# sim_evans: Simulation of an Evans (1991) bubble process In exuber: Econometric Analysis of Explosive Time Series

## Description

Simulation of an Evans (1991) rational periodically collapsing bubble process.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```sim_evans( n, alpha = 1, delta = 0.5, tau = 0.05, pi = 0.7, r = 0.05, b1 = delta, seed = NULL ) ```

## Arguments

 `n` A positive integer specifying the length of the simulated output series. `alpha` A positive scalar, with restrictions (see details). `delta` A positive scalar, with restrictions (see details). `tau` The standard deviation of the innovations. `pi` A positive value in (0, 1) which governs the probability of the bubble continuing to grow. `r` A positive scalar that determines the growth rate of the bubble process. `b1` A positive scalar, the initial value of the series. Defaults to `delta`. `seed` 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 saved as "seed" attribute of the returned value. The default, NULL, will not change rng state, and return .Random.seed as the "seed" attribute. Results are different between the parallel and non-parallel option, even if they have the same seed.

## Details

`delta` and `alpha` are positive parameters which satisfy 0 < δ < (1+r)α. `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] ≤ α the bubble grows at an average rate of 1 + r:

B[t+1]= (1+r)*B[t]*u[t+1].

When B[t] > α the bubble expands at the increased rate of (1+r)π^{-1}:

B[t+1] = δ*(1+r)/π* (B[t]-δ/(1+r))) *u[t+1],

where θ theta is a binary variable that takes the value 0 with probability 1-π and 1 with probability π. In the second phase, there is a (1-π) probability of the bubble process collapsing to `delta`. 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.

## Value

A numeric vector of length `n`.

## References

Evans, G. W. (1991). Pitfalls in testing for explosive bubbles in asset prices. The American Economic Review, 81(4), 922-930.

## See Also

`sim_psy1`, `sim_psy2`, `sim_blan`

## Examples

 ```1 2``` ```sim_evans(100, seed = 123) %>% autoplot() ```

### Example output

```
```

exuber documentation built on Dec. 18, 2020, 9:06 a.m.