For a 28-day-ahead forecast, how do recursive and direct multi-step strategies differ?

## The problem A supervised forecasting model maps a feature row to one target. Asking for 28 days of forecasts from one origin means producing 28 targets, and there are two structurally different ways to get them. ## Recursive (iterated) forecasting Train one model for a horizon of one step: features are the lags and rolling aggregates available at time `t`, and the target is `y[t+1]`. Every historical row is a training example, so the model sees the maximum amount of data. To forecast further ahead, run it in a loop. Predict `yhat[T+1]`. Treat that prediction as if it were the observation for `T+1`, rebuild the lag row for `T+2` — where lag-1 is now the prediction rather than a measurement — and predict again. Repeat 28 times. The attraction is simplicity and data efficiency: one model, one training run, one artifact, one set of hyperparameters, and every horizon benefits from all the data. The cost is that the model is used outside the conditions it was trained under. It learned the mapping from *observed* lags to the next value; from step two onward at least some of its inputs are its own estimates. Two things follow. First, errors compound: an error at step one enters the input for step two, and the process is autoregressive in error as well as in signal. Second, predictions are systematically smoother and less variable than real observations — a conditional mean has lower variance than the thing it estimates — so the feature distribution drifts towards the middle of the training range, and with it the forecasts drift towards the series mean. Long recursive paths are usually flatter than reality, and that flattening is a bias, not just added noise. ## Direct forecasting Train a separate model for each horizon. The model for horizon `h` maps features available at the origin to `y[t+h]`. Its feature set may only include lags of `h` or more: for `h = 28`, lag-1 is unavailable, so the shortest usable autoregressive column is lag-28, and the row leans much more on calendar structure and long rolling aggregates. Nothing is ever fed back, so nothing compounds; each model is trained and evaluated on exactly the task it will perform. Each horizon can also specialise — short horizons weight recent lags, long horizons weight seasonal and calendar terms — which frequently beats forcing one mapping to serve both. The costs are real. You now have as many models as horizons to fit, tune, deploy, version and monitor. Each is trained on fewer *usable* rows, because long lags require long history and the early part of every series drops out. The 28 forecasts come from 28 independently fit functions, so the path can be jagged or internally inconsistent — day 14 above day 13 and day 15 for no structural reason — which matters when a human reads the curve or when the numbers feed an optimiser that assumes a smooth trajectory. ## The middle ground Several practical hybrids exist and mentioning one shows range. **Pooled direct with a horizon feature.** Stack the training data for all horizons into one table, add `h` as a column, and fit a single model. It behaves like direct — no feedback — but shares statistical strength across horizons and yields one artifact. The features must still be restricted to what is available at the origin for that `h`, which usually means building the row from origin-relative lags. **Multi-output models.** One model emitting a vector of 28 values at once, sharing all internal structure and producing a coherent path. **DirRec.** A hybrid that extends the feature set at each step while training a separate model per horizon; it appears in the literature and is worth naming, though it is rarely the pragmatic choice. **Blending.** Recursive for the near horizons where autoregression dominates, direct for the far ones where calendar structure dominates, stitched with a smooth handover. ## Choosing between them Favour recursive when the horizon is short relative to the strength of the autocorrelation, when data per series is scarce, when the one-step model is well calibrated, and when operational simplicity matters more than a few points of accuracy at the tail. Favour direct when the horizon is long, when the drivers at long range are deterministic (calendar, holidays, planned promotions) rather than autoregressive, when compounding has been observed to hurt, or when different horizons genuinely deserve different features. Long-horizon retail and demand planning problems usually land here, which is why competition solutions on that shape of data commonly train per-horizon or pooled-direct models. The empirically honest position is that neither strategy dominates in general; the reliable way to decide is to evaluate both over the horizons that matter and inspect where the error curve separates. An interviewer is usually satisfied by a candidate who names the compounding mechanism, names the availability restriction on lags at long horizons, and treats the choice as a measurable one rather than a doctrinal one.