The following data generating process is similar to sim_dgp1, with the difference that
there are two episodes of mildly explosive dynamics.
sim_dgp2(n, te1 = 0.2 * n, tf1 = 0.2 * n + te1, te2 = 0.6 * n,
tf2 = 0.1 * n + te2, c = 1, alpha = 0.6, sigma = 6.79)A strictly positive integer specifying the length of the simulated output series.
A scalar in (0, n) specifying the observation in which the first bubble originates.
A scalar in (te1, n) specifying the observation in which the first bubble collapses.
A scalar in (tf1, n) specifying the observation in which the second bubble originates.
A scalar in (te2, n) specifying the observation in which the second bubble collapses.
A positive scalar determining the autoregressive coefficient in the explosive regime.
A positive scalar in (0, 1) determining the value of the expansion rate in the autoregressive coefficient.
A positive scalar indicating the standard deviation of the innovations.
A numeric vector of length n.
The data generating process is described by:
$$X_t = X_{t-1}1\{t \in N_0\}+ \delta_T X_{t-1}1\{t \in B_1 \cup B_2\} + \left(\sum_{k=\tau_{1f}+1}^t \epsilon_k + X^*_{\tau_{1f}}\right) 1\{t \in N_1\} $$
$$ + \left(\sum_{l=\tau_{2f}+1}^t \epsilon_l + X^*_{\tau_{2f}}\right) 1\{t \in N_2\} + \epsilon_t 1\{t \in N_0 \cup B_1 \cup B_2\}$$
where the autoregressive coefficient \(\delta_T\) is given:
$$\delta_T = 1 + cT^{-a}$$
with \(c>0\), \(\alpha \in (0,1)\), \(\epsilon \sim iid(0, \sigma^2)\), \(X_{\tau_{1f}} = X_{\tau_{1e}} + X^*\) and \(X_{\tau_{2f}} = X_{\tau_{2e}} + X^*\). We use the notation \(N_0 = [1, \tau_{1e})\), \(B_1 = [\tau_{1e}, \tau_{1f}]\), \(N_1 = (\tau_{1f}, \tau_{2e})\), \(B_2 = [\tau_{2e}, \tau_{2f}]\), \(N_2 = (\tau_{2f}, \tau]\), where \(\tau\) is the last observation of the sample. After the collapse of the first bubble, \(X_t\) resumes a martingale path until time \(\tau_{2e}-1\), and a second episode of exuberance begins at \(\tau_{2e}\). The expansion process lasts until \(\tau_{2f}\) and collapses to a value of \(X^*_{\tau_{2f}}\). The process then continues on a martingale path until the end of the sample period \(\tau\). The expansion duration of the first bubble is assumed to be longer than that of the second bubble, i.e. \(\tau_{1f}-\tau_{1e}>\tau_{2f}-\tau_{2e}\).
For further details the user can refer to Phillips et al., (2015) p. 1055.
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.
# NOT RUN {
# 100 periods with bubble origination dates 20/60 and termination dates 40/70 respectively
sim_dgp2(n = 100)
# 200 periods with bubble origination dates 40/120 and termination dates 80/140 respectively
sim_dgp2(n = 200)
# }
Run the code above in your browser using DataLab