Conditional Moments of Noncausal Alpha-Stable Processes and the Prediction of Bubble Crash Odds
研究了非因果厚尾过程的条件矩，给出了泡沫爆发期间预测分布的闭式公式，可用于计算事前崩盘概率，对金融泡沫预警有参考价值。
Noncausal, or anticipative, heavy-tailed processes generate trajectories featuring locally explosive episodes akin to speculative bubbles in financial time series data. For (Xt) a two-sided infinite α-stable moving average (MA), conditional moments up to integer order four are shown to exist provided (Xt) is anticipative enough, despite the process featuring infinite marginal variance. Formulas of these moments at any forecast horizon under any admissible parameterization are provided. Under the assumption of errors with regularly varying tails, closed-form formulas of the predictive distribution during explosive bubble episodes are obtained and expressions of the ex ante crash odds at any horizon are available. It is found that the noncausal autoregression of order 1 (AR(1)) with AR coefficient ρ and tail exponent α generates bubbles whose survival distributions are geometric with parameter ρ^α. This property extends to bubbles with arbitrarily shaped collapse after the peak, provided the inflation phase is noncausal AR(1)-like. It appears that mixed causal–noncausal processes generate explosive episodes with dynamics à la Blanchard and Watson which could reconcile rational bubbles with tail exponents greater than 1.