Description Usage Arguments Details Value References See Also Examples
The following data generating process is similar to sim_psy1
, with the difference that
there are two episodes of mildly explosive dynamics.
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n |
A positive integer specifying the length of the simulated output series. |
te1 |
A scalar in (0, n) specifying the observation in which the first bubble originates. |
tf1 |
A scalar in (te1, n) specifying the observation in which the first bubble collapses. |
te2 |
A scalar in (tf1, n) specifying the observation in which the second bubble originates. |
tf2 |
A scalar in (te2, n) specifying the observation in which the second 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. |
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
|
The two-bubble data generating process is given by (see also sim_psy1
):
X[t]=X[t-1] 1{t in N[0]}+ δ[T] * X[t-1] 1{t in B[1] union B[2]} + (∑[k=t1f+1]^t ε[k] + X'[t1f]) 1{t in N[1]} +
(∑[l=t2f+1]^t ε[l] + X'[t2f]) 1{t in N[2]} + ε[t] 1{t in N[0] union B[1] union B[2]},
where the autoregressive coefficient δ[T] is:
δ[T] = 1 + c*T^{-a},
with c>0, α in (0,1), ε - iid(0, σ^2), N0 = [1, t1e), B1 = [te1, t1f], N0 = (t1f, t2e), N0 = [t2e, t2f], N0 = [t2f, t], where t is the last observation of the sample. The observations te1 = [T re1] and tf = [T r1f] are the origination and termination dates of the first bubble; te2 = [T re2] and tf = [T r2f] are the origination and termination dates of the second bubble. After the collapse of the first bubble, X[t] resumes a martingale path until time t2e - 1, and a second episode of exuberance begins at t2e. Exuberance lasts lasts until t2f at which point the process collapses to a value of X[t2f]. The process then continues on a martingale path until the end of the sample period t. The duration of the first bubble is assumed to be longer than that of the second bubble, i.e. t1f - t1e > t2f - t2e.
For further details you can refer to Phillips et al., (2015) p. 1055.
A numeric vector of length n
.
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.
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