The hidden Markov models (HMMs) are state-space models widely applied in time series analysis. Well-known Bayesian state estimation methods designed for HMMs, such as the Baum-Welch algorithm and the Viterbi algorithm, allow state estimation with a complexity linear in the sample size. We consider recent extensions of HMMs, specifically the pairwise Markov models (PMMs) and the triplet Markov models (TMMs), in which the Baum-Welch algorithm also has a complexity linear in the sample size. However, the state process is not necessarily Markovian in PMMs and TMMs, which offers a considerable flexibility of modeling. This study explores potential performance gains achievable if PMMs and TMMs are used to describe the state-space system rather than HMMs. This is done through extensive comparative Monte-Carlo experiments among HMMs, PMMs and TMMs in the case of discrete state space models. A simple comparative example of the use of PMMs and HMMs to predict market direction is also given. These experiments confirm the interest of PMMs and TMMs in the time series modeling: specifically, the classification rate can be improved by nearly fifty percent. These findings mean that PMMs and TMMs may be more suitable than classic HMMs for real-world applications