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In a Markov model the transition probabilities between states do not depend on the time spent in the current state. The present paper explores two ways of selecting the states of a discrete-time Markov model for a system partitioned into categories where the duration of stay in a category affects the probability of transition to another category. For a set of panel data, we compare the likelihood fits of the Markov models with states based on duration intervals and with states defined by duration values. For hierarchical systems, we show that the model with states based on duration values has a better maximum likelihood fit than the baseline Markov model where the states are the categories. We also prove that this is not the case for the duration-interval model, under conditions on the data that seem realistic in practice. Furthermore, we use the Akaike and Bayesian information criteria to compare these alternative Markov models. The theoretical findings are illustrated by an analysis of a real-world personnel data set.

An open hierarchical (manpower) system divided into a totally ordered set of k grades is discussed. The transitions occur only from one grade to the next or to an additional (k+1)th grade representing the external environment of the system. The model used to describe the dynamics of the system is a continuous-time homogeneous Markov chain with k+1 states and infinitesimal generator R = (rij) satisfying rij = 0 if i > j or i + 1 < j ≤ k (i, j = 1,…,k+1), the transition matrix P between times 0 and 1 being P = expR. In this paper, two-wave panel data about the hierarchical system are considered and the resulting fact that, in general, the maximum-likelihood estimated transition matrix cannot be written as an exponential of an infinitesimal generator R having the form described above. The purpose of this paper is to investigate when this can be ascribed to the effect of sampling variability.

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