Mining and Visualizing Local Dependence
among Financial Data
Teruko Takada
∗Graduate School of Business, Osaka City University
Address: 3-3-138, Sugimoto, 3-3-138, Sumiyoshi-ku, Osaka 558-8585, Japan Phone/Fax: +81 6 6605 2226 Email: [email protected]
Financial markets periodically experience sudden large price and volume changes, and the behavior of asset returns in such extreme periods might be quite different from those in the central tendency. In measuring the level of dependence between the asset returns and other variables, therefore, it is desirable to investigate the dependence at each level of asset returns. However, conventional correlation coefficient analysis measures the global dependence between two stochastic variables, which gives large weight on the behavior in the center part of the distributions.
While the local dependence analysis has a big potential to uncover interesting non-central relationships, most literature has focused on the analysis of global dependence. This is partly because the investigation of tail relationship requires high level accuracy in the estimation to make the result meaningful. Accordingly, there is relatively little literature on nonlinear local dependence. In the case of financial asset returns, estimation of the density is generally difficult due to tail fatness, and the problem is enlarged in bivariate setting where the number of required data increases in the squared order.
The objective of this article is to propose a framework for mining and visualizing the local dependence focusing on the tail relations. Local dependence is measured by the mutual information which is defined as the logarithm of the ratio between the joint density, fx!y(x, y), and the product of the two marginal densities, fx(x)fy(y). The pointwise
mutual information is estimated by a kernel-based density estimator, therefore it may be regarded as the population version of contingency table analysis as to the independence of two random variables. Considering applications to finance, the accuracy and robustness in investigating tail behavior of fat tailed distributions is explored. Our approach has advantage over the parametric copula approach, as a tool for exploring the true dependence structure of the target variables.
Because of the information theoretic background and excellence as the general measure of the dependence, global average mutual information has been applied to various field. However, the application in the field of economics and finance is hardly seen. Moreover, the focus of most literature of mutual information has been on the global average values. Application of pointwise mutual information is hardly seen other than the computation of collocations in linguistics or text-mining related problems.
The contribution of this article are three-folds. First, we propose a new framework for local dependence analysis, which is enabled by the achievement of accurate estimation of the pointwise mutual information. By the proposed local dependence analysis, the nonlinear tail relations can be exploited. The graphical analysis of the pointwise mutual information provides the complete view of the local dependence structure. The local impulse response analysis reveals the dynamic local relations including causality. Second, the significance of the local dependence can be assessed by comparing with the confidence
bands of the independent null profile obtained by the bootstrap method; Finally, several new empirical regularities are found in addition to supporting some prior contributions.
1. Estimation of Mutual Information
1-1. Estimation of mutual information with high precision
Accuracy in the joint density estimation is a critical issue in the local dependence analysis, especially if we are investigating tail dependence. Considering the application to financial asset returns which are known to distribute as fat-tailed, we employ adaptive kernel density estimator of Breiman, Meisel and Purcell (1977) and Abramson (1982) whose bandwidth locally varies depending on the mass of the density. Its superior performance of estimating fat-tailed densities is shown by comparative study of Takada (2001, 2008) and Hwang, Lay, Lippman (1994). For improving the accuracy, robust sphering method of Rousseeuw and Zomeren (1990) and several other devices are implemented.
The proposed framework of the local dependence analysis is explained by using the application example of the method to the daily New York Stock Exchange (NYSE) index returns and volume from 1965 to 2008, where the number of adjusted data used for the estimation is 10, 358.
1-2. Graphical analysis of pointwise mutual information
Figure 1-1 illustrates the pointwise mutual information estimates of the NYSE returns and volume on the same trading day. The green line implies no dependence between returns and volume. Yellow to orange means positive mutual information, implying that volume tends to be high (low) when returns are very high (low). Blue area means negative mutual information implying that volume tends not to be high (low) when returns are very high (low). Figure 1-2 is the pointwise mutual information estimates of a sample of no dependence case which shows the noise level of this estimation, and the null profile is obtained by randomizing the order of the given data. From Figure 1-1, we find that the effect of negative past returns to the volume is in the opposite direction depending on the size of the price downfall.
