Common mistake 1 with STL forecasting

Using it on annual data. Annual data has no season and the module will refuse.

Common mistake 2 with STL forecasting

Forgetting the log transform when the seasonal amplitude grows with the level - the additive decomposition then leaves a pattern in the remainder.

Common mistake 3 with STL forecasting

Treating the seasonally adjusted series as if it were the truth. It is an estimate, and it is revised as new data arrives.

Common mistake 4 with STL forecasting

Turning robustness off on a series with a genuine one-off shock: that shock will bleed into the seasonal estimate for every year.

EconStatsCraft › STL forecasting

STL forecasting

The workhorse for strongly seasonal data. STL pulls out a seasonal shape that is allowed to drift and is robust to outliers, the remainder is forecast, and the season is added back - everything a fixed seasonal ARIMA term cannot do.

What it needs

Series to forecast

The intuition

Seasonality is often the largest and most predictable part of a series, and also the part that gets in the way of modelling everything else. STL forecasting takes the sensible route: strip the seasonality out with a robust decomposition, forecast the seasonally adjusted remainder with whatever method suits it, then add the seasonal pattern back. Each step does one job.

When to use it

With strongly seasonal data, especially when the seasonal pattern changes slowly or the series contains outliers that would distort a fixed seasonal term.

Where the estimator comes from

STL - seasonal and trend decomposition using loess - iterates two smoothers. An inner loop alternates between estimating the seasonal component by loess on the cycle-subseries (all the Januaries together, all the Februaries, and so on) and estimating the trend by loess on the deseasonalised series. An outer loop computes robustness weights from the remainder, w_t=B\!\left(|r_t|/(6\,\mathrm{median}|r|)\right) with B the bisquare, and repeats - so a handful of outliers get near-zero weight and stop distorting the seasonal shape. The forecast is \hat{y}_{T+h}=\hat{S}_{T+h}+\hat{A}_{T+h}, where the seasonal part is taken as the last estimated cycle repeated and the adjusted part comes from ARIMA or ETS.

The model

y_t=S_t+T_t+R_t,\qquad \hat y_{n+h}=\hat S_{n+h}+\widehat{(T+R)}_{n+h}

What it assumes

The components are additive. Take logs first if the seasonal swing grows with the level.
At least two full seasonal cycles of data, and preferably many more.
The seasonal pattern changes slowly rather than jumping.
The seasonal period is correct and constant. STL takes the period as given, and a series with two seasonalities - weekly and annual in daily data - needs a method that admits both.

Statistical properties

STL allows the seasonal pattern to CHANGE over time, at a rate the seasonal window controls. Classical decomposition and seasonal dummies force it to be constant, which is usually false over long samples.
The robust variant is not optional on real data. One strike, one holiday shift or one data error will otherwise bend the seasonal shape for every year.
Only additive decomposition is supported directly. For seasonality that grows with the level, take logs first, forecast, then exponentiate - and remember the back-transform is biased for the mean.
The seasonal component is extrapolated by repeating the last cycle, so the method assumes seasonality is stable going forward even though it allowed it to drift historically.
Prediction intervals come only from the model fitted to the adjusted series, and therefore ignore uncertainty in the seasonal estimate. They are slightly too narrow.
STL needs at least two full cycles and works poorly on short series, where the seasonal and trend smoothers cannot be separated.

How to read the output

Look at the seasonal panel: if the shape visibly drifts across the sample, that is exactly what STL is for and a fixed SARIMA term would have missed it.
The forecast intervals come from the model fitted to the adjusted series, and do NOT include uncertainty about the seasonal component itself - so they are slightly too narrow.
A remainder with obvious structure left in it means the inner model is too simple.
The seasonally adjusted series is the input to the forecast, so anything STL leaves in the seasonal component is not forecast by the model - it is added back at the end, unchanged.

A worked reading

On a monthly series with a drifting seasonal pattern, seasonal dummies force one fixed set of twelve coefficients while STL lets the July effect grow across the sample. The forecast difference is small at one step and substantial at twelve, because a fixed seasonal that was fitted to the average of the whole history is wrong at both ends by construction. The trade-off is honest and visible: STL follows recent seasonality more closely and will follow a spurious recent change just as faithfully.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Seasonal shape changing sharply year to year	Seasonal window too short	Lengthen it, or use periodic seasonality
Remainder showing seasonal structure	Under-smoothed seasonal extraction	The decomposition has not finished the job
Outliers bending the seasonal figure	Robust weights not used	Turn robustness on
Seasonality growing with the level	Additive model on multiplicative data	Take logs first
Fewer than two full cycles	STL cannot identify the components	Use seasonal dummies instead
Intervals too narrow	Seasonal uncertainty excluded by construction	Known limitation; widen if it matters

Common mistakes

Using it on annual data. Annual data has no season and the module will refuse.
Forgetting the log transform when the seasonal amplitude grows with the level - the additive decomposition then leaves a pattern in the remainder.
Treating the seasonally adjusted series as if it were the truth. It is an estimate, and it is revised as new data arrives.
Turning robustness off on a series with a genuine one-off shock: that shock will bleed into the seasonal estimate for every year.

Where to read this properly

Cleveland, Cleveland, McRae and Terpenning (1990), "STL: A Seasonal-Trend Decomposition Procedure Based on Loess", Journal of Official Statistics 6(1).
Hyndman and Athanasopoulos, Forecasting: Principles and Practice (2021), ch. 3 and 7.
Findley et al. (1998), on X-12-ARIMA, the official-statistics alternative.

Related Time series - one series

Time-varying coefficients (Kalman) · Unit root tests · Correlogram (ACF / PACF) · ARIMA / SARIMA · Rolling-origin cross-validation · Theta method · Unobserved components · Forecast combination · Markov regime switching · Change frequency (resample) · ARCH / GARCH volatility · Decompose & filter · Spectral analysis · Exponential smoothing · Forecast accuracy

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STL forecasting

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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