Catalin Starica

Multivariate extremes for models with constant conditional correlations

Abstract Models with constant conditional correlations are versatile tools for describing the behavior of multivariate time series of financial returns. Mathematically speaking, they are solutions of a special class of stochastic recurrence equations (SRE). The extremal behavior of general solutions of SRE has been studied in detail by Kesten [Kesten, H., 1973. Random difference equations and renewal theory for products of random matrices. Acta Mathematica 131, 207–248] and Perfekt [Perfekt, R., 1997. Extreme value theory for a class of Markov chains with values in R d . Advances in Applied Probability 29, 138–164]. The central concept to understanding the joint extremal behavior of such multivariate time series is the multivariate regular variation spectral measure . In this paper, we propose an estimator for the spectral measure associated with solutions of SRE and prove its consistency. Our estimator is the tail empirical measure of the multivariate time series. Successful use of the estimator depends on a good choice of k , the number of upper order statistics contributing to the empirical measure. We introduce a new criteria for the choice of k based on a scaling property of the spectral measure. We investigate the performance of our estimation technique on exchange rate time series from HFDF96 data set. The estimated spectral measure is used to calculate probabilities of joint extreme returns and probabilities of large movements in an exchange rate conditional on the occurrence of extreme returns in another exchange rate. We find a high level of dependence between the extreme movements of most of the currencies in the EU. We also investigate the changes in the level of dependence between the extreme returns of pairs of currencies as the sampling frequency decreases. When at least one return is extreme, a strong dependence between the components is present already at the 4-hour level for most of the European currencies.

Consistency of Hill's Estimator for Dependent Data

Consider a sequence of possibly dependent random variables having the same marginal distribution F, whose tail 1-F is regularly varying at infinity with an unknown index - a < 0 which is to be estimated. For i.i.d. data or for dependent sequences with the same marginal satisfying mixing conditions, it is well known that Hills estimator is consistent for a -I and asymptotically normally distributed. The purpose of this paper is to emphasize the central role played by the tail empirical process for the problem of consistency. This approach allows us to easily prove Hills estimator is consistent for infinite order moving averages of independent random variables. Our method also suffices to prove that, for the case of an AR model, the unknown index can be estimated using the residuals generated by the estimation of the autoregressive parameters.

Second-order regular variation, convolution and the central limit theorem

Second-order regular variation is a refinement of the concept of regular variation which is useful for studying rates of convergence in extreme value theory and asymptotic normality of tail estimators. For a distribution tail 1 - F which possesses second-order regular variation, we discuss how this property is inherited by 1 - F2 and 1 - F*2. We also discuss the relationship of central limit behavior of tail empirical processes, asymptotic normality of Hills estimator and second-order regular variation.

Asymptotic behavior of Hill's estimator for autoregressive data

Consider a stationary, pth order autoregression {X n } satisfying formula math whose innovation sequence {Z n } is iid with regularly varying tail probabilities of index -α. From observations X 1 ,..., X n , one may estimate α -1 by applying Hills estimator to X 1 ,..., X n . Alternatively, a second procedure is to use X 1 ,..., X n to get estimates ? 1 ,...,? p of the autoregressive coefficients and then to estimate the residuals by formula math and then to apply Hills estimator to the estimated residuals. We show that from the point of asymptotic variance, the second procedure is superior.

Empirical Testing of the Infinite Source Poisson Data Traffic Model

The infinite source Poisson model is a fluid queue approximation of network data transmission that assumes that sources begin constant rate transmissions of data at Poisson time points for random lengths of time. This model has been a popular one as analysts attempt to provide explanations for observed features in telecommunications data such as self-similarity, long range dependence and heavy tails. We survey some features of this model in cases where transmission length distributions have (a) tails so heavy that means are infinite, (b) heavy tails with finite mean and infinite variance and (c) finite variance. We survey the self-similarity properties of various descriptor processes in this model and then present analyses of four data sets which show that certain features of the model are consistent with the data while others are contradicted. The data sets are 1) the Boston University 1995 study of web sessions, 2) the UC Berkeley home IP HTTP data collected in November 1996, 3) traces collected in end of 1997 at a Customer Service Switch in Munich, and 4) detailed data from a corporate Ericsson WWW server from October 1998. #Research supported by the Gothenburg Stochastic Centre, by the EU TMR network ERB-FMRX-CT96-0095 on “Computational and statistical methods for the analysis of spatial data” and by the Knut and Aline Wallenburg Foundation. Sidney Resnicks research was also partially supported by NSF grant DMS-97-04982 and NSA Grant MDA904-98-1-0041 at Cornell University.

A simple non-stationary model for stock returns

The aim of the present paper is to show by the example of the S&P 500 return series that a simple non-stationary model seem to fit the data significantly better than conventional GARCH-type models outperforming them also in forecasting the distribution of tomorrow’s return. Instead of a complex endogenous specification of the conditional variance, we assume that the volatility dynamics is exogenous. Since no obvious candidates for explanatory exogenous variables are at hand, we model the volatility as deterministic. This approach leads to a structurally simple regression-type model. Special attention is paid to the accurate description of the tails of the innovations. AMS Subject Classification: primary: 62P20; secondary: 91B28, 91B70, 91B84

Smoothing the Moment Estimator of the Extreme Value Parameter

Let {Xn be a sequence of i.i.d. random variables whose common distribution F belongs to the domain of attraction of an extreme value law. A semi-parametric estimator of the extreme value parameter is the Dekkers, Einmahl and de Haan [8] moment estimator. Practical use of this estimator requires the problematic choice of a number k=k(n) of upper order statistics and there are few reliable guidelines for this choice. An averaging or smoothing technique is proposed for this estimator yielding a less volatile function of k which in practice aids estimation.

Testing for independence in heavy tailed and positive innovation time series

For time series with positive innovations, a test is given to distinguish between data coming from a stationary process where the variables are dependent versus the model being independent identically distributed random variables. The techniques are suitable either when the innovation distribution has heavy right tails or regularly varying left tails and is based on linear programming estimators given by Feigin and Resnick (1992, 1994). Examples using teletraffic data and the lynx data are discussed. The method has applications to model confirmation where the fit of a model is also examined by gauging whether the residuals are independent or not