In a hidden Markov model (HMM), the system goes through a hidden Markovian sequence of states (X) and produces a sequence of emissions (Y). We define the hidden Gaussian Markov model (HGMM) as the HMM where the hidden process is Gaussian and is affected by a normal white noise. The Kalman filter (KF) is a fast optimal statistical estimation method for the HGMMs and is very popular among the practitioners. However, the classic HGMM formulation is too restrictive. It extends to recent pairwise Gaussian Markov model (PGMM) where we assume that the pair (X, Y) is Gaussian Markovian. Moreover, there exists a KF version for the PGMM. The PGMM is more general than HGMM and in particular, the PGMM hidden process is not necessarily Markovian. The authors share their findings on about enhancing the KF when improving HGMM to PGMM. We discover singular cases where HGMM is at least ten times less accurate than the PGMM. On average, PGMM outperforms HGMM by twenty percent