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Abstract:
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The anticipative, or noncausal, alpha-stable autoregression of order 1 (AR(1)) is a stationary Markov process undergoing short-lived explosive episodes akin to bubbles in financial time series data: recurrently, it diverges away from central values at exponential speed and brutally collapses. Although featuring infinite variance, conditional moments up to integer order four may exist. Little is known about their forms and this impedes understanding of the dynamics of anticipative processes and the ability to forecast them. We provide the functional forms of the conditional expectation, variance, skewness and kurtosis at any forecast horizon under any admissible parameterisation of the process. During bubble episodes, the moments become equivalent to that of a weighted Bernoulli distribution charging complementary probabilities to two polarly-opposite deterministic paths: pursued explosion or collapse. These results extend to the continuous time analogue of the AR(1), the anticipative alpha-stable Ornstein-Uhlenbeck. The proofs build heavily on and extend properties of arbitrary, not necessarily symmetric alpha-stable bivariate random vectors.
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