An analysis of cross-correlations in an emerging market

Abstract

We apply random matrix theory to compare correlation matrix estimators $C$ obtained from emerging market data. The correlation matrices are constructed from 10 years of daily data for stocks listed on the Johannesburg stock exchange (JSE) from January 1993 to December 2002. We test the spectral properties of $C$ against random matrix predictions and find some agreement between the distributions of eigenvalues, nearest neighbour spacings, distributions of eigenvector components and the inverse participation ratios for eigenvectors. We show that interpolating both missing data and illiquid trading days with a zero-order hold increases agreement with RMT predictions. For the more realistic estimation of correlations in an emerging market, we suggest a pairwise measured-data correlation matrix. For the data set used, this approach suggests greater temporal stability for the leading eigenvectors. An interpretation of eigenvectors in terms of trading strategies is given, as opposed to classification by economic sectors.

Introduction

Correlation matrices are common to problems involving complex interactions and the extraction of information from series of measured data. Our aim is to determine empirical correlations in price fluctuations of daily sampled price data of distinct shares in a reliable way. Our investigation is based on 10 years of daily data for 250–350 traded shares listed on the JSE Main Board from January 1993 to December 2002.

There are several aspects to the question of how to calculate correlations in financial time series. In particular, missing data and thin trading (no prices changes for a stock over several time periods) may be significant. Random correlations in price changes are likely to arise in an ensemble of several shares. Furthermore, for a portfolio of $N$ distinct assets, there will be $N(N-1)/2$ entries in a correlation matrix which has been determined from time series of length $L$. When $L$ is not large, the calculated covariance matrix may be dominated by measurement noise. Hence, it is necessary to understand effects of: (i) noise, (ii) finiteness of time series, (iii) missing data, and (iv) thin trading in determination of empirical correlation.

The properties of random matrices first became known with Wigner's seminal work in the 1950s for application in nuclear physics in the study of statistical behaviour of neutron resonances and other complex systems of interactions [1], [2], [3]. More recently random matrix theory has been applied to calibrate and reduce the effects of noise in financial time series and to investigate constraints on rational (empirically based) decision making (cf. Refs. [4], [5], [6], [7], [8], [9], [10], [11], [28], [29], [34]). Correlation matrices are computed for the data under investigation and quantities associated with these matrices may be compared to those of random matrices. The extent to which properties of the correlation matrices deviate from random matrix predictions clarifies the status of the information derived from the computation of covariances. In several studies of shares traded in the S&P 500 and DAX, it was found that, aside from a small number of leading eigenvalues, the eigenvalue spectra for the measured data coincide with theoretic random matrix predictions, i.e., it was found that the estimation of covariances is dominated by random noise. Ref. [12] postulates a model for the correlations to explain the observed spectral properties. RMT has also been shown to yield an improved estimation technique: an estimated correlation matrix can be filtered by removing the contributions of eigenvalues which lie in the RMT range. In Ref. [13] it is shown that noise levels in the correlation matrix depend on the ratio $N:L$, where $N$ denotes the number of stocks and $L$ denotes the length of the time-series.

In this paper we consider the problems of missing data and thin trading in determination of empirical correlation in daily sampled price fluctuations. We analyze the data base containing prices $S_i(t)$, the prices of assets $i=1,\dots ,N$ at time $t$ as follows. We first find the change in asset prices

$$r_i(t)=\ln S_i(t+\mathrm{\Delta }t)-\ln S_i(t)\text{.}$$

The usual cross-correlation matrix for idealised data (non-zero price fluctuations and no missing data) is given by

$$C_{\mathit{ij}}:=\frac{〈r_i{r}_j〉-〈r_i〉〈r_i〉}{{\sigma }_i{\sigma }_j}\text{,}$$

where $〈\dots 〉$ denotes average over period studied and ${\sigma }_i^2:=〈r_i^2〉-〈r_i{〉}^2$ is the variance of the price changes of asset $i$. Alternatively, one could write

$$C_{\mathit{ij}}=\frac{1}{L}\sum _{t=1}^L{R}_i(t)R_j(t)\text{,}$$

where $L$ denotes the uniform length of the time series and $R_i(t)$ denotes the price change of asset $i$ at time $t$ such that the average values of the $R_i$'s have been subtracted off and the $R_i$'s are rescaled so that they all have constant volatility ${\sigma }_i^2:=〈R_i^2〉=1.$ This is written as $\mathbf{C}=(1/L){\mathbf{MM}}^T$ where $\mathbf{M}$ is a $N\times L$ matrix and ${\mathbf{M}}^T$ is its transpose (cf. Ref. [5]).

The pairwise measured-data cross-correlation matrix using the pairwise deletion method [14], [15] for the case when there is missing data in time series of returns is computed as follows:

$${\mathcal{C}}_{\mathit{ij}}:=\frac{〈{\rho }_i{\rho }_j〉-〈{\rho }_i〉〈{\rho }_i〉}{{\sigma }_i{\sigma }_j}\text{,}$$

where ${\rho }_i$ and ${\rho }_j$ denote subseries of $r_i$ and $r_j$ such that there exists measured data for both ${\rho }_i$ and ${\rho }_j$ at every time period in the subseries, and $〈\dots 〉$ denotes average over period studied, ${\sigma }_i^2:=〈{\rho }_i^2〉-〈{\rho }_i{〉}^2$ is the variance of the price changes of asset $i$.