decreasing thresholds, where $s_1=\sum_{l=1}^pr_l$, $K=1$ constantly throughout the $s_1$ iterations. The threshold is $\delta:=\delta_{r_1}$ in the first $r_1$ iterations, $\delta:=\delta_{r_2}$ in the next $r_2$ iterations etc. In summary the first $r_1$ iterations use $(\delta,K):=(\delta_{r_1},1)$ and in general during iterations $(r_l:r_{l+1})$ we use $(\delta,K):=(\delta_{r_l},1)$. During the last $r_p$ iterations of ABC-MCMC we keep track of the maximum value $\mathrm{max}\{\pi_{\delta_{r_p}}(\theta|y)\}$ of the approximated posterior $\pi_{\delta_{r_p}}(\theta|y)$ and corresponding $\tilde{\theta}:=\mathrm{argmax}\pi_{\delta_{r_p}}(\theta|y)$: this is easily accomplished and cheap to implement by initializing $\mathrm{max}\{\pi_{\delta_{r_p}}(\theta|y)\}:=0$ just before setting $(\delta,K):=(\delta_{r_p},1)$. Then, whenever we have $J_{\delta}(y,z^{\#})\pi(\theta^{\#})>\mathrm{max}\pi_{\delta_{r_p}}(\theta|y)$ for the current maximised value of the posterior kernel, we set $\mathrm{max}\{\pi_{\delta_{r_p}}(\theta|y)\}:=J_{\delta}(y,z^{\#})\pi(\theta^{\#})$ and $\tilde{\theta}:=\theta^{\#}$. This search for the maximum has to be performed only when $\delta\equiv\delta_{r_p}$ as we are not interested in the maximum obtained for poorer approximations of the posterior. An alternative approach to the search for a mode $\tilde{\theta}$ is given by adjusting the output of ABC-MCMC as from [@beaumont2002approximate], hence step $5'$ denoted as “optional” in algorithm \[alg:abc-dc-dynam\]; this step is described in section \[sec:regression-adjustment\].

**ABC-MCMC stage** 1. Initialization: Fix a starting value $\theta^*$ or generate it from its prior $\pi(\theta)$ and set $\theta_1=\theta^*$. Set $j:=1$, $\delta:=\delta_{r_1}$ and $\mathrm{max}\pi_{\delta_{r_p}}:=0$. 2. Generate $X^{*}$ from $p(X|\theta^{*})$ and a corresponding $z^{*}$ from . Compute 