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Consider a situation where there are multiple subjects and each subject has multiple measurements (response, covariates) over time. The goal is to identify a statistical regression model which allows past responses to influence future responses without creating statistical problems.

When the response is discrete, there are many modelling options for stochastic processes (discrete time and continuous time):

When the response is continuous, some choices include:

  • Time series models with exogenous variables
  • Dynamic Panel Models such as those based on the Arellano-Bond GMM estimator

But is it common to base a regression model on a Stochastic Differential Equation (SDE)?

For example, I found this interesting paper (https://gatesopenresearch.org/articles/4-71/v2) here where the authors seem to have adapted a SDE where the drift and diffusion components are themselves regression models.

Using SDEs seems to be the most natural approach as it directly models the response dynamics in continuous time/space as a function of changing response/covariates.

I would be interesting in reading more references on this and see how prevalent this is in statistics.

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    $\begingroup$ These references are special cases of considerable complexity. The link between DEs and probability models estimable by regression are more fundamental. If I recall correct, Sheldon Ross' text "Introduction to Probability Models" is an excellent primer. $\endgroup$ Commented Nov 10 at 18:31
  • $\begingroup$ @AdamO yes! Great book (the Ross book). DEs play a vital role in his overview of stochastic processes. For the OP: even older versions of the book are good and can be obtained for very very cheap! $\endgroup$ Commented Nov 11 at 0:58
  • $\begingroup$ I looked at this book and it seems to have a introduction to some of these topics. But I am looking for something which discusses how parameters of SDE's can be estimated from data $\endgroup$ Commented Nov 11 at 2:03
  • $\begingroup$ Some models may fit into a state space framework. $\endgroup$ Commented Nov 11 at 4:26
  • $\begingroup$ "Parameter Estimation" techniques often involve an iterative numerical solutions of diffeqs wrapped inside of algorithms that tweak the parameters. Also, many statistical approaches involve optimizations of an error function, which often involves taking the derivatives and setting them equal to zero. $\endgroup$ Commented Nov 11 at 16:42

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There is a task view for R centered around differential equations: https://cran.r-project.org/web/views/DifferentialEquations.html

This gives an overview of R packages that have been developed to help solve differential equations as well as pointers to some books and other resources.

Even if you do not use R, this could point you in directions that others are already working on.

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Much of Physics and engineering is about predicting future of continuous variables, and most of that is done using differential equations. In some cases one includes noise in these equations, which makes them very similar to SDE, e.g. https://en.wikipedia.org/wiki/Langevin_equation. That's one place to start. In general, look for terms like 'noise term' or 'noise analysis' or 'noise spectrum' or 'fluctuations' in physics/engineering

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