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A computationally simple approach to inference in state space models is proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation of an intractable likelihood by matching summary statistics for the observed data with statistics computed from data simulated from the true process, based on parameter draws from the prior. Draws that produce a 'match' between observed and simulated summaries are retained, and used to estimate the inaccessible posterior. With no reduction to a low-dimensional set of sufficient statistics being possible in the state space setting, we define the summaries as the maximum of an auxiliary likelihood function, and thereby exploit the asymptotic sufficiency of this estimator for the auxiliary parameter vector. We derive conditions under which this approach -including a computationally efficient version based on the auxiliary score -achieves Bayesian consistency. To reduce the well-documented inaccuracy of ABC in multi-parameter settings, we propose the separate treatment of each parameter dimension using an integrated likelihood technique. Three stochastic volatility models for which exact Bayesian inference is either computationally challenging, or infeasible, are used for illustration. We demonstrate that our approach compares favorably against an extensive set of approximate and exact comparators. An empirical illustration completes the paper.

Summary We propose parametric copulas that capture serial dependence in stationary heteroskedastic time series. We suggest copulas for first‐order Markov series, and then extend them to higher orders and multivariate series. We derive the copula of a volatility proxy, based on which we propose new measures of volatility dependence, including co‐movement and spillover in multivariate series. In general, these depend upon the marginal distributions of the series. Using exchange rate returns, we show that the resulting copula models can capture their marginal distributions more accurately than univariate and multivariate generalized autoregressive conditional heteroskedasticity models, and produce more accurate value‐at‐risk forecasts.

Dynamic jumps in the price and volatility of an asset are modelled using a joint Hawkes process in conjunction with a bivariate jump diffusion. A state space representation is used to link observed returns, plus nonparametric measures of integrated volatility and price jumps, to the specified model components; with Bayesian inference conducted using a Markov chain Monte Carlo algorithm. An evaluation of marginal likelihoods for the proposed model relative to a large number of alternative models, including some that have featured in the literature, is provided. An extensive empirical investigation is undertaken using data on the S&P500 market index over the 1996 to 2014 period, with substantial support for dynamic jump intensities -including in terms of predictive accuracy -documented.alternatives to the state space form.As in Bandi and Reno (2016) spot price data only is used to analyse all models, with the results unaffected as a consequence by the nature of -and potential dynamics in -volatility and jump risk premia (see Bollerslev, Gibson andManeesoonthorn, Martin, Forbes andGrose, 2012, for analyses in which such specifications do feature). However, and in contrast with Bandi and Reno, data measured at the daily frequency (including that which aggregates to the daily level over intraday observations) underpins the analysis. In common with the large part of the relevant literature (Bollerslev, Kretschmer, Pigorsch andTauchen, 2009, andLiu, Patton and Sheppard, 2015, amongst many others) we also choose to construct all measures using within-day observations only, thereby avoiding the need to model closeto-open movements in the index (as in, for example, Ahoniemi, Fuertes and Olmo, 2015, and Andersen, Bollerslev and Huang, 2011) and any specific dynamic movements therein. (See Hansen and Lunde, 2005, and Takahashi, Omori and Watanabe, 2009, for earlier discussions on the role played by non-trading periods in the construction of high frequency measures).The remainder of the paper is organized as follows. Section 2 describes our proposed asset price model and its main properties. The continuous time representation is presented first, followed by the discrete time state space structure adopted for inference. Details are given of the high frequency measures of volatility and price jumps that are used to supplement daily returns in defining the state space model. The Bayesian inferential approach is then outlined in Section 3, including the way in which the alternative specifications are to be assessed, relative to the most general model, both in terms of marginal likelihoods and cumulative log scores. Results from the extensive empirical analysis of the S&P500 index are presented and discussed in Section 4. The benefits of allowing for a very flexible dynamic specification for price and variance jumps are confirmed by both the within-sample and predictive assessments, with the bivariate Hawkes specification given strong support by the data, relative to other more restrictive models. The empirical results also indic...

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