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
\(X_{\tau_f} = X_{\tau_e} + X^*\).
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.