When building a time series model is there a difference both from a theoretical perspective and a practical performance perspective to train a one-step ahead prediction model and forecast one-by-one in to the future for N steps vs to train directly an N-step ahead model?

If the purpose is to forecast N steps into the future, would an N-step ahead model have any performance advantages?

**Answer**

If the model is correct, then the optimal forecast is given by the iterated forecast (i.e. when you forecast each intermediate $y_{T+k}$ to finally produce $\hat y_{T+h}$). The direct forecast (when you estimate the model with $y_t$ as a function of $y_{t-h}$ in which the ‘one’-step-ahead forecast is now a $h$-step ahead forecast in ‘physical’ time) is less efficient in this case, but on the upside it is more robust to model misspecification.

Marcellino, Stock and Watson investigated this (in the AR context) in more detail and the abstract reads:

“Iterated” multiperiod ahead time series forecasts are made using a

one-period ahead model, iterated forward for the desired number of

periods, whereas “direct” forecasts are made using a horizon-specific

estimated model, where the dependent variable is the multi-period

ahead value being forecasted. Which approach is better is an

empirical matter: in theory, iterated forecasts are more efficient if

correctly specified, but direct forecasts are more robust to model

misspecification. This paper compares empirical iterated and direct

forecasts from linear univariate and bivariate models by applying

simulated out-of-sample methods to 171 U.S. monthly macroeconomic time

series spanning 1959 – 2002. The iterated forecasts typically

outperform the direct forecasts, particularly if the models can select

long lag specifications. The relative performance of the iterated

forecasts improves with the forecast horizon.

A free version of their paper is available here: https://www.princeton.edu/~mwatson/papers/hstep_3.pdf

Massimiliano Marcellino, James H. Stock, Mark W. Watson (2006)

“A comparison of direct and iterated multistep AR methods for forecasting macroeconomic time series”, *Journal of Econometrics*, (135):1–2, 499-526, https://doi.org/10.1016/j.jeconom.2005.07.020.

**Attribution***Source : Link , Question Author : Cagdas Ozgenc , Answer Author : hejseb*