Economists and other practitioners are sometimes interested
in extracting the global trends and business cycles of a time series,
free from the effect of known seasonality. Small movements in the
trend can be masked by a *seasonal component*,
a trend with fixed and known periodicity (e.g., monthly or quarterly).
The presence of seasonality can make it difficult to compare relative
changes in two or more series.

Seasonal adjustment is the process of removing a nuisance periodic
component. The result of a seasonal adjustment is a *deseasonalized* time
series. Deseasonalized data is useful for exploring the trend and
any remaining irregular component. Because information is lost during
the seasonal adjustment process, you should retain the original data
for future modeling purposes.

Consider decomposing a time series, *y _{t}*,
into three components:

Trend component,

*T*_{t}Seasonal component,

*S*with known periodicity_{t}*s*Irregular (stationary) stochastic component,

*I*_{t}

The most common decompositions are additive, multiplicative, and log-additive.

To seasonally adjust a time series, first obtain an estimate of the seasonal component, $${\widehat{S}}_{t}$$. The estimate$${\widehat{S}}_{t}$$ should be constrained to fluctuate around zero (at least approximately) for additive models, and around one, approximately, for multiplicative models. These constraints allow the seasonal component to be identifiable from the trend component.

Given $${\widehat{S}}_{t}$$, the deseasonalized series is calculated by subtracting (or dividing by) the estimated seasonal component, depending on the assumed decomposition.

For an additive decomposition, the deseasonalized series is given by $${d}_{t}={y}_{t}-{\widehat{S}}_{t}.$$

For a multiplicative decomposition, the deseasonalized series is given by $${d}_{t}={y}_{t}/{\widehat{S}}_{t}.$$

To best estimate the seasonal component of a series, you should first estimate and remove the trend component. Conversely, to best estimate the trend component, you should first estimate and remove the seasonal component. Thus, seasonal adjustment is typically performed as an iterative process. The following steps for seasonal adjustment resemble those used within the X-12-ARIMA seasonal adjustment program of the U.S. Census Bureau [1].

Obtain a first estimate of the trend component, $${\widehat{T}}_{t},$$ using a moving average or parametric trend estimate.

Detrend the original series. For an additive decomposition, calculate $${x}_{t}={y}_{t}-{\widehat{T}}_{t}$$. For a multiplicative decomposition, calculate $${x}_{t}={y}_{t}/{\widehat{T}}_{t}$$.

Apply a seasonal filter to the detrended series,$${x}_{t}$$, to obtain an estimate of the seasonal component, $${\widehat{S}}_{t}$$. Center the estimate to fluctuate around zero or one, depending on the chosen decomposition. Use an

*S*_{3×3}seasonal filter if you have adequate data, or a stable seasonal filter otherwise.Deseasonalize the original series. For an additive decomposition, calculate $${d}_{t}={y}_{t}-{\widehat{S}}_{t}$$. For a multiplicative decomposition, calculate $${d}_{t}={y}_{t}/{\widehat{S}}_{t}.$$.

Obtain a second estimate of the trend component, $${\widehat{T}}_{t},$$, using the deseasonalized series $${d}_{t}.$$ Consider using a Henderson filter [1], with asymmetric weights at the ends of the series.

Detrend the original series again. For an additive decomposition, calculate $${x}_{t}={y}_{t}-{\widehat{T}}_{t}$$. For a multiplicative decomposition, calculate $${x}_{t}={y}_{t}/{\widehat{T}}_{t}$$.

Apply a seasonal filter to the detrended series, $${x}_{t}$$, to obtain an estimate of the seasonal component, $${\widehat{S}}_{t}$$. Consider using an

*S*_{3×5}seasonal filter if you have adequate data, or a stable seasonal filter otherwise.Deseasonalize the original series. For an additive decomposition, calculate $${d}_{t}={y}_{t}-{\widehat{S}}_{t}$$. For a multiplicative decomposition, calculate $${d}_{t}={y}_{t}/{\widehat{S}}_{t}.$$ This is the final deseasonalized series.

[1] Findley, D. F., B. C. Monsell, W. R. Bell,
M. C. Otto, and B.-C. Chen. "New Capabilities and Methods of
the X-12-ARIMA Seasonal-Adjustment Program." *Journal
of Business & Economic Statistics*. Vol. 16, Number
2, 1998, pp. 127–152.

- Moving Average Trend Estimation
- Seasonal Adjustment Using a Stable Seasonal Filter
- Seasonal Adjustment Using S(n,m) Seasonal Filters
- Parametric Trend Estimation

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