scholarly journals Auto-tail Dependence Coefficients for Stationary Solutions of Linear Stochastic Recurrence Equations and for GARCH(1,1)

2011 ◽  
pp. 327-339
Author(s):  
Raymond Brummelhuis
Keyword(s):  
Tail Dependence ◽  
Stationary Solutions ◽  
Recurrence Equations ◽  
Stochastic Recurrence

scholarly journals CHARACTERIZATION OF THE TAIL BEHAVIOR OF A CLASS OF BEKK PROCESSES: A STOCHASTIC RECURRENCE EQUATION APPROACH

2021 ◽  
pp. 1-34
Author(s):  
Muneya Matsui ◽  
Rasmus Søndergaard Pedersen
Keyword(s):  
Stationary Solutions ◽  
Tail Index ◽  
Recurrence Equation ◽  
Multivariate Garch ◽  
Tail Behavior ◽  
Stochastic Recurrence ◽  
Garch Processes ◽  
Strict Stationarity ◽  
Daily Returns

Abstract We consider conditions for strict stationarity and ergodicity of a class of multivariate BEKK processes $(X_t : t=1,2,\ldots )$ and study the tail behavior of the associated stationary distributions. Specifically, we consider a class of BEKK-ARCH processes where the innovations are assumed to be Gaussian and a finite number of lagged $X_t$ ’s may load into the conditional covariance matrix of $X_t$ . By exploiting that the processes have multivariate stochastic recurrence equation representations, we show the existence of strictly stationary solutions under mild conditions, where only a fractional moment of $X_t$ may be finite. Moreover, we show that each component of the BEKK processes is regularly varying with some tail index. In general, the tail index differs along the components, which contrasts with most of the existing literature on the tail behavior of multivariate GARCH processes. Lastly, in an empirical illustration of our theoretical results, we quantify the model-implied tail index of the daily returns on two cryptocurrencies.

Non-causal strictly stationary solutions of random recurrence equations

2014 ◽  
Vol 94 ◽  
pp. 113-118
Author(s):  
Dirk-Philip Brandes ◽  
Alexander Lindner
Keyword(s):  
Stationary Solutions ◽  
Recurrence Equations

scholarly journals Fractional Moments of Solutions to Stochastic Recurrence Equations

2013 ◽  
Vol 50 (4) ◽  
pp. 969-982 ◽  
Author(s):  
Thomas Mikosch ◽  
Gennady Samorodnitsky ◽  
Laleh Tafakori
Keyword(s):  
Stationary Solution ◽  
Numerical Study ◽  
Recurrence Equation ◽  
Erlang Distribution ◽  
Recurrence Equations ◽  
Stochastic Recurrence ◽  
Fractional Moments ◽  
Independent And Identically Distributed

In this paper we study the fractional moments of the stationary solution to the stochastic recurrence equation Xt = AtXt−1 + Bt, t ∈ Z, where ((At, Bt))t∈Z is an independent and identically distributed bivariate sequence. We derive recursive formulae for the fractional moments E|X0|p, p ∈ R. Special attention is given to the case when Bt has an Erlang distribution. We provide various approximations to the moments E|X0|p and show their performance in a small numerical study.

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