exuber (version 0.2.1)

sim_dgp1: Simulation of a single-bubble process

Description

The following function generates a time series which switches from a martingale to a mildly explosive process and then back to a martingale.

Usage

sim_dgp1(n, te = 0.4 * n, tf = 0.15 * n + te, c = 1, alpha = 0.6,
  sigma = 6.79)

Arguments

n

A strictly 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.

c

A positive scalar determining the autoregressive coefficient in the explosive regime.

alpha

A positive scalar in (0, 1) determining the value of the expansion rate in the autoregressive coefficient.

sigma

A positive scalar indicating the standard deviation of the innovations.

Value

A numeric vector of length n.

Details

The data generating process is described by the following equation: $$X_t=X_{t-1}1\{t<{\tau }_e\}+{\delta }_T{X}_{t-1}1\{{\tau }_e\le t\le {\tau }_f\}+(\sum _{k={\tau }_f+1}^t{ϵ}_k+X_{{\tau }_f}^{\ast })1\{t>{\tau }_f\}+{ϵ}_{t}1\{t\le {\tau }_f\}$$

where the autoregressive coefficient ${\delta }_T$ is given by:

$${\delta }_T=1+c{T}^{-a}$$

with $c>0$, $\alpha \in (0,1)$, $ϵ\sim iid(0,{\sigma }^2)$ and $X_{{\tau }_f}=X_{{\tau }_e}+X^{\ast }$. During the pre- and post- bubble periods, $N_0=[1,{\tau }_e)$, X is a pure random walk process. During the bubble expansion period $B=[{\tau }_e,{\tau }_f]$ is a mildly explosive process with expansion rate given by the autoregressive coefficient ${\delta }_T$, and continues its martingale path for the subsequent period $N_1=({\tau }_f,\tau ]$.

For further details the user can refer to Phillips et al. (2015) p. 1054.

References

Phillips, P. C. B., Shi, S., & Yu, J. (2015). Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500. International Economic Review, 5 6(4), 1043-1078.

See Also

sim_dgp2, sim_blan, sim_evans

Examples

# NOT RUN {
# 100 periods with bubble origination date 40 and termination date 55
sim_dgp1(n = 100)

# 200 periods with bubble origination date 80 and termination date 110
sim_dgp1(n = 200)

# 200 periods with bubble origination date 100 and termination date 150
sim_dgp1(n = 200, te = 100, tf = 150)
# }