  State space models contain time-indexed parameters, termed states, as well as
static parameters, simply termed parameters. The problem of inferring both
static parameters as well as states simultaneously, based on time-indexed
observations, is the subject of much recent literature. This problem is
compounded once we consider models with intractable likelihoods. In these
situations, some emerging approaches have incorporated existing likelihood-free
techniques for static parameters, such as approximate Bayesian computation
(ABC) into likelihood-based algorithms for combined inference of parameters and
states. These emerging approaches currently require extensive manual
calibration of a time-indexed tuning parameter: the acceptance threshold.
  We design an SMC$^2$ algorithm (Chopin et al., 2013, JRSS B) for
likelihood-free approximation with automatically tuned thresholds. We prove
consistency of the algorithm and discuss the proposed calibration. We
demonstrate this algorithm's performance with three examples. We begin with two
examples of state space models. The first example is a toy example, with an
emission distribution that is a skew normal distribution. The second example is
a stochastic volatility model involving an intractable stable distribution. The
last example is the most challenging; it deals with an inhomogeneous Hawkes
process.
