An external regressor in the mean specification can be added to the mean specification, i.e. $$r_t = \mu + \varepsilon_t + \theta x_t $$. An external regressor in the variance specification can be added to the variance specification, i.e. $$\sigma^2_t = \omega + \alpha \sigma_{t-1}^2 + \beta \varepsilon_{t-1}^2 + \theta x_{t}$$


I am not sure I perfectly understand your question, the concept of "time series with varying density over time" is not very clear. One thing is for sure, the optimal way to "feed" a neural network is a function of the type of NNet itself and of the learning method you have chosen. For time series either you believe your data are iid vectors, and you can ...


If you are using the "rugarch" package in R, you can include these terms via the argument external.regressors within the argument mean.model in the ugarchspec function. From CRAN: external.regressors A matrix object containing the external regressors to include in the mean equation with as many rows as will be included in the data (which is passed in the ...

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