Do Interest Rates Really Follow Continuous-time Markov Diffusions?

Interest rates have traditionally been modeled in the literature as following continuous-time Markov processes, and more specifically diffusions. By contrast, recent term structure models often imply non-Markovian continuous-time dynamics. Can discretely sampled interest rate data help decide which continuous-time models are sensible? First, how reasonable is the Markovian assumption? A test of this hypothesis will be proposed. Second, if the process is Markovian, can it be identified further as a diffusion, as has been assumed by most of the theoretical literature? A second test will be proposed, which tests the diffusion hypothesis under the maintained Markovian assumption. Within the Markovian world, diffusion processes are characterized by the continuity of their sample paths. It is immediately obvious that this condition cannot be verified from the observed sample path: by nature, even if the sample path were continuous, the discretely sampled interest rate data will appear as a sequence of discrete changes. This paper examines whether the discontinuities observed in the discrete data are the result of the discreteness of sampling, or rather evidence of genuine non-diffusion dynamics of the continuous-time interest rate process. The issue is to isolate the observable implications for the data of being an incomplete discrete sample from a continuous-time diffusion. This paper’s answer relies on testing a necessary and sufficient restriction on the conditional densities of diffusions, at the sampling interval of the observed data. This restriction characterizes the continuity of the unobservable complete sample path. The distribution of the test statistics, as well as their consistency and power properties, are derivted. We find empirically that: (i) neither the short rate nor the long rate can be characterized individually as Markov processes; (ii) jointly, they form a Markovian system; (iii) the slope of the yield curve is a univariate Markov process; (iv) and a diffusion. As a caveat, these preliminary empirical results are sensitive to the choice of dataset.