Simulate log-returns from a stochastic volatility model

This function draws the initial log-volatility (h_t) from its stationary distribution, meaning that h_0 is drawn from a gaussian distribution with mean zero and standard deviation sigma_h / sqrt(1 - phi^2). h_{t+1} is then simulated from its conditional distribution given h_t, which is N(phi*h_t, sigma_h). Log-returns (y_t) is simulated from its conditional distribution given the latent process h. If model = "gaussian", then y_t given h_t is gaussian with mean zero and standard deviation equal to sigma_y*exp(h_t / 2). Heavy tail returns can be obtained by simulating from the t-distribution by setting model = "t". How heavy of a tail is specified by the degree of freedom parameter df. Note that the observations are scaled by sqrt((df-2)/2) so that the error term has variance equal to one. Asymmetric returns are obtained from the "skew_gaussian" model. How asymmetric is governed by the skewness parameter alpha. The so called leverage model, where we allow for correlation between log-returns and volatility can be simulated by setting model to "leverage" and specifying the correlation parameter rho. In the "skew_gaussian_leverage" model the return shock z_t is skew gaussian with mean zero, unit variance and shape alpha, and it drives the volatility shock: h_{t+1} = phi * h_t + sigma_h * (rho * z_t + sqrt(1 - rho^2) * e_t), where e_t is standard gaussian.

Usage

sim_sv(
  param = list(phi = 0.9, sigma_y = 0.4, sigma_h = 0.2, df = 4, alpha = -2, rho = -0.7),
  nobs = 1000L,
  seed = NULL,
  model = "gaussian"
)

Arguments

param

List of parameters. This includes the standard deviation of the observations, sigma_y, the standard deviation of the latent volatility process, sigma_h, the persistence parameter phi. If model = "t", the degree of freedom df must be specified. If model = "skew_gaussian", the shape parameter alpha must be specified (see the section on skew gaussian models in estimate_parameters for how alpha relates to the skewness) and if model = "leverage", the correlation rho between the latent error term and the observational error has to be specified. If model = "skew_t", the degrees of freedom of the left and the right tail, df_left and df_right, must be specified. The "skew_t_leverage" model needs df_left, df_right and rho; like "skew_gaussian_leverage", the skew t return shock drives the volatility shock.

nobs

Length of time series.

seed

Seed to reproduce simulation.

model

Distribution of error term.

Value

data.table with columns y (observations) and h (latent log-volatility).