Simulate GARCH(1,1)/TGARCH(1,1) innovations

Generates shocks z_t = sqrt(h_t) * eps_t from a GARCH(1,1) recursion with an optional threshold (leverage) term, for use as sim_psy1(..., e = sim_vol_garch(...)).

Usage

sim_vol_garch(n, omega = 0.1, alpha = 0.1, beta = 0.8, gamma = 0, seed = NULL)

Arguments

n: Number of innovations to generate.
omega, alpha, beta: Positive GARCH(1,1) parameters. The defaults (omega = 0.1, alpha = 0.1, beta = 0.8) match Whitehouse, Harvey & Leybourne (2025) and Harvey, Leybourne, Taylor & Zu (2024).
gamma: Non-negative TGARCH leverage parameter. The NASDAQ calibration of Monschang & Wilfling (2021) is omega = 0.4387, alpha = 0, beta = 0.9319, gamma = 0.1306.
seed: An object specifying if and how the random number generator (rng) should be initialized. It is either NULL or an integer, which is passed to set.seed before the simulation. If you set it, the value is saved as the "seed" attribute of the returned value. The default, NULL, leaves the state of the rng unchanged and returns .Random.seed as the "seed" attribute. Results are reproducible across the parallel and the non-parallel option when you use the same seed.

Value

A numeric vector of length n.

Details

$$z_t = \sqrt{h_t}\,\epsilon_t,\quad h_t = \omega + \alpha z_{t-1}^2 + \beta h_{t-1} + \gamma z_{t-1}^2 1\{z_{t-1}<0\}$$

with $\epsilon_t \sim NIID(0,1)$ and $h_0 = z_0 = 0$. gamma = 0 (the default) gives plain GARCH(1,1), and gamma > 0 adds the TGARCH leverage effect, a larger response to negative shocks.

References

Whitehouse, E.J., Harvey, D.I. & Leybourne, S.J. (2025). "Real-time monitoring of explosive financial bubbles." Monschang, V. & Wilfling, B. (2021). "Sup-ADF-style bubble-detection methods under test." Empirical Economics, 61, 145-172.

Examples

sim_vol_garch(199, seed = 1) %>%
  autoplot()

# NASDAQ-calibrated TGARCH (Monschang & Wilfling 2021)
sim_vol_garch(199, omega = 0.4387, alpha = 0, beta = 0.9319, gamma = 0.1306, seed = 1) %>%
  autoplot()