Supplementary Appendix Intraday Stochastic Volatility in Discrete Price Changes: the Dynamic Skellam Model

We first present a short review of Monte Carlo techniques for likelihood evaluation for state space models and the efficient importance sampler (EIS) of Liesenfeld and Richard (2003) and Richard and Zhang (2007). Next, we present our bivariate extension of the numerically accelerated importance sampling techniques (NAIS) of Koopman, Lucas, and Scharth (2014). NAIS: likelihood evaluation and importance sampling The non-Gaussian nonlinear state space model in the main paper is introduced as yt ∼ p(yt|θt;ψ), θt = ct +Ztαt, αt+1 ∼ pg(αt+1|αt;ψ), t = 1,..., n, where yt is the observation, θt is the unobserved signal vector composed of a constant vector ct and a linear function of a fixed matrix Zt and the dynamic, stochastic state vector αt. The time index is t and we assume to have n observations for yt. The fixed unknown parameter vector is ψ and will be estimated via the numerical optimization of the likelihood function that is evaluated by a simulation method as discussed below. We assume that for given realizations of the signal θ ′ = (θ′1,...,θ n) the observations y = (y1,..., yn) ′ are conditionally independent. In vector notation, we have θ = c+Zα with c ′ = (c′1,..., c n), α = (