# Time Series One Step Ahead vs N-Step Ahead

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?

• You can search for iterated vs direct forecasting. Generally the former is optimal if the model is correct whereas the latter is often more robust. – hejseb Apr 24 '18 at 15:48

## 1 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.

• Admittedly, taking into account estimation uncertainty may mean that you are often better off using a misspecified simpler model than a correct (but more complicated) model. There is a paper by Rob Hyndman which contains some results and references if you didn't see it already: robjhyndman.com/papers/rectify.pdf – hejseb Apr 25 '18 at 10:50
• @CagdasOzgenc You'd only shift the sample. In an AR(1) that would mean to regress $y_{3}$ on $y_1$, $y_4$ on $y_2$ and so on. Despite introducing new problems (as you say), this is often still better. However, I believe it's very model and application specific so I don't think you will find anything more conclusive than "it depends." – hejseb Apr 25 '18 at 11:11
• I have already done exactly what you have described for an AR(1) data. When shifted one step the residuals show autocorrelation when a t+2 model estimated. As far as I know this yields incorrect (inflated) t-stat for the estimated coefficient. Of course one may not care about the t-stat and focus on predictability. So the question is how important is this? – Cagdas Ozgenc Apr 25 '18 at 11:37
• hejseb - great reference, thanks. I posted a related question a few months back with no responses - care to weigh in on it or provide a reference as well? stats.stackexchange.com/questions/324540/… – Skander H. Apr 25 '18 at 18:29