Common mistake 1 with Decompose & filter

Using a multiplicative model on data that can be zero or negative.

Common mistake 2 with Decompose & filter

Trusting the HP filter at the very start and end of the sample, where it is known to be unreliable.

Common mistake 3 with Decompose & filter

Treating an HP cycle as an economic fact. Hamilton (2018) showed the filter can manufacture cycles that are not in the data.

Common mistake 4 with Decompose & filter

Interpreting the seasonally adjusted series near the end of the sample. The filters are two-sided, so the most recent points use asymmetric weights and are revised as new data arrive.

EconStatsCraft › Decompose & filter

Decompose & filter

Split a series into trend, seasonal and irregular parts, or extract the business cycle with the Hodrick–Prescott, Baxter–King and Christiano–Fitzgerald filters.

What it needs

Series

The intuition

Most macro and business series are three things at once: where the level is heading, a pattern that repeats every year, and everything else. Separating them lets you ask the right question - "is unemployment genuinely falling, or is it just always low in December?"

When to use it

Any seasonal series before modelling, and any macro series where the question is about the cycle rather than the level.

Where the estimator comes from

Classical decomposition estimates the trend with a centred moving average of length equal to the seasonal period, averages the detrended values by position in the cycle to get the seasonal factors, and calls the remainder irregular. STL replaces the moving average with iterated LOESS smoothing, which lets the seasonal shape evolve and makes the whole thing robust to outliers. The HP filter instead solves \min_{\tau}\sum_t (y_t-\tau_t)^2+\lambda\sum_t\left[(\tau_{t+1}-\tau_t)-(\tau_t-\tau_{t-1})\right]^2, trading fit against smoothness of the trend.

The model

y_t=T_t+S_t+R_t \quad\text{(additive)}\qquad y_t=T_t\times S_t\times R_t \quad\text{(multiplicative)}

What it assumes

You know the seasonal period: 4 for quarterly, 12 for monthly.
Classical decomposition assumes the seasonal pattern never changes shape; STL allows it to evolve.
Multiplicative decomposition needs strictly positive data.
The trend is smooth relative to the seasonal cycle. Where a genuine level shift occurs, the trend component absorbs part of it and the remainder absorbs the rest, and neither is then interpretable.

Statistical properties

Classical decomposition loses period/2 observations at each end; STL does not.
A multiplicative model is just an additive model in logs, and needs strictly positive data.
The HP filter is a two-sided filter, so its end-of-sample values are unreliable - exactly where policymakers want them.
Hamilton (2018) showed the HP filter induces spurious dynamics that are artefacts of the filter, not features of the data.
Band-pass filters (Baxter–King, Christiano–Fitzgerald) target an explicit frequency band; BK loses K observations at each end, CF does not.

How to read the output

Trend: the underlying direction with seasonality removed.
Seasonal: the repeating deviation. Its amplitude tells you how much seasonality matters.
Remainder: if it still shows visible structure, your period is probably wrong.
HP filter cycle: deviations from trend - recessions are the troughs.

A worked reading

STL on quarterly inflation: the trend explains 71% of the variance, seasonality 12%, remainder 17%. Peak at position 2 and trough at position 4 means Q2 inflation is systematically above trend and Q4 below - so comparing raw Q2 to raw Q4 would overstate any slowdown.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Remainder still shows a clear pattern	Wrong period	Try 4 for quarterly, 12 for monthly, 7 for daily
Seasonal amplitude grows with the level	Multiplicative seasonality	Use the multiplicative model, or take logs first
Trend wiggles implausibly	λ too small	Raise λ, or use a band-pass filter
Cycle looks different at the sample ends	End-point problem	Do not interpret the last few periods; say so
STL and classical disagree	The seasonal pattern is evolving	Trust STL and report that seasonality changed

Common mistakes

Using a multiplicative model on data that can be zero or negative.
Trusting the HP filter at the very start and end of the sample, where it is known to be unreliable.
Treating an HP cycle as an economic fact. Hamilton (2018) showed the filter can manufacture cycles that are not in the data.
Interpreting the seasonally adjusted series near the end of the sample. The filters are two-sided, so the most recent points use asymmetric weights and are revised as new data arrive.

Where to read this properly

Cleveland, Cleveland, McRae & Terpenning (1990) - STL
Hodrick & Prescott (1997)
Baxter & King (1999)
Christiano & Fitzgerald (2003)
Hamilton (2018), "Why You Should Never Use the Hodrick-Prescott Filter"

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 · STL forecasting · Forecast combination · Markov regime switching · Change frequency (resample) · ARCH / GARCH volatility · Spectral analysis · Exponential smoothing · Forecast accuracy

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Each example is simulated from a known model, so you can check whether the estimator recovers the truth - and nothing here is anyone's private data.

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Drag a variable into a slot on the right - or just click it.

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Decompose & filter

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

See also

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