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, seed = NULL)
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
.
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
set.seed
before simulation. If set, the value is save as "seed" attribute
of the returned value. The default, NULL will note change the rng state, and
return .Random.seed as the "seed" attribute.
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.