# What are the underlying statistical differences between the vector autoregression model and Prophet?

I am trying to understand the underlying fundamental/statistical differences between vector autoregression models and Facebook's Prophet, with regards to multivariate time series forecasting.

I am very new to time series forecasting, but I am really looking to understand the differences between these models in terms of how they approach fitting a trend and forecasting future values. Most comparisons I can find online are in terms of their actual performance differences, that is not what I am looking for.

This is all very new so any pointers to resources which explain how Prophet and VAR work – for someone without a deep mathematical background – would be excellent, and then I can derive the answer myself from there. I cannot find very much about what Prophet really is, beyond the description.

My experience is in machine learning methods as opposed to time series forecasting, so I would appreciate any explanation which accounts for my very basic subject matter knowledge.

• More details on the Prophet model can be found in their paper, "Forecasting at scale" doi.org/10.7287/peerj.preprints.3190v2. Simply put, the Prophet model is a generalized additive model (GAM) which uses predictors commonly used in time series forecasting problems. The model using three time series components - growth: $g(t)$, seasonality: $s(t)$, and holidays: $h(t)$. The terms are added with an Normal error $\varepsilon(t)$ to give: $y(t) = g(t) + s(t) + h(t) + \varepsilon(t)$. $g(t)$ is modelled using piecewise linear trends, $s(t)$ with fourier terms, and $h(t)$ as dummies. Aug 2, 2020 at 9:11
• Thank you @MitchellO'Hara-Wild, I know this is beyond the scope of my original question and I can create a new question if that would be more appropriate: why/when would someone choose to use VAR instead of Prophet and vice versa? Aug 2, 2020 at 10:30
• VAR and Prophet models are different model classes, and one is not always better than the other. A simple approach to choosing models for forecasting, is to try both and use which one produces the most accurate results on test data. To touch on your original question, a VAR is a multivariate model while Prophet is a univariate model. So a VAR will produce multivariate forecasts with uncertainty jointly determined by errors of multiple series (covariance matrix), while Prophet will produce univariate forecasts with uncertainty based on errors from a single series (variance scalar). Aug 2, 2020 at 11:49
• Thanks @MitchellO'Hara-Wild, it is interesting to learn a bit more about the differences. I believe Prophet can be implemented as a multivariate forecast using the add_regressor method: facebook.github.io/prophet/docs/… Aug 3, 2020 at 6:58
• By multivariate model above, I am referring to models that have multiple response variables. Using add_regressor with Prophet will use an additional time series variable(s) to predict a single response variable. Aug 4, 2020 at 0:01