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Let's say I have a stock that pays dividend once a year. I know how much did it pay in 2014, and at which level was stock trading when the dividend decision was made. I'd like to use these data to estimate dividend of 2015 as a function of the stock price: $D = f(S)$. Clearly, $f(S) \leq S$ and a naive assumption would be that $f$ is monotonic. Is there any research done on this topic, I mean what are other natural properties $f$ shall have, and examples of it?

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As a first Idea I would propose to incorporate basic ideas of Behavioural Finance and Dividend Theory into your considerations; for reference, look at:

Baker, Malcolm, and Jeffrey Wurgler. Behavioral corporate finance: An updated survey. No. w17333. National Bureau of Economic Research, 2011.

They state that investors prefer rather smooth dividend payments, this should indicate an high persistence.

This reasoning could be enough to start with a simple regression of the lagged dividend payments. You already stated a upper limit of the payoff function, a lower limit can also be approximated at least for several reporting regimes.

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