Recursive vs Direct Multi-Horizon Forecasting

Forecasting one step ahead is straightforward: train a model that maps recent history to the next value. Forecasting ten steps ahead is not obviously the same problem, because there is no ground truth for steps 2 through 10 available at prediction time the way there is for step 1. Two genuinely different strategies exist for extending a one-step model into a multi-step one, and they aren't just implementation details — they trade off error compounding against structural mismatch, and the wrong choice quietly costs accuracy no amount of tuning fixes.

The analogy: dead reckoning versus pre-planned waypoints

A ship without GPS can navigate by dead reckoning: at each moment, use your current estimated position plus speed and heading to estimate the next position, then repeat from that new estimate. Simple, and it works — but any error in one estimate feeds the next one, so small errors compound the further you sail. Alternatively, a navigator could instead plan a fixed set of waypoints directly from the start — "in 2 hours here, in 4 hours here, in 6 hours here" — each computed independently from the current known position. That avoids compounding, but a waypoint 6 hours out is a much harder, noisier estimate to make directly, and the waypoints aren't guaranteed to be consistent with each other the way a dead-reckoned path automatically is.

The mechanics: recursive versus direct

Recursive (also called iterated) forecasting trains one model $f$ for one step ahead, then applies it repeatedly, feeding each output back in as if it were an observed input:

$$\hat{y}_{t+1} = f(y_t, y_{t-1}, \dots), \qquad \hat{y}_{t+2} = f(\hat{y}_{t+1}, y_t, \dots), \qquad \hat{y}_{t+h} = f(\hat{y}_{t+h-1}, \dots)$$

In words: the same one-step model is reused $h$ times, and from the second application onward it's fed its own previous predictions instead of real data — exactly the mechanism behind Teacher Forcing and Exposure Bias, so recursive forecasting inherits that compounding-error problem directly.

Direct forecasting instead trains a separate model $f_h$ for each horizon $h$, each mapping the same known history straight to that specific future step, with no model's output feeding another model's input:

$$\hat{y}_{t+1} = f_1(y_t, y_{t-1}, \dots), \qquad \hat{y}_{t+2} = f_2(y_t, y_{t-1}, \dots), \qquad \dots, \qquad \hat{y}_{t+h} = f_h(y_t, y_{t-1}, \dots)$$

Every $f_h$ conditions only on real, observed history — no self-generated inputs, so no error compounding across horizons. The cost is that each $f_h$ must learn the harder, noisier task of predicting further into the future directly, and because the $h$ models are trained independently, nothing forces $\hat{y}_{t+1}$ and $\hat{y}_{t+2}$ to be mutually consistent (e.g. a monotonically rising true series could get a non-monotonic multi-horizon forecast).

Worked example 1: compounding error under recursive forecasting

Suppose a one-step model has RMSE of 2 units per prediction, and errors behave roughly independently step to step. Recursively forecasting 5 steps ahead means the 5th prediction is built on 4 previous predictions, each carrying its own error that feeds into the next. If errors added in quadrature (a simplification, but illustrative): $\sqrt{2^2 \times 5} \approx 4.47$ — more than double the one-step error, purely from compounding, even though the same model with the same one-step accuracy is used throughout.

Worked example 2: direct forecasting's independent-horizon cost

Now suppose training data gives 10,000 examples total. A recursive approach uses essentially all 10,000 to train its single one-step model. A direct approach with a 5-step horizon must split effort across 5 separate models $f_1, \dots, f_5$, and $f_5$'s task (predicting 5 steps out from only currently-known data) is inherently harder to learn well with limited data — direct forecasting's later-horizon models can end up less accurate than a well-tuned recursive one, purely because each sees a harder mapping and gets no help from the others.

Watch how a single simulated path that updates its own state each step resembles recursive forecasting, while several independent snapshots at fixed future times resemble direct forecasting's lack of inherited error.

What this means in practice

Neither strategy dominates: recursive forecasting is preferred when the one-step model is highly accurate and horizons are short (error compounding stays small), direct forecasting is preferred when horizons are long or the underlying dynamics change with horizon (e.g. a demand model where the drivers of next-week versus next-quarter demand genuinely differ). Architectures like N-BEATS Interpretable Forecasting and DeepAR Probabilistic Forecasting actually offer a middle path — a single model producing an entire multi-step output block at once (sometimes called sequence-to-sequence or "multi-output" forecasting), sidestepping both pure recursion and fully independent per-horizon models.

Practice

1. If a recursive model's one-step RMSE is 3 and you model error accumulation in quadrature over 4 steps, what is the approximate 4-step RMSE?
2. Give one concrete forecasting scenario where direct forecasting's independence from compounding is worth its harder per-horizon learning task.
3. Explain why a direct-forecasting approach's outputs across horizons are not guaranteed to be mutually consistent.