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We propose the discretization of real-valued financial time series into few ordinal values and use non-linear likelihood modeling for sparse Markov chains within the framework of generalized linear models for categorical time series. We analyze daily return and volume data and estimate the probability structure of the process of extreme lower, extreme upper and the complementary usual events. Knowing the whole probability law of such ordinal-valued vector processes of extreme events of return and volume allows us to quantify non-linear associations. In particular, we find a (new kind of) asymmetry in the return-volume relationship which is a partial answer to a research issue given by Karpoff (1987). We also propose a simple prediction algorithm which is based on an empirically selected model.

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