When fitting (stationary) time series models, such as ARIMA models, the standard approach is to minimise the one-step ahead forecasting error, which is equivalent to performing maximum likelihood estimation of the Gaussian likelihood. Sometimes however, the main purpose of a model is for it to make good predictions on a considerable timescale (larger then one step, for example 4 or 12 steps). I am thus wondering if approaches exist that estimate a model by minimising the sum of the one-, two, ... and h-step ahead forecasts. I can imagine one would like for such a sum to be weighted, such that one could adhere more importance to 'early' errors.
I have searched for quite some time now, but can only find some papers on "Multi-Step Ahead Estimation methods", which minimise (only) the h-step ahead forecasting error. Maybe I am using the wrong search keywords?
Besides the fact that the approach I describe here seems intuitive to me, I also feel it could be used to put more emphasis on the data trend (and less on the noise) when estimating, potentially offering an alternative to an initial (and sometimes undesirable) denoising/smoothing of the data.