Conditional Moments of Noncausal Alpha-Stable Processes and the Prediction of Bubble Crash Odds

Abstract

Noncausal, or anticipative, heavy-tailed processes generate trajectories featuring locally explosive episodes akin to speculative bubbles in financial time series data.For () t X a two-sided infinite α-stable moving average (MA), conditional moments up to integer order four are shown to exist provided () t X is anticipative enough, despite the process featuring infinite marginal variance. Formulae of these moments at any forecast horizon under any admissible parameterisation are provided. Under the assumption of errors with regularly varying tails, closed-form formulae 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   . 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 (1982) which could reconcile rational bubbles with tail exponents greater than 1.

Forecasting bubbles with mixed causal-noncausal autoregressive models

“…We use the alpha-stable distribution as the data-generating process (DGP). For implementations of the alpha-stable distribution in noncausal models, see Fries and Zakoian (2019) and Fries (2021). The probability density function of the alpha-stable is…”

“…For the same distribution, Andrews, Davis, and Breidt (2006) proposes the estimation of the all-pass ARMA models, where all roots of the autoregressive polynomial are reciprocals of roots of the moving average polynomial and vice versa. MLE for autoregressive models using the alpha-stable distribution has also been implemented in Andrews, Calder, and Davis (2009); Fries (2021).…”

Forecast performance and bubble analysis in noncausal MAR(1, 1) processes

“…8 See also Fries (2019) for a related result derived from conditional power moments behavior resembles the martingale tree 9 reported in Blanchard and Watson (1982).…”

Is Climate Change Time-Reversible?

“…A mixed causal and noncausal model represented in this way is denoted as MAR(r,s), where ϕ(L −1 ) is the noncausal polynomial of order s and φ(L) is the causal polynomial of order r. Exactly as representation (3), r + s = p is true even in this case. Purely causal and purely noncausal models are obtained setting, respectively, ϕ(L −1 ) = 1 and φ(L) = 1 (see Gouriéroux and Zakoian 2013;Hencic and Gouriéroux 2015;Hecq et al 2016;Fries and Zakoian 2019;Hecq and Voisin 2021;Giancaterini and Hecq 2022;Fries 2021). In (8), both causal and noncausal polynomials have their roots outside the unit circle, such that:…”