Stationarity of the transition probabilities in the Markov chain formulation of owner payments on projects

Abstract

Markov chain theory has been used to model the likelihood of payment to contractors based on historical owner payment practices. An important assumption in this modelling of owner payment behaviour is that the transition probability matrices are stationary. We study this assumption: the null hypothesis postulated is that the transition matrices are stationary; the alternate hypothesis is that they are not. This article also explores the impact of this assumption on the performance of the model. The outcomes of the model, in the form of payment likelihood of an outstanding amount against its age, given by both stationary and non-stationary approaches are compared. Tests performed on two project data sets show that the null hypothesis is rejected at the 5% level of significance. However, the payment probabilities estimated using non-stationary transition matrices are shown to approach a steady state after a relatively short fluctuation. These steady state values of payment probabilities are almost identical to those estimated under a stationary assumption. Therefore, we recommend users of the model adopt the stationary transition matrix approach to avoid the extra mathematical complication caused by non-stationarity. This article reinforces the validity of the existing Markov chain formulation of owner payments and its assumption of stationarity. The analysis presented, although based on case study data, can be translated to any project, provided data in the right form are available. References

• T. W. Anderson and L. A. Goodman, Statistical Inference about Markov Chains. Annals of Mathematical Statistics, Vol. 28, No. 1, pp. 89--110, 1957. doi:10.1214/aoms/1177707039
• M. S. Barlett, The frequency goodness of fit test for probability chains. Proceedings of the Cambridge Philosophical Society, Vol. 47, No. 1, pp. 86--95, 1951. doi:10.1017/S0305004100026402
• L. Betancourt, Using Markov chains to estimate losses from a porforlio of mortgages. Review of Quantitative Finance and Accounting, Vol. 12, No. 3, pp. 303--317, 1999. doi:10.1023/A:1008331016892
• P. Billingsley, Statistical methods in Markov chains. Annals of Mathematical Statistics, Vol. 32, No. 1, pp. 12--40, 1961. doi:10.1214/aoms/1177705136
• D. G. Carmichael, Engineering Queues in Construction and Mining. Ellis Horwood Ltd. (John Wiley and Sons Ltd.), Chichester, 1987.
• D. G. Carmichael and M. C. A. Balatbat, A contractor's analysis of the likelihood of payment of claims. Journal of Financial Management of Property and Construction, Vol. 15, No. 2, pp. 102--117, 2010. doi:10.1108/13664381011063412
• H. Cramer, Mathematical methods of statistics. Princeton University Press, Princeton, New Jersey, 1946
• B. D. Fielitz, On the stationarity of transition probability matrices of common stocks. Journal of Financial and Quantitative Analysis, Vol. 10, No. 2, pp. 327--339, 1975. doi:10.2307/2979039
• F. Hahary, N. Lipstein and G. P. H. Styan, A matrix approach to nonstationary chains. Operations Research, Vol. 18, No. 6, pp. 1168--1181, 1970. doi:10.1287/opre.18.6.1168
• M. C. Hallberg, Projecting the size distribution of agricultural firms. An application of a Markov process with non-stationary transition probabilities. American Journal of Agricultural Economics, Vol. 51, No. 2, pp. 289--302, 1969. doi:10.2307/1237580
• M. Hierro and A. Maza, Non-stationary transition matrices: An overlooked issue in intra-distribution dynamics. Economics Letters, Vol. 103, No. 2, pp. 107--109, 2009. doi:10.1016/j.econlet.2009.02.005
• P. G. Hoel, A test for Markoff chains. Biometrika, Vol. 413, No. 3/4, pp. 430--433, 1954. http://www.jstor.org/stable/2332723
• S. Jain, Markov chain model and its application. Computers and Biomedical Research, Vol. 19, No. 4, pp. 374--378, 1986. doi:10.1016/0010-4809(86)90049-2
• J. G. Kallberg and A. Saunders, Markov chain approaches to the analysis of payment behaviour of retail credit customers. Financial Management, Vol. 12, No. 2, pp. 5--14, 1983. http://www.jstor.org/stable/3665204
• N. M. Kiefer and C. E. Larson, Testing statistical hypotheses. John Wiley and Sons, New York, 1959.
• E. L. Lehmann, Testing statistical hypotheses. John Wiley and Sons Inc., New York, 1959.
• M. C. Pardo, Testing stationary distributions of Markov chains based on Rao's divergence. Applied Mathematics Letters, Vol. 12, No. 1, pp. 31--36, 1999. doi:10.1016/S0893-9659(98)00122-0
• E. Parzen, Stochastic Processes. Holden-Day Inc., San Francisco, 1962
• M. S. Salkin, R. E. Just and O. A. Cleveland, Estimation of nonstationary transition probabilities for agricultural firm size projection. The Annals of Regional Science, Vol. 10, No. 1, pp. 71--82, 1976. doi:10.1007/BF01291236
• M. Z. Sirdari, M. A. Islam and N. Awang, A stationary test on Markov chain models based on marginal distribution. Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications, Kuala Lumper, Malaysia, 2010. http://research.utar.edu.my/CMS/ICMSA2010/ICMSA2010_Proceedings/files/statistics/ST-Zang.pdf
• G. P. H. Styan and H. Smith, Markov chains applied to marketing. Journal of Marketing Research, Vol. 1, No. 1, pp. 50--55, 1964. http://www.jstor.org/stable/3150320
• M. Tainiter, Esimation, hypothesis testing and parameter correlation for Markov chains. Transaction on Reliability, Vol. R-12, No. 4, pp. 26--36, 1963. doi:10.1109/TR.1963.5218225
• H. Tran, D. G. Carmichael and M. C. A. Balatbat, Tree form classification of owner payment behaviour. Proceedings of the 4th International Conference on Construction Engineering and Project Management, Sydney, Australia, 2011.
• H. Tran and D. G. Carmichael, A contractor's classification of owner payment practices. Engineering, Construction and Architectural Management, accepted for publication, 2012.
• R. Weissbach, P. Tschiersch and C. Lawrenz, Testing time-homogeneity of rating transtions after origination of debt. Empirical Economics, Vol. 36, No. 3, pp. 575--596, 2009.
• R. Weissbach and R. Walter, A likelihood ratio test for stationarity of rating transitions. Journal of Econometrics, Vol. 155, No. 2, pp. 188--194, 2010. doi:10.1016/j.jeconom.2009.10.016