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).
Usage
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
)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 < 1explosive,lambda2 > 1mean-reverting).- sigma_b
A positive scalar, the bubble-innovation standard deviation.
- b0
Starting value.
- s0
Starting regime,
1Lor2L.- 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.seedbefore 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.
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).
