Seasonal Adjustment

When the government announces that unemployment "fell by 20,000 last month," the number they quote is almost never the raw count. It is seasonally adjusted. Raw employment always jumps in December (holiday hiring) and slumps in January (those temporary jobs end) — every single year, like clockwork. If you reported the raw figures, every January would look like an economic catastrophe and every summer like a boom, when in fact nothing had changed except the calendar. Seasonal adjustment is the routine, high-stakes act of removing the seasonal component so the genuine underlying movement becomes visible.

Formally, for an additive series you subtract the estimated season; for a multiplicative one you divide it out:

y^{\text{SA}}_t = y_t - \hat S_t \qquad\text{(additive)}, \qquad y^{\text{SA}}_t = \frac{y_t}{\hat S_t}\quad\text{(multiplicative)}.

What is left, y^{\text{SA}}_t \approx T_t + R_t, is trend plus irregular — the signal, stripped of its predictable calendar rhythm.

Seasonal indices

The engine of adjustment is a set of s numbers called the seasonal indices — one per position in the cycle (one per month, per quarter, per weekday). For a multiplicative model each index is a ratio: "March typically runs at 0.92 of the local average, December at 1.35." For an additive model each is an offset in the original units. Adjustment is then just: look up this period's index and divide (or subtract) it out.

There are exactly s indices, one per phase of the cycle, and they repeat every s steps.
Multiplicative indices are normalised to average 1 (equivalently sum to s); additive indices are normalised to sum to 0.
The normalisation guarantees adjustment neither inflates nor deflates the overall level — it only reshapes the within-year profile.

The ratio-to-moving-average method

The classic way to estimate multiplicative seasonal indices, and the historical backbone of official adjustment, is ratio-to-moving-average. It threads together everything from the last three pages:

Estimate the trend-cycle with a centered 2×s moving average — this smooths out the season and the noise, leaving \hat T_t.
Form the ratios y_t / \hat T_t. Dividing the raw series by its trend removes the trend and leaves (approximately) season × irregular.
Average the ratios within each season slot — all the Januaries, all the Februaries — so the irregular cancels and a clean seasonal ratio survives for each phase.
Normalise the s figures to average 1. These are the seasonal indices; divide the raw series by them to seasonally adjust.

This is precisely classical decomposition put to work for a single purpose — recover \hat S_t and take it out.

Raw versus adjusted

The jagged line is a raw monthly series with a strong yearly swing; the smoother line is the same series after the seasonal component has been divided out. The calendar ripple vanishes and the underlying trend — a steady climb with a mild dip in the middle — steps clearly into view. Every "is it really getting better?" question is answered on the smooth line, never the jagged one.

Note what adjustment keeps: the trend and the genuinely unusual months are still there. It removes only the predictable part of the calendar, never the news.

X-11, X-13ARIMA-SEATS and the industrial version

National statistics offices do not adjust by hand. The lineage runs from the US Census Bureau's X-11 (1960s) through X-11-ARIMA to today's X-13ARIMA-SEATS, the world standard used by the Census Bureau, Eurostat and central banks. The core is still ratio-to-moving-average with refined Henderson moving averages, but wrapped in serious machinery: an ARIMA model first extends the series with forecasts and backcasts so the moving averages reach right to the endpoints (curing the classical end-effect); trading-day and moving-holiday (Easter, Ramadan) regressors are removed; and outliers are detected and down-weighted. "SEATS" offers an alternative, model-based route to the same goal. It is decomposition, industrialised.

Adjustment feels like it just "cleans" the data, but it is a filter, and filters have side effects. Two to respect. First, the symmetric moving averages inside X-11 must switch to lopsided one-sided filters at the most recent points — so the latest adjusted values get revised as more data arrives, and an apparent turning point (a recession starting, say) can appear, move, or vanish across successive releases. Second, if the seasonal pattern is itself changing and the method assumes it is fixed, some season leaks through — residual seasonality — or, worse, real signal gets absorbed into the seasonal factor and quietly removed. The lesson: a seasonally adjusted series is an estimate, not ground truth, and its freshest points are the least trustworthy — precisely the ones the headlines shout about.

A tempting shortcut to "adjust" for seasonality is to report only year-on-year changes — this December vs last December — which is really the seasonal difference y_t - y_{t-12}. It does cancel a stable season, and it needs no model. But it is a blunt instrument: it blends twelve months of change into one number, reacts a full year late to genuine turning points, and doubles the noise. Proper seasonal adjustment estimates the seasonal shape and removes just that, keeping the month-to-month resolution intact — which is why agencies invest in X-13 instead of quoting year-on-year and calling it done.