Simulation of an Evans (1991) rational periodically collapsing bubble process.
sim_evans(n, alpha = 1, delta = 0.5, tau = 0.05, pi = 0.7,
r = 0.05, b1 = delta)A strictly positive integer specifying the length of the simulated output series.
A positive scalar, with restrictions (see details).
A positive scalar, with restrictions (see details).
The standard deviation of the innovations.
A positive value in (0, 1) which governs the probability of the bubble continuing to grow.
A positive scalar that determines the growth rate of the bubble process.
A positive scalar, the initial value of the series. Defaults to delta.
A numeric vector of length n.
delta and alpha are positive parameters which satisfy \(0 < \delta < (1+r)\alpha\).
delta represents the size of the bubble after collapse.
The default value of r is 0.05.
The function checks whether alpha and delta satisfy this condition and will return an error if not.
The Evans bubble has two regimes. If \(B_t \leq \alpha\) the bubble grows at an average rate of \(1 + r\):
$$B_{t+1} = (1+r) B_t u_{t+1},$$
When \(B_t > \alpha\) the bubble expands at an increased rate of \((1+r)\pi^{-1}\):
$$B_{t+1} = [\delta + (1+r)\pi^{-1} \theta_{t+1}(B_t - (1+r)^{-1}\delta B_t )]u_{t+1},$$
where \(\theta\) is an indicator function taking a value of 0 with probability \(1-\pi\) and 1 with probability \(\pi\).
In this secondary phase there is a probability (\(1-\pi\)) that the bubble collapses to delta and the process starts again.
By modifying the values of delta, alpha and pi the user can change the frequency at which bubbles appear, the mean duration of a bubble before collapse and the scale of the bubble.
Evans, G. W. (1991). Pitfalls in testing for explosive bubbles in asset prices. The American Economic Review, 81(4), 922-930.