There are countless ways to develop a model to predict the future behavior of almost any data that can be defined as a time series. The precision of these methods is persistently getting better and better. However, there is one question in my mind to which I've not been able to find an answer, and that is why we use exogenous inputs in time series modeling?
The reasoning behind an autoregressive time series model is pretty solid. It can and certainly have in most cases resulted in acceptable models, not particularly excellent but acceptable and we can use the outputs of the same model as inputs for future time steps.
There is a fact that introducing an exogenous input to the model, for example, using a set of precipitation data to an autoregressive river discharge estimation model, can significantly increase the precision of the model, for both the training and testing phase. But using another variable in a model that is also autocorrelated seems really wrong. Now, to forecast the future we need to develop another model to predict that exogenous variable as well, since we don't have future data for that excess input! Even if a time series model can be trained to tremendously good precision, we have essentially taken away its forecasting capability and I think this makes the model completely useless. What is the reasoning behind such an approach? Am I missing something?
You are right. Even if, in principle, adding more information like an exogenous variable (I would not speak of an exogenous parameter) can improve the forecasting model, it is not sure that you can improve the forecasts made using that model in general. There are exceptions when you can use a lagged exogenous variable and you are interested in small horizons. Then, the past values of that variable are available and your short-term forecasts can benefit from the added information. In your example, using upstream precipitations one or two days before can help, I suppose. In another example, like sales of a product, the existence of promotions is known in advance and can therefore be used.
Dear Guy Mélard thank you for your kind response. It's good to know that I was not completely wrong.
As mentioned by Guy Mélard above, it is not guaranteed that adding an exogenous parameter will improve modelling. A parameter that does not correlate with future outcomes will only make predictions more diffuse and reduce modelling accuracy. However, if you have a parameter (a leading indicator) that correlates with future outcomes and you can observe this parameter at the moment of forecast origin, it is worth considering this parameter as an additional input to modelling. However, be careful of spurious correlation, see more about G-causality here: https://en.wikipedia.org/wiki/Granger_causality
In fact, you can even use judgment as an exogenous parameter. Here's an article describing this approach:
Technical Report A joint Bayesian forecasting model of judgment and observed data
The benefit here is the reduction of uncertainty and improvements in accuracy given that judgmental inputs are adequate.
Regarding the question of how adding an exogenous parameter improves modelling performance in general: One good approach is to perform rolling-origin out-of-sample evaluation. Here it is important to select correct metrics to evaluate out-of-sample accuracy and bias.
Let me recommend forecast evaluation workflows proposed in this article (FEW-L1 and FEW-L2). page 65:
Article Assessing Point Forecast Bias Across Multiple Time Series: M...
The idea is that you can compare modelling performance with and without an exogenous parameter in order to empirically evaluate so-called Forecast-Value-Added (FVA), see the above article (pp. 47-48) for more about the FVA approach and the criteria for assessing modelling performance. So, if there's no FVA or the gains (expressed using metrics such as the AvgRelMAE or AvgRelMSE, see p. 56) are not worth the resources needed to collect and process the additional parameter, there's no need to complicate your model.
In addition to other answers, you can also consider the quality of information, the analysis behind them and the time distance to the future forecast.
Regards
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