Monitoring the cross-covariances of a multivariate time series

Abstract

In this paper sequential procedures are proposed for jointly monitoring all elements of the covariance matrix at lag 0 of a multivariate time series. All control charts are based on exponential smoothing. As a measure of the distance between the target values and the actual values the Mahalanobis distance is used. It is distinguished between residual control schemes and modified control schemes. Several properties of these charts are proved assuming the target process to be a stationary Gaussian process. Within an extensive Monte Carlo study all procedures are compared with each other. As a measure of the performance of a control chart the average run length is used. An empirical example about Eastern European stock markets illustrates how the autocovariance and the cross-covariance structure of financial assets can be monitored by these methods. Copyright Springer-Verlag 2005Statistical process control, Multivariate time series, Simultaneous control charts, Exponential smoothing, Financial application,

Interpolation in Time Series: An Introductive Overview of Existing Methods, Their Performance Criteria and Uncertainty Assessment

“…The ARIMA (0,1,1) is also called the simple smoothing exponential [4], or the Exponentially Weighted Moving Average (EWMA, in [44]), and is presented in Equations (11a) and (11b).…”

Surveillance of the covariance matrix of multivariate nonlinear time series

“…A generalization of the control chart of Lowry et al [11] to multivariate time series was proposed by Kramer and Schmid [14]. While all these papers dealt with mean shifts,Śliwa and Schmid [17] considered EWMA charts for the detection of a change in the autocovariance matrix at lag 0 of a Gaussian process. This is a very general family of stochastic processes including all stationary ARMA processes with normal innovations.…”

“…This relaxes the strong assumption imposed by several authors on the distribution of the analysed process [e.g., refs. [16,17]]. …”

Properties and Use of the Shewhart Method and Its Followers

“…In several papers by Schmid and coworkers, multivariate economic surveillance is studied. An example is Sliwa and Schmid (2005) where the autocovariance and cross-covariance structures of financial assets in the Eastern European stock markets are monitored. Reviews of multivariate surveillance are given for example by Woodall and Montgomery (1999) and Sonesson and Frisén (2005).…”

“…In both the vector and the scalar accumulation approaches, the correlations between the variables are used in the transformation. Rosolowski and Schmid (2003) and Sliwa and Schmid (2005) use the Mahalanobis distance to measure the distance between the target values and the actual values together with vector accumulation.…”