Say I want to predict the total rainfall for the next 365 days based on a set of predictors and daily historical observations.
I could build a model that predicts annual rainfall and make a single prediction.
Or I could build a model that predicts n-step ahead daily rainfall, make 365 predictions, and sum the result.
Are there theoretical or practical justifications for choosing one approach over the other when the objective is to produce the most accurate prediction?
You shouldn't do either the one or the other. You should do both and combine the predictions.
Your time granularity forms a hierarchy: there are 365 days to a year. Abstractly, this is no different to any other hierarchical prediction problem, such as forecasting sales in a product hierarchy, or tourism to the different states of Australia and to Australia as a whole. In such situations, it is often best to forecast separately on all levels of your hierarchy and combine the forecasts, by considering the sum-inconsistent original forecasts as a regression-type problem. I have repeatedly found that this improves prediction accuracy across all levels of the hierarchy.
The original paper on this approach was by Hyndman et al. (2011, Computational Statistics & Data Analysis). Additional info can be found in Athanasopoulos et al. (2009, IJF) Hyndman et al. are the authors of the hts package for R, which implements their optimal combination approach. Here is a blog post on the package by Rob Hyndman. Hyndman & Athanasopoulos devote a section of their free online forecasting textbook to hierarchical forecasts and their optimal combination approach in particular.
Kourentzes et al. (2014, IJF) in fact have looked at the specific use case of temporal hierarchy. Their approach is to calculate forecasts on different time granularities, by aggregating the original data first, then to combine these forecasts using Hyndman et al.'s approach. Sound familiar? Here is a non-gated presentation. Kourentzes & Petropoulos are the authors of the mapa package for R, which implements this algorithm.
