Direct versus Recursive Multi-Horizon Forecasting

You have a model that predicts tomorrow. The business needs the next 28 days. There are only two honest ways to get there, and the choice between them changes the error profile of every number you ship.

The recursive (or iterated) strategy trains one model \(f\) for a one-step-ahead target and then feeds its own output back in: predict \(\hat{y}_{T+1}\), append it to the history, predict \(\hat{y}_{T+2}\) from a window that now contains a prediction rather than an observation, and repeat. The direct strategy trains a separate model \(f_h\) for each horizon \(h\), each one mapping the same observed window to the target \(h\) steps out. Twenty-eight horizons means twenty-eight fitted models, none of which ever consumes its own output.

The bias-variance argument, and why it points both ways

Recursion is efficient when the one-step model is correctly specified. All the data trains a single parameter set, and the multi-step forecast is the correct implication of that model applied repeatedly. The catch is that recursion propagates error multiplicatively. For a stationary AR(1), \(y_t = \phi y_{t-1} + \varepsilon_t\), the \(h\)-step recursive forecast is \(\hat{y}_{T+h} = \hat{\phi}^h y_T\), so an estimation error in \(\hat{\phi}\) is raised to the power \(h\). A 2% error in \(\phi\) at \(h = 28\) is roughly a 75% error in the coefficient that actually produces the forecast.

Direct estimation has no such compounding, because each \(f_h\) is fitted against the real target at that horizon. In exchange, each model sees the same number of rows as the recursive model but must learn a harder, noisier mapping, and nothing enforces coherence between horizons: a direct system can predict 100 at \(h = 5\) and 40 at \(h = 6\) with no mechanism telling it that is implausible. The textbook summary is that direct is more robust to model misspecification and recursive is more efficient under correct specification (Chevillon, 2007, Direct Multi-Step Estimation and Forecasting, Journal of Economic Surveys 21(4)).

Empirically, the robustness argument loses more often than you would expect. Marcellino, Stock and Watson ran both strategies over 171 US monthly macroeconomic series from 1959 to 2002 and found iterated forecasts typically beat direct ones, with the advantage growing at longer horizons and widening when the model was allowed long lag specifications (Marcellino, Stock & Watson, 2006, Journal of Econometrics 135(1-2), 499-526). The plain reading is that the variance cost of fitting many separate long-horizon models usually exceeds the bias cost of compounding.

What changes when the model is global

The macro result was about univariate AR models on a few hundred series. Modern practice fits one global model across thousands of series, which moves the tradeoff:

• Data scarcity is no longer the binding constraint on direct models. With 30,000 series, each horizon-specific model still sees 30,000 training rows per time origin.
• Feature engineering makes the strategies visibly different. A recursive gradient-boosted model needs its lag features regenerated at every step from predicted values, so lag-1 at \(h = 10\) is nine parts prediction. A direct model at \(h = 10\) uses only lags of 10 or more, all observed.
• Neural architectures usually sidestep the question by emitting the whole horizon vector at once from a single forward pass, which is multi-output direct: one model, one parameter set, \(H\) outputs, trained on the joint loss. N-BEATS, DeepAR's decoder variants and the Temporal Fusion Transformer all do some form of this, and it is why they scale to long horizons without 28 model artefacts.

Multi-output direct is the pragmatic default in large systems because it keeps direct's freedom from compounding while sharing statistical strength across horizons, and because deploying one model is operationally cheaper than deploying \(H\).

Hybrids

The strategies are endpoints, not a dichotomy. DirRec trains a model per horizon, as direct does, but lets model \(h\) use the predictions from models \(1 \ldots h-1\) as inputs, which restores cross-horizon information at the cost of reintroducing some error propagation. Rectify starts from the biased recursive base forecast and fits a direct model to correct it at each horizon, so the base carries the shared structure and the correction carries the horizon-specific bias; its authors report it is always better than, or comparable to, the better of recursive and direct (Ben Taieb & Hyndman, 2012, Recursive and direct multi-step forecasting: the best of both worlds, Monash Working Paper 19/12). The honest summary of the strategy literature is that no strategy dominates across data-generating processes, and the ranking flips with noise level, horizon and sample size.

When it breaks

Recursion with exogenous inputs needs forecasts of the inputs. If the model consumes yesterday's price, a recursive 28-day forecast needs 28 days of predicted prices, and you have now built a second forecasting problem with no evaluation of its own.

Direct models drift out of coherence. Nothing in the training objective ties \(h\) and \(h+1\) together, so aggregating a direct horizon vector to a weekly total can produce a shape no single series ever exhibits. Reconciliation across horizons is a real post-processing step, not a formality.

Backtesting must match the strategy. Evaluating a recursive model by feeding it true history at every step (teacher forcing) measures a model you will never deploy, and typically overstates accuracy by a wide margin at long horizons. The evaluation loop has to close the recursion exactly as production does.

Cost scales differently. Direct costs \(H\) fits and \(H\) artefacts but one inference pass. Recursive costs one fit but \(H\) sequential inference passes, which for an autoregressive neural model means \(H\) forward passes per series and can dominate serving latency at scale.