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Methodology: Seasonal Adjustment of the Transportation Services Index Time Series Data


Because the principal purpose of the index is to reflect monthly shifts in transportation services output and analyze short-term trends, the variation introduced by normal seasonal changes must be removed from the data. Transportation is highly seasonal, and without adjustment, the index would not give an accurate picture of underlying changes in transportation output.

The data underlying the TSI are seasonally adjusted using X12-ARIMA, Release 0.21. X12-ARIMA adjusts, when specified, for the effects of trading day, moving holidays, and data outliers and then decomposes the time series into three components: trend (including cyclic phenomena), seasonal, and irregular. By a series of iterative steps, the seasonal effects are isolated and removed from the original data series. In applying this methodology to the transportation services time-series data, we found that each element of the TSI—rail (passenger and freight), pipeline (petroleum and natural gas), transit, waterborne, trucking, and aviation (passenger and freight)—displays strong seasonal patterns and some but not all are affected by trading days and moving holidays. A brief description of trading-day, holiday, and seasonal effects follows along with a description of the models used to seasonally adjust the TSI data series.



Monthly time series that are totals of daily economic activities are frequently influenced by the weekday composition of the month. Trading-day effects reflect the number of days in the month and the number of times each day of the week occurs in the month, which can affect the monthly totals of output services. Recurring effects associated with weekday composition in monthly (or quarterly) economic time series are called trading-day effects.


Certain kinds of transportation services and their associated time series, such as aviation (passenger and freight), are affected significantly by holidays. Effects from holidays, such as Christmas, that always occur on the same date of a month each year are seasonal components of a time series. Effects associated with holidays that are not always on the same date of a month, such as Labor Day, Thanksgiving, and Easter, are called moving holiday effects.


The seasonal effect in a time series is any effect that is reasonably stable in terms of annual timing, direction, and magnitude. Seasonal adjustment is the process of estimating and removing the seasonal effects from a time series after adjustments have been made for trading days and moving holidays. Because the seasonal effects can disguise important features of economic series such as direction, turning points, and consistency between other economic indicators, seasonal adjustment can also be thought of as focused noise reduction.


The time series data used to create the TSI have varying amounts of historical data. Earlier data are useful for historical purposes, but are of little help in seasonally adjusting data in recent years. Data in the most recent years carry the most weight in seasonal adjustment; data from the beginning of the series have only a marginal impact. For this reason, all time series used in the current (2013) TSI begin at January 2000. This is more than sufficient time to obtain a good seasonal adjustment.
The models for the seasonal adjustment of the TSI inputs are specified in the table below. A multiplicative decomposition was selected, instead of an additive decomposition, when the magnitude of the seasonal variation fluctuated with the level of the series. Trading-day and holiday effects were included in the decomposition, when present with statistical significance, and removed from the original data series. The remaining seasonal component of the data series was removed through the use of an appropriate Autoregressive Integrated Moving Average (ARIM A) model. The ARIMA model describes the relationship between the data points in the time series and the way in which controlled for in decomposing the data into the trend, seasonal, and irregular component.

1 X-12-ARIMA was created by the U.S. Department of Commerce, U.S. Census Bureau. (For details on X-12 ARIMA, now X-13, see:

Updated: Thursday, October 25, 2018