Simulation of a Markov-switching present-value bubble

sim_msbubble(
  n,
  p11 = 0.98,
  p22 = 0.9,
  lambda1 = 0.98,
  lambda2 = 1.03,
  sigma_b = 0.05,
  b0 = 0,
  s0 = 1L,
  seed = NULL
)

Simulation of Chan & Santi (2021)'s bubble component of a present-value state-space model: an AR(1) whose persistence switches between a "surviving" (explosive) and a "collapsing" (mean-reverting) regime under a first-order Markov chain, rather than at deterministic dates (sim_psy1) or a fixed-probability mixture (sim_blan).

Arguments

n	A positive integer specifying the length of the simulated output series.
p11, p22	Regime-1-to-1 and regime-2-to-2 transition probabilities, in (0, 1).
lambda1, lambda2	Regime persistence parameters (lambda1 < 1 explosive, lambda2 > 1 mean-reverting).
sigma_b	A positive scalar, the bubble-innovation standard deviation.
b0	Starting value.
s0	Starting regime, 1L or 2L.
seed	An object specifying if and how the random number generator (rng) should be initialized. It is either NULL or an integer, which is passed to set.seed before the simulation. If you set it, the value is saved as the "seed" attribute of the returned value. The default, NULL, leaves the state of the rng unchanged and returns .Random.seed as the "seed" attribute. Results are reproducible across the parallel and the non-parallel option when you use the same seed.

Value

A numeric vector of length n, with a "regime" attribute (the simulated S_t path).

Details

$$b_t = \frac{1}{\lambda_{S_t}}b_{t-1}+\epsilon_t^b,\quad \epsilon_t^b \sim iid\,N(0,\sigma_b^2)$$ with $S_t \in \{1,2\}$ a Markov chain with transition probabilities p11 = P(S[t]=1|S[t-1]=1), p22 = P(S[t]=2|S[t-1]=2). Regime 1 ("surviving") uses lambda1 < 1 (so 1/lambda1 > 1, explosive); regime 2 ("collapsing") uses lambda2 > 1 (mean-reverting). Note: the source's own eq. 16 indexes the coefficient by $S_{t+1}$; this implementation uses the contemporaneous $S_t$ instead (an indexing-convention simplification, not a change to the qualitative Markov-switching mechanism).

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