Continuous-Time Markov Chains

MARKOV CHAIN MODELS widely used in the social sciences to describe the movement of agents across states. If there n states a Markov chain model has n2 parameters pi>, arranged in an n x n transition matrix P, indicating the probability that an agent in state i one period will be in state j the succeeding period. Estimates of the pij from survey data can be used to address a variety of questions. Given two years of panel data on the income of young females, let state 1 in each year be an income below the poverty line. Then by separating the sample into blacks and whites '11 for the two groups can be compared. However, to address such questions as are white young females more mobile than black young females? requires the comparison of the matrices P for the two groups. Questions of this nature have led to the development of mobility indices, which map the matrix P into a scalar M(P). Consider two transition matrices, P1 and P2. Ideally a mobility index should yield the ranking M(P1) > M(P2) for any PI and P2 for which reasonable individuals agree that P1 implies greater mobility than P2. Since the concept of mobility comparisons is vaguely defined the problem of comparing mobility indices is ill posed, unless the criteria inherently used by reasonable individuals in the comparison of transition matrices can be quantified. An attractive, systematic approach is to postulate properties which a reasonable index of mobility ought to satisfy, and then evaluate proposed indices against these properties (as Lovell (1962) did for seasonal adjustment procedures). Shorrocks (1978) took this approach, but found logical inconsistencies between apparently plausible criteria and left unresolved the question of whether the criteria consistent for an interesting subset of all possible transition matrices P. Alternative indices of mobility were also discussed in Shorrocks (1978) and evaluated against these criteria. In this paper we extend Shorrocks' treatment in several ways. We begin, in