−0.03 0.00 0.02 −0.5 0.0 0.5 Return (ΔPt) Volume (V t ) 1. ΔPt→ Vt Return (ΔPt) Volume (V t ) −0.03 0.00 0.02 −0.5 0.0 0.5 2. No dependence (A sample) Return (ΔPt) Volume (V shuffled ) −0.03 0.00 0.02 −0.5 0.0 0.5 3. No dependence (2.5% quantile) Return (ΔPt) Volume (V shuffled ) −0.03 0.00 0.02 −0.5 0.0 0.5 4. No dependence (97.5% quantile) Return (ΔPt) Volume (V shuffled )
Figure 1: Pointwise mutual information of NYSE returns and volume
1-3. Local impulse response analysis
We divided the data range into nine sub-regions and computed the average mutual infor-mation for each region, which we call local average mutual inforinfor-mation. In Figure 1-2, the subregion (ΔP+, V+) illustrates the local dependence when large price increase induces large volume in the future. The four sub-regions at the corners of Figure 2 depict that the change in the impact of large price change to the future volume level. For example, a large
price significantly induces a large volume in the following days, but the effect disappears shortly. −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 MI(local average) (ΔP− ,V+) ΔPt→ Vt+lag (ΔPm ,V+) (ΔP+ ,V+) −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 MI(local average) (ΔP− ,Vm) (ΔPm ,Vm) (ΔP+ ,Vm) 0 5 10 20 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 MI(local average) lag (ΔP− ,V−) 0 5 10 20 lag (ΔPm ,V−) 0 5 10 20 lag (ΔP+ ,V−)
Figure 2: Local impulse response analysis from NYSE returns to future volume. The data size is fixed as the same for all the lag length.
2. Evaluation of Significance of Local Dependence
The significance of the detected profile can be statistically validated by comparing the estimated results with the confidence band of the null profile. If the resulting estimate falls within the null confidence band, insignificance of the result is indicated. We have constructed 95% null confidence band by bootstrapping based on 1, 001 replicated null samples obtained by shuffling the order of the given data.
Figure 1-3 and 1-4 are 2.5% and 97.5% sample quantile points of the 1, 001 null profiles at each grid points of the data range, respectively. In other words, if the color is found to be darker than those shown in Figure 1-3 and 1-4, the corresponding dependence is considered to be significant. The graphical confidence band is wider at the corners, reflecting the large noise level due to small number of available data points in the tails.
In Figure 2, the two blue dotted lines are the 95% null confidence band for local average mutual information estimates corresponding to each sub-region. If the length of the red bar indicating the local average dependence exceeds the confidence band in absolute value, the dependence is considered to be significant. Reflecting the number of available samples, the confidence bands in the subregions at the corners are wide, while those of the center subregion shrink to one line.
3. Empirical Regularities Found from the Application to Stock Price Changes and Volume
By applying the proposed method to the daily NYSE index returns and volume from 1965 to 2008, the local dependence structure among returns and volume for all directions are
investigated. We found the following empirical regularities: (1) The effect of negative past returns to the volume in the same and the next day is in the opposite direction depending on the size of the downfall; (2) Large price downfall weakly induces persisting low volume, while large price increase significantly induces large volume which disappears in a shorter period; (3) While the impact of price changes is symmetric after the shock, the effect of large price increase disappears in one or two days, but the effect of large price decrease persists; (4) High past volume induces high positive returns for several days.
References
Abramson, I. S (1982). On bandwidth variation in kernel estimates – A square root law, Annals of Statistics,10, 1217–1223.
Breiman, L., W. Meisel and E. Purcell (1977). Variable kernel estimates of multivariate densities, Technometrics,19, 135–144.
Hwang, J. N., S. R. Lay and A. Lippman (1994). Nonparametric multivariate density estimation: a comparative study. IEEE Transactions on Signal Processing, 42, 2795–2810.
Rousseeuw, P. J. and B. C. van Zomeren (1990). Unmasking multivariate outliers and leverage points, Journal of the American Statistical Association, 85, 633–639. Takada, T. (2001). Density Estimation for Robust Financial Econometrics, PhD thesis,
University of Illinois at Urbana-Champaign.
Takada, T. (2008). Asymptotic and qualitative performance of nonparametric density estimators: A comparative study. The Econometrics Journal. doi: 10.1111/j.1368-423X.2008.00249.x.
