The data generating process is described by the following equation:
$$X_t = X_{t-1}1\{t < \tau_e\}+ \delta_T X_{t-1}1\{\tau_e \leq t\leq \tau_f\} +
\left(\sum_{k=\tau_f+1}^t \epsilon_k + X_{\tau_f}\right) 1\{t > \tau_f\} + \epsilon_t 1\{t \leq \tau_f\}
$$
where the autoregressive coefficient \(\delta_T\) is given by:
$$\delta_T = 1 + cT^{-a}$$
with \(c>0\), \(\alpha \in (0,1)\),
\(\epsilon \sim iid(0, \sigma^2)\) and
\(X_{\tau_f} = X_{\tau_e} + X'\) with \(X' = O_p(1)\),
\(\tau_e = [T r_e]\) dates the origination of the bubble,
and \(\tau_f = [T r_f]\) dates the collapse of the bubble.
During the pre- and post- bubble periods, \([1, \tau_e)\),
\(X_t\) is a pure random walk process. During the bubble expansion period
\(\tau_e, \tau_f]\) becomes a mildly explosive process with expansion rate
given by the autoregressive coefficient \(\delta_T\); and, finally
during the post-bubble period, \((\tau_f, \tau]\) \(X_t\) reverts to a martingale.
For further details see Phillips et al. (2015) p. 1054.