Direct vs Recursive Multi-Step Forecasting

A model that predicts tomorrow's price is easy to train. A model that needs to predict prices five days out has a real design choice to make, and it's not obvious at first glance: does it predict one day at a time and feed its own guesses back in as if they were real data, or does it train a separate model for each horizon from the start? These are the recursive and direct strategies, and they trade off in almost opposite ways.

Recursive: one model, fed its own output

The recursive (also called iterated) strategy trains a single model to predict one step ahead — given the last $k$ values, predict $x_{t+1}$. To forecast further out, the model's own prediction $\hat{x}_{t+1}$ is appended to the input window in place of the (unknown) true value, and the model is run again to predict $x_{t+2}$, and so on, five times to reach a five-day-ahead forecast. This uses only one model, trained once, which is simple and data-efficient. The cost is that any error in the first prediction becomes part of the input to the second prediction, so errors compound: a model that's off by a small amount at step one is working from a wrong starting point at step two, and the gap tends to widen the further out the forecast reaches.

Direct: a separate model per horizon

The direct strategy instead trains a distinct model for every horizon: model $h_1$ is trained to predict $x_{t+1}$ directly from the window ending at $t$, model $h_2$ is trained to predict $x_{t+2}$ directly from that same window, and so on, none of them ever touching another model's prediction as an input. This avoids compounding error entirely — each forecast is a fresh, direct estimate from real, known data — but it costs $h$ times the training effort for an $h$-step forecast, and because each horizon's model is trained independently, their predictions are not guaranteed to be consistent with each other: the 3-day-ahead forecast and the 4-day-ahead forecast come from two unrelated models and can imply a strange, non-smooth path between them.

Worked example: watching error compound versus stay flat

Suppose a recursive one-step model has a typical one-step prediction error with standard deviation 1.0, and — as a simplifying approximation often used to reason about this tradeoff — treat each step's error as roughly independent and additive. Forecasting $h$ steps ahead recursively, the accumulated error's standard deviation grows roughly as $\sqrt{h} \times 1.0$: about 1.41 at two steps, 1.73 at three steps, 2.0 at four steps, and 2.24 at five steps. A direct model trained specifically for the 5-day horizon, by contrast, is trained against real 5-day-ahead outcomes throughout, so its error doesn't inherit this compounding structure — suppose it has been measured with a standard deviation of 1.8 at that horizon. At five steps out, the direct model (1.8) beats the naive recursive estimate (2.24); at two steps out, the recursive model (1.41, assuming it were also measured directly, might be closer to 1.2 in practice) is likely still competitive, because there's been little distance for errors to compound over.

What this means in practice

A hybrid, sometimes called DirRec, exists precisely because neither extreme is always right: it trains a separate model per horizon like the direct strategy, but each horizon's model is also allowed to use the previous horizon's prediction as an extra input feature, gaining some of the direct strategy's accuracy while still letting information flow forward between horizons. In quant forecasting specifically, the choice often comes down to how many horizons genuinely matter — a desk that only cares about the 1-day and 20-day return has little reason to build the full chain of intermediate recursive steps, and a direct model at exactly those two horizons is usually simpler and more accurate than an artificially extended recursive chain.

Practice

1. A trading desk needs forecasts at exactly two horizons: 1 day and 10 days. Would you recommend a recursive chain or two separate direct models? Justify using the compounding-error tradeoff above.
2. Explain, without using the word "consistent," what it means for a set of direct multi-horizon forecasts to be inconsistent with each other, and why that can matter operationally.