یک روش عضو رابط بر پویایی وابستگیهای آماری در بازار سهام ایالات متحده

We analyze the statistical dependence structure of the S&P 500 constituents in the 4-year period from 2007 to 2010 using intraday data from the New York Stock Exchange’s TAQ database. Instead of using a given parametric copula with a predetermined shape, we study the empirical pairwise copula directly. We find that the shape of this copula resembles the Gaussian copula to some degree, but exhibits a stronger tail dependence, for both correlated and anti-correlated extreme events. By comparing the tail dependence dynamically to the market’s average correlation level as a commonly used quantity we disclose the average level of error of the Gaussian copula, which is implied in the calculation of many correlation coefficients.

The measurement of statistical dependence is often broken down to the calculation of a correlation coefficient, such as the Pearson coefficient [1] or the Spearman coefficient [2]. Correlation coefficients are widely used in various disciplines of science. It is also often included in financial modeling, e.g., in the Capital Assets Pricing Model (CAPM) [3] or Noh’s model [4]. The Pearson correlation coefficient, however, only accounts for linear statistical dependence assuming that the observables are nearly normal distributed. Due to the central limit theorem, this might be justified in some cases, but often the statistical dependence is much more complex. In these cases, the statistical dependence cannot be represented by a single number. The joint probability distribution, of course, holds all information of the statistical dependence. Certainly, the joint probability distribution also contains the individual marginal probability distributions. These can have different shapes depending on the underlying process. The statistical dependence of different systems usually cannot be directly compared with this approach. Copulae, first introduced by Sklar in 1959 [5] and [6], permit a separation between the pure statistical dependence and the marginal probability distributions. This allows to compare the statistical dependence of diverse systems. The usage of copulae is well established in statistics and finance. There are many classes of analytical copula functions that meet various properties [7]. Several studies of financial markets are devoted to developing suitable copulae or fitting existing ones to empirical data [8], [9] and [10] or are based on a small subset of assets [11]. In this study, we choose a different approach. We perform a large-scale empirical study to disclose the structure of the average pairwise copula of the US stock returns. As the copula does not depend on the shape of the return distribution, we are able to average over the copula of different stock pairs although their marginal distributions’ shape may differ, i.e., exhibits stronger or weaker tails. In particular, we study the intraday stock market returns of the 428 continuous S&P 500 constituents in 2007–2010 based on intraday data from the New York Stock Exchange’s TAQ database. This empirical study enables us to disclose the average level of error involved in the Gaussian copula. We then compare the tail dependence of the empirical copula and the Gaussian copula dynamically based on a moving window of 2-weeks. We map this tail dependence to the market’s average correlation level and thereby provide an error estimate for the Gaussian copula based on the current general market situation. This article is organized as follows. In Section 2, we give a brief introduction to the concept of copulae. We discuss the average pairwise copula in Section 3, which is extended by dynamical aspects in Section 4. We conclude our results in Section 5.

In a large-scale empirical study of the S&P 500 stocks’ copula, we disclosed important features of the statistical dependence structure. This gives the opportunity to isolate the statistical dependence structure from features of the probability distributions, such as heavy tails. In general, the overall average pairwise copula of the 4-year period feature stronger tails than the Gaussian copula. Extreme events are much more correlated than assumed by a linear correlation. Moreover, the empirical copula indicates the presence of anti-correlated extreme events. Despite the large differences between the Gaussian marginal distribution and the distribution of high frequency returns, the dependence structure in the central part of the distributions is quite similar. This explains why techniques to reduce risk involving correlations work well in “quiet times”, as this is represented by the center region of the copula. It also gives insight into the reason why these approaches often fail during stock market crashes. The probability of simultaneous extreme events, both in correlated and anti-correlated manner, is underestimated. In a further study, where we calculated the time-dependent empirical copula in the resolution of 2-weeks, we showed that on small return intervals the Gaussian copula, in particular, systematically underestimates the tail dependence. The market reacts sensible to large negative returns resulting in a collective downward motion. The evolution of the copula in the 4-year period discloses a strong relation between the market’s average correlation level and the tail dependence. For return intervals of 4 h and in the center region of the distribution, the Gaussian copula describes the situation fairy well. However, when using smaller return intervals or estimating the tail regions, the fluctuations in the correlation–tail-dependence relation become strong. By comparing the empirical copula to the Gaussian copula using the market’s average correlation level we disclose the degree of error involved in the usage of correlation coefficients. By identifying this degree of error one can enhance the estimation of risk in models that use correlation coefficients significantly.