Quasi-likelihood SV(1) via the Kalman filter

The stochastic volatility model $r_t = \exp(h_t/2) z_t$ with $z_t$ standard normal and $h_t$ a Gaussian AR(1) is not a linear Gaussian state space model, so the exact likelihood needs simulation. Squaring and taking logs linearises it, $$\log r_t^2 = h_t + \log z_t^2,$$ at the price of a measurement error $\log z_t^2$ that is the log of a chi-square with one degree of freedom, and so is very far from normal: it is sharply left-skewed. Treating it as normal anyway, with its true first two moments $E[\log z^2] = \psi(1/2) + \log 2 = -\gamma - \log 2 = -1.2703628$ and $Var[\log z^2] = \psi'(1/2) = \pi^2/2 = 4.9348022$, turns the model into a linear Gaussian one whose Kalman filter prediction errors give a quasi likelihood. Maximising it is consistent and asymptotically normal but not efficient, and the reported ll is a quasi log-likelihood, not a log-likelihood: it is not comparable with the likelihood of a model estimated exactly.

Usage

Volsv(r, init = NULL, sweeps = 25L, offset = 1e-08)

Arguments

r: Numeric vector of returns, not squared and not logged.
init: Optional starting c(mu, phi, sigma_eta). Defaults to a method of moments start read off $\log r_t^2$.
sweeps: Number of coordinate-ascent sweeps.
offset: Added to $r_t^2$ before taking logs.

Value

Named list with mu, phi, sigma_eta, ll, n, sweeps, offset, init, method.

Details

The filter runs on the state $h^*_t = h_t - \mu$, with stationary prior $Var(h^*_0) = \sigma_\eta^2/(1 - \phi^2)$ and measurement intercept $\mu + E[\log z^2]$.

Maximisation is a deterministic coordinate ascent: golden-section line search on each of $\mu$, $\phi$ and $\log \sigma_\eta$ in turn, for a fixed number of sweeps. There is no random restart and no convergence tolerance, so the result is a deterministic function of the data and of init.

Zero returns make $\log r_t^2$ infinite. offset is added to $r_t^2$ before the log to keep the filter finite; it is the usual remedy and it biases the estimate slightly, so it is reported.

Mirrors morie.fn.volsv on the Python side.

References

Harvey A C, Ruiz E & Shephard N (1994). Multivariate stochastic variance models. Review of Economic Studies 61(2), 247–264.

Examples

set.seed(3)
Volsv(rnorm(200) * 0.5, sweeps = 5)$phi
#> [1] 0.2336336