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.
We give the theoretical basis of a possible explanation for two stylized facts observed in long log-return series: the long-range dependence (LRD) in volatility and the integrated GARCH (IGARCH). ...Both these effects can be explained theoretically if one assumes that the data are nonstationary.
Nonstationarities in Stock Returns Stărică, Cătălin; Granger, Clive
The review of economics and statistics,
08/2005, Letnik:
87, Številka:
3
Journal Article
Recenzirano
The paper outlines a methodology for analyzing daily stock returns that relinquishes the assumption of global stationarity. Giving up this common working hypothesis reflects our belief that ...fundamental features of the financial markets are continuously and significantly changing. Our approach approximates the nonstationary data locally by stationary models. The methodology is applied to the S&P 500 series of returns covering a period of over seventy years of market activity. We find most of the dynamics of this time series to be concentrated in shifts of the unconditional variance. The forecasts based on our nonstationary unconditional modeling were found to be superior to those obtained in a stationary long-memory framework and to those based on a stationary Garch(1,1) data-generating process.
Nonstationarities in stock returns Starica, Catalin; Granger, Clive
The review of economics and statistics,
08/2005, Letnik:
LXXXVII, Številka:
3
Journal Article
Recenzirano
The paper outlines a methodology for analyzing daily stock returns that relinquishes the assumption of global stationarity. Giving up this common working hypothesis reflects our belief that ...fundamental features of the financial markets are continuously and significantly changing. Our approach approximates the nonstationary data locally by stationary models. The methodology is applied to the S&P 500 series of returns covering a period of over seventy years of market activity. We find most of the dynamics of this time series to be concentrated in shifts of the unconditional variance. The forecasts based on our nonstationary unconditional modeling were found to be superior to those obtained in a stationary long-memory framework and to those based on a stationary Garch(1,1) data-generating process. Reprinted by permission of the MIT Press
A popular estimator of the index of regular variation in heavy-tailed models is Hill's estimator. We discuss the consistency of Hill's estimator when it is applied to certain classes of heavy-tailed ...stationary processes. One class of processes discussed consists of processes which can be appropriately approximated by sequences of m-dependent random variables and special cases of our results show the consistency of Hill's estimator for (i) infinite moving averages with heavy-tail innovations, (ii) a simple stationary bilinear model driven by heavy-tail noise variables and (iii) solutions of stochastic difference equations of the form $Y_t = A_t Y_{t-1} + Z_t, - \infty < t < \infty$ where $\{(A_n, Z_n), - \infty < n < \infty\}$ are iid and the Z's have regularly varying tail probabilities. Another class of problems where our methods work successfully are solutions of stochastic difference equations such as the ARCH process where the process cannot be successfully approximated by m-dependent random variables. A final class of models where Hill estimator consistency is proven by our tail empirical process methods is the class of hidden semi-Markov models.
We examined the impact of including sustainability-related constraints in optimal portfolio decision-making. Our analysis covered an investment set containing the components of the S&P500 index from ...1993 to 2008. Optimizations were performed according to the classic mean-variance approach, while sustainability constraints were introduced by eliminating, from the investment pool, those assets that do not comply with the given social responsibility criteria (screening). We compared the efficient frontiers with and without screening. The analysis focused on the three main dimensions of sustainability, namely the environmental, social and governance ones. We found that socially responsible screening gives rise to a small loss in terms of the Sharpe ratio even though it has a great impact on the market capitalization of the optimal portfolio. The spanning test showed that the ex-post differences between the two frontiers, when short selling is not allowed, are significant only in the case of environmental screening.
The asymptotic theory for the sample autocorrelations and extremes of a GARCH(1, 1) process is provided. Special attention is given to the case when the sum of the ARCH and GARCH parameters is close ...to 1, that is, when one is close to an infinite variance marginal distribution. This situation has been observed for various financial log-return series and led to the introduction of the IGARCH model. In such a situation, the sample auto-correlations are unreliable estimators of their deterministic counterparts for the time series and its absolute values, and the sample autocorrelations of the squared time series have nondegenerate limit distributions. We discuss the consequences for a foreign exchange rate series.
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 − α < 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 Hill's estimator is consistent for α−1 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 Hill's 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.
Smoothing the Hill Estimator Resnick, Sidney; Stărică, Cătălin
Advances in applied probability,
03/1997, Letnik:
29, Številka:
1
Journal Article
Recenzirano
Odprti dostop
For sequences of i.i.d. random variables whose common tail 1 – F is regularly varying at infinity wtih an unknown index –α < 0, it is well known that the Hill estimator is consistent for α–1 and ...usually asymptotically normally distributed. However, because the Hill estimator is a function of k = k(n), the number of upper order statistics used and which is only subject to the conditions k →∞, k/n → 0, its use in practice is problematic since there are few reliable guidelines about how to choose k. The purpose of this paper is to make the use of the Hill estimator more reliable through an averaging technique which reduces the asymptotic variance. As a direct result the range in which the smoothed estimator varies as a function of k decreases and the successful use of the esimator is made less dependent on the choice of k. A tail empirical process approach is used to prove the weak convergence of a process closely related to the Hill estimator. The smoothed version of the Hill estimator is a functional of the tail empirical process.
Smoothing the Hill Estimator Resnick, Sidney; Stărică, Cătălin
Advances in applied probability,
03/1997, Letnik:
29, Številka:
1
Journal Article
Recenzirano
Odprti dostop
For sequences of i.i.d. random variables whose common tail 1 –
F
is regularly varying at infinity wtih an unknown index –
α
< 0, it is well known that the Hill estimator is consistent for α
–1
and ...usually asymptotically normally distributed. However, because the Hill estimator is a function of
k = k
(
n
), the number of upper order statistics used and which is only subject to the conditions
k
→∞,
k/n →
0, its use in practice is problematic since there are few reliable guidelines about how to choose
k.
The purpose of this paper is to make the use of the Hill estimator more reliable through an averaging technique which reduces the asymptotic variance. As a direct result the range in which the smoothed estimator varies as a function of
k
decreases and the successful use of the esimator is made less dependent on the choice of
k.
A tail empirical process approach is used to prove the weak convergence of a process closely related to the Hill estimator. The smoothed version of the Hill estimator is a functional of the tail empirical process.
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