Methodology and Results
The dataset we use is one of the most liquid financial assets in the world, that is the Standard and Poor (S&P) 500 stock index futures from 1993 to 2007, for a total of N = 4344 days. By using the futures instead of the cash index, we avoid the nonsynchronous trading bias (15). We have all highfrequency information, but to avoid microstructure effects we use a grid of n = 84 5minute logarithmic returns per day, interpolated according to the previoustick scheme (the price at time t is the last observed price before t). These choices are the standard ones in this kind of application.
Denote by R_{t} the closetoclose return at day t. Let us assume to have r_{1, t},…,r_{n, t} intraday logarithmic returns. To quantify volatility, we construct daily realized variance measures defined as the cumulative sum of squared intraday 5minute returns (16): Because volatility has been shown to be approximately lognormal (17), with powerlaw deviations in the tail events (9, 10), we use the logarithm of RV_{t} to obtain distributions which are close to normal.
The LeBaron effect (12) can be interpreted as the negative relation between volatility forecasts at time t, obtained with observables up to time t − 1, and the product R_{t}R_{t+1}. We improve on the original LeBaron methodology in 2 ways. First, to obtain volatility forecasts, we borrow from recent advancements in financial econometrics, since we cannot ignore the fact that volatility is well known to display longrange dependence. One effective way to accommodate for this stylized fact without resorting to the estimation burden of a long memory model is the heterogeneous autoregressive (HAR) model of ref. 14. Following the heterogeneous market hypothesis of refs. 18–22, which recognizes the presence of heterogeneity in traders' horizon and the asymmetric propagation of volatility cascade from long to short time periods (23) with respect to that from short to long time periods (24), the basic idea that emerges is that heterogeneous market structure generates an heterogeneous volatility cascade. Hence, ref. 14 proposed a stochastic additive cascade of 3 different realized variance components, which explains the long memory observed in the volatility as the superimposition of few processes operating at different time scales. These processes mirror the 3 typical time horizons operating in the financial market: daily, weekly, and monthly. This stochastic volatility cascade leads to a simple ARtype model in the realized variance with the feature of considering realized volatilities defined over heterogeneous time periods (the HAR model):
where η_{t} is a zeromean estimation error and
Although the HAR model does not formally belong to the class of longmemory models, it generates apparent power laws and long memory, i.e. it is able to reproduce a memory decay which is indistinguishable from that observed in the empirical data. It has been used in many applications in financial economics (25–28). We estimate the HAR model with ordinary least squares, and use the estimated coefficients
Second, we test the LeBaron effect by measuring the dependence of serial correlation from volatility forecasts by using a NadarayaWatson estimator:
with h = 3 · std(σ_{p,t}) ·
Motivated by this finding, we investigate the presence of the LeBaron effect at intraday level (for data sampled at 5minute frequencies) by studying the relation between realized volatility and highfrequency correlation. To measure the latter, we borrow from ref. 30 by using a modified overlapped variance ratio. Define where We define the variance ratio as follows: The use of the power transformation f(x) = x^{β} makes the distribution closer to a normal one in small samples (33). The expression of Eq. 11 is, when the return process is a martingale difference with timevarying bounded variance (see ref. 33 for additional technical assumptions), asymptotically normal with mean 1 and given standard deviation. β is given by where W_{k} is the Fejer kernel: and k = q − 1, λ_{j} = 2πj/n.
Intuitively, the variance ratio expresses the ratio of variances computed at 2 different frequencies whose ratio is given by q. If there is no serial correlation in the data, VR(q) should be close to one. In the presence of positive serial correlation, the variance
We start by first studying the relation between intraday serial correlation and contemporaneous realized volatility by using the simple linear regression and then inserting lagged volatility as well: As in ref. 30, when we use the variance ratio VR(q)_{t} as dependent variable, we also add as explanatory variable 5 lags of VR(q) to remove the autocorrelation of the residuals, that is:
Lagged volatility is, however, a very poor volatility forecast. Thus, we resort again to the HAR model by estimating the regression which fully takes into account heterogeneity, long memory, and heteroskedasticity of financial market volatility. We finally estimate the extension in which unexpected volatility is inserted as an additional explanatory variable. Note that in the regression in Eq. 17, we add a contemporaneous variable σ_{u, t}, which cannot be used for prediction.
Estimation results with q = 2,3,4,5,6 are in Table 1, and Fig. 3 shows the estimated coefficients of regressions 14, 15, and 17 with q = 2,6 on a rolling window with a length of 5 years.
Discussion
Estimates of the model in Eq. 14 depicted in the first row of Fig. 3 may look disappointing, showing no significant correlation between variance ratio and contemporaneous realized variance. Moreover, this correlation tends to be slightly positive (even if not significantly) instead of negative, especially in the first part of the sample.
However, the second row of Fig. 3 shows that the coefficient of lagged volatility on variance ratio is negative and significant across the entire sample. The same can be seen more clearly from the estimate of regressions 16 and 17, reported in Table 1 and the third row of Fig. 3.
Most interestingly, we find that contemporaneous volatility is significantly and positively correlated with the variance ratio. Hence, estimation results for Eq. 15 indicate a sharp difference in the relation between intraday serial correlation and volatility: strongly positive for contemporaneous volatility and strongly negative for lagged one. Such antithetical behavior of the relation is even more puzzling considering the wellknown stylized fact of volatility to be highly persistent. How could we explain this result? By our heterogeneous “rotation” of the regressors, we can rewrite Eq. 15 in the form of Eq. 17. This provides the separation between predictable and unexpected volatility illustrated in Fig. 1. The new specification greatly helps in shedding light on this result, providing a precise economic interpretation. Hence, as ref. 30 suggested, we can now provide an explanation in term of predictable and unexpected volatility: Because volatility is known to be predictable by market participants, it has a different impact with respect to its unpredictable component.
The third row of Fig. 3 shows indeed that the predictable volatility, now defined by means of the HAR model, is negatively correlated with the variance ratio (more with higher q) and that the unpredictable volatility is positively correlated with the variance ratio (more with higher q).
The full sample estimates in Table 1 corroborate this finding. We can then rephrase our results as follows: Intraday serial correlation is negatively correlated with the expected volatility. Moreover, we can conclude that serial correlation is instead positively correlated with unexpected volatility, which is a previously unrecognized empirical feature of financial returns. Our finding suggests that the usual explanation of the LeBaron effect in terms of feedback trading (35) is at least incomplete, advocating for a broader theory on the link between volatility and the way information is spread to heterogeneous market components. It is particularly interesting that a market anomaly like serial correlation is associated with higher unexpected volatility, typically due to unexpected news.
Acknowledgments
We thank the Biomath Program at the College of William and Mary for financial support.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: sbianco{at}wm.edu

Author contributions: S.B., F.C., and R.R. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.
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