We consider two discrete-time processes X and Y, the pair (X,Y) being Gaussian, homogeneous and Markovian. Such models, called "Gaussian homogeneous pairwise Markov models" (GH-PMMs), extend the classical Gaussian homogeneous hidden Markov models (GH-HMMs), also called Gaussian homogeneous state space models. In GH-PMMs, neither X nor Y is necessarily Markovian, and the problem addressed is to find conditions on the GH-PMM parameters for X (or Y) to be Markovian. We give necessary and sufficient conditions for real-valued X and Y, and necessary conditions for the general multivariate case. The advantage of GH-PMMs over GH-HMMs is that they are more general and still allow various treatments such as smoothing, filtering or forecasting. A practical application of the proposed contributions is that when, for a given smoothing, filtering or forecasting problem, the specified conditions cannot be justified, GH-PMMs should be used rather than GH-HMMs.