Suppose I have a set of measurements taken at regular intervals, and I want to predict future values of one of those measurements. There are relationships between the variables being measured. For example, some variables may tend to rise together, or one variable might tend to fall when another rises. What are the state-of-the-art methods for this type of forecasting?
As you explicitly ask for state of the art models: some newer and more advanced forms of recurrent neural nets, like a long-short-term-memory net might be interesting to you. Be aware that a) those tend to need huge computation power and b) (though this is undergoing a strong change at the moment) there are still no/very few ready-to-use-out-of-the-box implementations of such models. Most evolved from research prototypes that have been made available to the public. Take a look at e.g. Keras, or http://lstm.iupr.com/ or http://deeplearning4j.org/lstm.html for hints on other open source implementations.