The Estimation of Nonstationary Transition Probabilities

Abstract

The estimation of an underlying Markov process of constant transition probabilities which explains aggregated market share movements over time has received much attention from theorists and applied researchers. Summaries of the work to date are given in [Dent, Warren T., Richard Ballintine. 1971. A review of the estimation of transition probabilities in Markov chains. Australian J. Agricultural Econom. 15 and Lee, T. C., G. G. Judge, A. Zellner. 1970. Estimating the Parameters of the Markov Probability Model from Aggregate Time Series Data. North-Holland, Amsterdam.]. The problem of estimating a nonstationary process however has not been solved, although Hallberg [Hallberg, M. C. 1969. Projecting the size distribution of agricultural firms—an application to a Markov process with non-stationary transition probabilities. J. Farm Econom. 51.] has looked at the problem in the case of micro data, that is, when movements between periods are observable. Harary, Lipstein, and Styan [Harary, F., B. Lipstein, G. P. H. Styan. 1970. A matrix approach to nonstationary chains. Oper. Res. 18.] have investigated nonstationary chain properties in the case of aggregated data but have not looked at estimation. We present an attempt to formalize Harary, Lipstein, and Styan's model and to develop appropriate solution techniques for estimation of its unknown parameters. An iterative quadratic programming procedure is suggested.