This paper extends the basic stochastic volatility (SV) model in order to incorporate the realized variance (RV) as an additional measure for the latent daily volatility. The particular model we use explicitly accounts for the dependency between daily returns and measurement errors of the realized volatility estimate. Within a simulation study we investigate the form of the dependency. In order to capture the long memory property of asset volatility, we explore different autoregressive dynamics for the latent volatility process, including heterogeneous autoregressive (HAR) dynamics and a two-component approach. We estimate the model using simulated maximum likelihood based on efficient importance sampling (EIS), producing numerically accurate parameter estimates and filtered state sequences. The model is applied to daily asset returns and realized variances of New York Stock Exchange (NYSE) traded stocks. Estimation results indicate that accounting for the dependency of returns and realized measures significantly affects the estimation results and improves the model fit for all autoregressive dynamics.
Appendix: Implementation of EIS for the extended SV model with leverage and endogeneity
Since the extended SV model with leverage and endogeneity nests all model specifications discussed in the paper, we restrict our attention to this general specification. The model implies
where ω=(γ, δ, ση, σu, ξ, ρ, θ) denotes the vector of parameters to be estimated. EIS aims to find an auxiliary density m(·) with auxiliary parameters from which we can easily simulate trajectories and that provides a good match between the numerator and the denominator of the expression in Eq. (29). This requires to find and appropriate density kernel k for the auxiliary density m. The link between the auxiliary density m, its kernel k and the integrating constant χ is given by
An appropriate choice for the kernel should provide a good functional approximation to the products χ·f, where f denotes the joint density of R, Y and Λ. We follow the argument of Liesenfeld and Richard (2003) and Richard and Zhang (2007) and choose for m a parametric extension of f(λt|λt−1, Rt−1) such that the resulting kernel k(Λt; at) is Gaussian:
The explicit form of the chosen kernel is given by
with corresponding moments
Integrating k(Λt;at) w.r.t. λt yields
Given the expressions for k, f and χ, the optimal values for at are obtained by a simple backward-recursive least squares sequence:
for t=T→1 and χ(ΛT; aT+1)≡1. Note that f(λt|λt−1, Rt−1) cancels out in the LS problems. Following Liesenfeld and Richard (2003), the first set of trajectories is based on a Taylor-Series-Expansion of log(f(λt|λt−1, Rt−1)) around zero. The complete EIS-algorithm for evaluating the MC estimate in Eq. (29) reads as follows:
Step 1: Draw S trajectories from the initial sampler obtained by a TSE of log(f(λt|λt−1, Rt−1)).
Step 2: Solve the LS-problems (55) backward-recursively for t=T→1 and obtain optimal values for
Step 3: Draw a new set of trajectories from m(â) and solve the LS-problems based on the new trajectories.
Step 4: Repeat Step 3 until convergence in order to obtain maximal efficient importance samplers.
Step 5: Use the last set of trajectories to evaluate the likelihood estimate (29).
Changes of the model structure require only minor adjustments of the EIS-algorithm.
We are grateful to Roman Liesenfeld for helpful comments and we would like to thank the editor and two anonymous referees.
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