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I have monthly data and my goal is to forecast values with an inclusion of exogenous variables. I've built SARIMA and Dynamic Harmonic Regression models, where the second one performs little bit better (but it produces smoother forecasts). Literature always stands that DHR models are for higher frequency series and series with weak or complex seasonality. Nevertheless I've never seen DHR models on monthly series. If there are any contraindications to not use DHR models on monthly series? Or is it only about getting most parsimonious models (DHR uses more terms than SARIMA)? How to choose between those two for monthly series?

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I'm not familiar with DHR models, however in general for choosing between two different models, it's up to a test on the performance.
You state that DHR is doing a little better - however not how you measure that. So starting from the measurement I would recommend taking a look Evaluating point forecast accuracy in Forecasting: Principle and Practice.
Depending on your use case you might also want to have a look at time series cross-validation.

Also, good way to ensure your models are appropriate is to validate them against a benchmark

Final point: if your models perform similarly, you should favor the simpler model - see more here Green, Kesten & Armstrong, J.. (2015). Simple Versus Complex Forecasting: The Evidence

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