Forecasting extreme trajectories using seminorm representations

For (X t ) 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 X t = (X t-m , . . . , X t , X t+1 , . . . , X t+h ), m ≥ 0, h ≥ 1, are multivariate αstable 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 sets {s ∈ R m+h+1 : ∥s∥ = 1} for ∥ • ∥ an adequate seminorm is proposed to describe the tail behaviour of vectors X t 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 (X t ) will have to be "anticipative enough" for X t 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.