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Usually we use time plot to examine the behaviour of time series data cause it reveals the chronological characteristic. Does it make sense that one looks at the data distribution using some non-parametric statistics (e.g. kernel density)? While looking at the kde of some iid data makes sense, I'm not sure does it apply to time series data which is not noise (iid), or even with structural breaks or conditional heteroscedasticity.

Possibly duplicate of Can I plot time series data in a histogram?, which doesn't have any answers.

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Yes, it makes complete sense in certain cases to examine the density of time-series, especially after mean-centring the original series. This relates directly to the concept of density forecasting and providing probabilistic forecasts of our continuous variables in the form of predictive densities functions. A famous reference on the matter is Gneiting et al. (2007) "Probabilistic forecasts, calibration and sharpness".

I would even note that in certain years the notion of looking in probabilistic forecasts against the standard idea of point-forecasts and their associated interval forecasts, is gaining a lot of traction (e.g. see Askanazi et al. (2018) "On the Comparison of Interval Forecasts"). Examining the density of our forecast (or that of our original time series) is therefore relevant even in the case where we do not have IID data. Especially when it comes to cases where we have conditional heteroscedasticity; for example Bao et al. (2007) "Comparing density forecast models" go to great lengths to showcase how GARCH and extensions of it (EGARCH, FIGARCH, etc.) measure up in terms for density forecasts for financial return series.

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