Simulation of a single-bubble process with multiple forms of collapse regime

sim_ps1(
  n,
  te = 0.4 * n,
  tf = te + 0.2 * n,
  tr = tf + 0.1 * n,
  c = 1,
  c1 = 1,
  c2 = 1,
  eta = 0.6,
  alpha = 0.6,
  beta = 0.5,
  sigma = 6.79,
  seed = NULL,
  e = NULL
)

The process differs from the sim_psy1 model in three respects (Phillips and Shi 2018). First, it includes an asymptotically negligible drift in the martingale path during normal periods. Second, the collapse is modeled directly as a transient mildly integrated process that covers an explicit period of market collapse. Third, it introduces a market recovery date to capture the return to normal market behavior. Three forms of collapse are available:

• sudden: with beta = 0.1 and tr = tf + 0.01*n
• disturbing: with beta = 0.5 and tr = tf + 0.1*n
• smooth: with beta = 0.9 and tr = tf + 0.2*n

To set the duration of the collapse period through tr = tf + 0.2n, you must also provide tf.

Arguments

n	A positive integer specifying the length of the simulated output series.
te	A scalar in (0, tf) specifying the observation in which the bubble originates.
tf	A scalar in (te, n) specifying the observation in which the bubble collapses.
tr	A scalar in (tf, n) specifying the observation in which market recovers
c	A positive scalar determining the drift in the normal market periods.
c1	A positive scalar determining the autoregressive coefficient in the explosive regime.
c2	A positive scalar determining the autoregressive coefficient in the collapse regime.
eta	A positive scalar (>0.5) determining the drift in the normal market periods.
alpha	A positive scalar in (0, 1) determining the autoregressive coefficient in the bubble period.
beta	A positive scalar in (0, 1) determining the autoregressive coefficient in the collapse period.
sigma	A positive scalar indicating the standard deviation of the innovations.
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.
e	An optional numeric vector of length n - 1 with innovations to use in place of rnorm(n - 1, sd = sigma). It lets the plain PSY equation above be driven by a shock sequence that is non-Gaussian, heteroskedastic or dependent instead of i.i.d. Gaussian noise. The generators sim_innov (heavy-tailed or skewed), sim_vol_break (permanent volatility break), sim_vol_garch (GARCH/TGARCH), sim_vol_cir and sim_vol_sv (stochastic volatility) and sim_fi (long memory) produce suitable sequences. The default NULL reproduces the plain i.i.d. Gaussian process exactly.

Value

A numeric vector of length n.

# Disturbing collapse (default)
disturbing <- sim_ps1(100)
autoplot(disturbing)

# Sudden collapse
sudden <- sim_ps1(100, te = 40, tf= 60, tr = 61, beta = 0.1)
autoplot(sudden)