Uniquement disponible en anglais.
Par Gilles de Truchis, Sébastien Fries et Arthur Thomas.
For (Xt) a two-sided α-stable moving average, this paper studies the conditional distribution of future paths given a piece of observed trajectory when the process is far from its central values. Under this framework, vectors of the form Xt = (Xt−m, …, Xt, Xt+1, …, Xt+h), m ≥ 0, h ≥ 1, are multivariate α-stable and the dependence between the past and future components is encoded in their spectral measures. A new representation of stable random vectors on unit cylinders {s ∈ R^(m+h+1) : ||s|| = 1} for ||·|| an adequate seminorm is proposed to describe the tail behaviour of vectors Xt when only the first m + 1 components are assumed to be observed and large in norm. Not all stable vectors admit such a representation, and (Xt) will have to be ‘anticipative enough’ for Xt to admit one. The conditional distribution of future paths can then be explicitly derived using the regularly varying tails property of stable vectors and has a natural interpretation in terms of pattern identification. Through Monte Carlo simulations, we develop procedures to forecast crash probabilities and crash dates and demonstrate their finite sample performances. As an empirical illustration, we estimate probabilities and reversal dates of El Niño and La Niña occurrences.