Simulation of a mixed causal-noncausal AR(1,1) bubble

Simulation of Blasques, Koopman, Mingoli & Telg (2025)'s mixed causal-noncausal autoregressive (MAR) bubble process, in which transient, self-terminating local bubbles arise autonomously from the noncausal (forward-looking) component – no scripted origination/ collapse dates, unlike sim_psy1.

Usage

sim_mar(
  n,
  phi1 = 0.7,
  psi1 = 0.7,
  dist = c("cauchy", "t"),
  df = 2,
  burn = 100,
  seed = NULL
)

Arguments

n: A positive integer specifying the length of the simulated output series.
phi1: Causal AR coefficient, in (0, 1).
psi1: Noncausal AR coefficient, in (0, 1).
dist: Innovation distribution: "cauchy" or "t" (with df degrees of freedom).
df: Degrees of freedom if dist = "t".
burn: Non-negative burn-in length applied at both ends (see Details).
seed: An object specifying if and how the random number generator (rng) should be initialized. Either NULL or an integer will be used in a call to set.seed before simulation. If set, the value is saved as "seed" attribute of the returned value. The default, NULL, will not change rng state, and return .Random.seed as the "seed" attribute. Results are reproducible across the parallel and non-parallel option when the same seed is used.

Value

A numeric vector of length n.

Details

$$(1-\phi_1 L)(1-\psi_1 L^{-1})y_t = \epsilon_t$$

Simulated by the standard two-sided-filtering method for MAR processes (Lanne & Saikkonen 2011; Gourieroux & Zakoian 2017): the noncausal component is generated by running $u_t=\psi_1 u_{t+1}+\epsilon_t$ backward from a zero boundary burn observations past the end of the sample, then the causal component by running $y_t=\phi_1 y_{t-1}+u_t$ forward from a zero boundary burn observations before the start; both burn-in windows are then dropped.

References

Blasques, F., Koopman, S.J., Mingoli, G. & Telg, S. (2025). "A Novel Test for the Presence of Local Explosive Dynamics." JTSA, 46(5), 966-980.

Examples

sim_mar(200, seed = 123) %>%
  autoplot()