Exponential Smoothing and the ETS State-Space Form

In the M5 competition, 5,507 teams forecast Walmart unit sales, and only 7.5% of their final submissions beat the strongest benchmark the organisers supplied. That benchmark was exponential smoothing fitted bottom-up, a method whose core recursion dates from the 1950s (Makridakis, Spiliotis & Assimakopoulos, 2022, M5 accuracy competition: Results, findings, and conclusions, IJF 38(4)). Knowing why something so simple is so hard to beat is most of what there is to know about forecasting baselines.

From a weighted average to three recursions

Simple exponential smoothing keeps one number, the level \(\ell_t\), and nudges it toward each new observation:

with \(0 < \alpha \le 1\). Unrolling the recursion shows the forecast is a weighted average of all past observations with weights \(\alpha(1-\alpha)^k\) on the value \(k\) steps back. With \(\alpha = 0.3\) the three most recent points carry weights 0.30, 0.21 and 0.147; a level at 100 that sees \(y_t = 110\) moves to 103. Small \(\alpha\) means long memory and smooth forecasts; \(\alpha = 1\) is the naive forecast.

Holt added a trend \(b_t\) smoothed with its own parameter \(\beta\) (Holt, 1957, reprinted 2004, Forecasting seasonals and trends by exponentially weighted moving averages, IJF 20(1)), and Winters added a seasonal component \(s_t\) of period \(m\) with parameter \(\gamma\) (Winters, 1960, Forecasting Sales by Exponentially Weighted Moving Averages, Management Science 6(3)). The additive Holt-Winters recursions are

with forecast \(\hat{y}_{t+h|t} = \ell_t + h\,b_t + s_{t+h-m(k+1)}\), where \(k = \lfloor (h-1)/m \rfloor\) picks the most recent matching season. Gardner and McKenzie's damped trend multiplies the trend by \(\phi^j\) at horizon \(j\), so long-range forecasts flatten rather than extrapolate a line forever (Gardner & McKenzie, 1985, Forecasting Trends in Time Series, Management Science 31(10)).

For forty years these were algorithms, not models. They produced point forecasts but no likelihood, no principled intervals and no way to choose between variants except by comparing error on held-out data.

The innovations state-space form

Hyndman, Koehler, Snyder and Grose showed that every method in the taxonomy produces the same point forecasts as a state-space model driven by a single error term (Hyndman et al., 2002, A state space framework for automatic forecasting using exponential smoothing methods, IJF 18(3)). For simple smoothing with additive errors, ETS(A,N,N):

The same \(\varepsilon_t\) appears in the measurement and state equations, which is why the form is called single source of error. The name ETS labels each model by its Error, Trend and Seasonal types, each none (N), additive (A), multiplicative (M) or damped (\(A_d\)).

The payoff is that a likelihood now exists. Parameters and initial states can be estimated by maximum likelihood, models can be compared by AIC, and intervals follow from the model. For ETS(A,N,N) the \(h\)-step forecast variance is \(\sigma^2[1 + \alpha^2(h-1)]\). With \(\sigma = 10\), \(\alpha = 0.3\) and \(h = 5\), the standard deviation is \(\sqrt{136} \approx 11.7\); a random walk (\(\alpha = 1\)) gives \(\sqrt{500} \approx 22.4\). The 2002 paper's automatic procedure fitted 24 candidate models to each series and selected by AIC, then was tested on the 1,001 M-competition and 3,003 M3 series, where it matched the best competition methods and was strongest at short horizons with seasonal data.

AIC can also choose between additive and multiplicative errors, which a point-forecast error metric cannot: both error types give identical point forecasts and differ only in their intervals.

A choice the field has not settled

The textbook most practitioners learn from drops multiplicative-trend models entirely because they tend to produce poor forecasts, leaving 9 trend-seasonal combinations and 18 models (Hyndman & Athanasopoulos, Forecasting: Principles and Practice, 3^rd ed., §8.4). The single-source-of-error form is also a position, not a consensus: structural time series models in the Harvey tradition give level, trend and season independent disturbances, estimated with the Kalman filter, and the two camps disagree about which is the more natural description of a series. The companion concept on Kalman filtering shows that for the local level model the two coincide in steady state.

When it breaks

Multiplicative components need strictly positive data. A series with zeros, which describes most store-level retail demand, makes multiplicative errors and seasonality undefined. Intermittent demand needs its own methods.

Long seasonal periods are expensive. Holt-Winters carries \(m\) seasonal states that must be initialised and estimated. Monthly data needs 12; hourly data with a weekly cycle needs 168, and there is only one seasonal period. Multiple seasonalities belong to decomposition-based or regression approaches, covered under seasonality and decomposition.

No covariates. Prices, promotions and holidays have no place in the recursions. In M5 an exponential smoothing variant with explanatory variables improved on the plain version by 5.7% on average, and the winning LightGBM ensemble beat the best smoothing benchmark by 22.4%, largely through covariates and cross-learning.

Multiplicative intervals have no closed form. For many ETS variants the forecast distribution must be simulated from the state-space model, and intervals are only as good as the Gaussian error assumption, which is routinely optimistic in the tails.