Notes regarding seasonal adjustment

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The purpose of seasonal adjustment is to remove the more or less regular within year patterns often found in economic time series data. This is done to highlight the underlying trend and short run effects of various economic phenomena on the series.

Users of seasonally adjusted data include government officials responsible for formulating economic policy; businesses concerned with economic trends within their industry; and economic researchers.


Appropriate shifting of supply and demand curves can cause seasonal effects in a price series. Consider a market for an agricultural commodity, like the one in the graph. Typically, supply will be restricted at Sw during the winter season. However, the curve will shift to the right as more firms enter the industry during the late summer and fall harvest season. Thus prices will be characteristically high or low during different seasons of the year.

The demand curve could also shift for various reasons. Example - heating oil prices increase during the winter due to an increase in demand caused by lower temperatures.


  • Developed by the U.S. Bureau of the Census in 1967 - Shiskin, Young and Musgrave.
  • First seasonal adjustment software package. It made seasonal adjustment practical in a large scale data production environment.
  • Monthly or quarterly data - usually need 8 to 10 years of continuous data.
  • X-11 assumes the data is decomposable in one of two ways:

Additive decomposition:

Xt = Tt + St + It

SAt = XtSt = Tt + It

Multiplicative decomposition:

Xt = Tt * St * It

SA_t = \frac{X_t}{S_t} = T_t * I_t


  • X = the original series
  • T = trend-cycle component
  • S = seasonal component
  • I = irregular or random component
  • SA= seasonally adjusted series

Note: These are statistical models. An econometric model of supply and demand would be too specialized and not manageable in large-scale data production.

As already discussed, the SEASONAL component models the within year pattern for the series. The TREND can be thought of as the long run or permanent component in the series. The IRREGULAR models the short run or transitory component in the series. A seasonally adjusted series is composed of the trend and irregular and has both the long run and short run effects.

Below are graphs of stylized series for these components: Image:02Graphs of trend, seasonal, irregular.PNG


Part A: The data may be adjusted by "prior adjustment factors" that are supplied by the user. Usually this option is used to adjust for the effects of exogenous events on the series. Example - the series to be seasonally adjusted is monthly production data, and a strike temporarily reduced the level of the series.

Part B: Preliminary trading-day adjustment factors (optional) are estimated in this section. Trading-day adjustment refers to cases where the number of weekdays in a month matters. Example - industrial production data. Preliminary weights for adjusting extreme values for the irregular are also estimated. The irregular component is meant to capture a variety of short run economic phenomena.

Part C: Final trading-day (optional) and extreme irregular adjustments are estimated for the data.

Part D: Final estimates of seasonal factors, trend and irregular components are calculated. Final seasonally adjusted data are also calculated.

Part E: Adjustments for extreme values in the original series and the seasonally adjusted series are made in this section. Typically these series are used for diagnostic purposes only.

Part F: Various diagnostic measures and quality control statistics are calculated.

Part G: Charts of seasonally adjusted series, final trend series, and other series.

5) Some Notes on X-11’s Estimation Methods

5.1) X-11’s Extreme Value Adjustment

• X-11 tries to identify extreme irregular values in parts B and C.

• The first step is to calculate standard errors of the irregular series using a 5-year moving sample of the irregular for each month (quarter).

• Upper and lower critical values for judging the irregulars can be chosen by the user, but the X-11 default values are 1.5 and 2.5 times the corresponding standard error for a given data point.

• Irregulars that are less than the lower critical value are viewed as normal and no adjustment is made. Values that are greater than the lower critical value are viewed as extreme and an adjustment is made. A full adjustment to the irregular is made when it exceeds the upper critical value and a linearly graduated adjustment is made between the lower and upper critical values.


• X-11’s adjustment to the series is:

X_t = \frac{1+w_t[I_t-1]}{I_t}

5.2) X-11’s Moving Average Estimators for the Trend and Seasonal Components

  • X-11 uses a variety of moving averages (MA) to estimate the trend and seasonal components. The irregular componen is estimated as a residual. The general form of a moving average is:

Z_t = \sum_{j=-m}^n{w_jX_{t+j}}

  • Regression techniques do exist for seasonally adjusting data. However, they do not account for "moving seasonality" (amplitude and phase shifts) as efficiently as moving average methods such as X-11.
  • Part D of X-11 contains the most important steps of the seasonal adjustment algorithm. These steps also illustrate the methods used in parts B and C.

• Tables are presented in matrix format - year by month (or quarter).

D1: This table contains the series adjusted for:

  • 1) user supplied "prior adjustment factors" from part A.
  • 2) trading-day effects, if requested.
  • 3) extreme irregular values as calculated in part C.

D2: A 2x12 MA of D1 is computed as a preliminary estimate of the trend component:

1) average every 12 consecutive observations: Y_t = \frac{1}{12}\sum_{j=-5}^6{X_{t+j}}

2) average every 2 consecutive values of the 12-term averages. Z_t = \frac{1}{2}(Y_t + Y_{t-1})

Substituting the 12-term average into the expression for the 2-term average, we obtain the 2x12 MA: Z_t = \frac{1}{24}X_{t+6} + \frac{2}{24}(\sum_{j=-5}^5X_{t+j}) + \frac{1}{24}X_{t-6} = MA(X_t)

We apply the 2x12 MA to the series: MA(Xt) = MA(Tt + St + It) = MA(Tt) + MA(St) + MA(It)

Under the assumptions:

1) \sum_{j=-5}^6 S_{t+j} = 0, then MA(St) = 0 -- the sum of every 12 consecutive seasonals is zero for a a process with ‘stable seasonals’.

2) Tt = a + bt, then MA(Tt) = a + bt - a linear trend component.

3) It is a purely random process, then MA(It)0.

Then MA(Xt) = a + bt = Tt. That is, table D2 approximates the trend.

D3: This table is not used.

D4: Trend adjusted series is calculated as:

D4 = D1 − D2 = S + I These are referred to as the SI sums (or ratios for a multiplicative model).

D5: A 3x3 MA (a 5-term MA) is applied to the D4 data for each month separately. This estimates the seasonal component from the SI sums.

D5 = MA(D4) = MA(S) + MA(I)

MA(St) = \frac{1}{9}S_{t+24} + \frac{2}{9}S_{t+12} + \frac{3}{9}{S_t} + \frac{2}{9}S_{t-12} +\frac{1}{9}S_{t-24}

MA(It) = 0

The weights of this MA have a triangular distribution, where more central observations on S are given more weight. Thus a "smoothed" estimate of S is calculated in this step. Note that this filter will exactly replicate a stable seasonal pattern for and given month because the sum of the weights is 1.

D6: D6 = D1 − D5 = T + I

This is the original series less the seasonal factor estimates, or a preliminary seasonally adjusted series.

D7: In this step the Henderson (1917) moving average for trend is applied to the data of D6. The Henderson moving average is a symmetric,linear, unbiased filter that maximizes the smoothness of the trend estimates. It exactly replicates a cubic function in time. Asymmetric trend filters in X-11 are derived by minimizing the revision between it and the symmetric Henderson filter. MacCauley (1932), Durbin and Kenny (1982), Laniel (1985), Buszuwski (1989), and Thompson and Gray (1995) further discuss these filters.

D8: D8 = (C19 or B1) - D7 = S + I

B1 = original series adjusted for the user supplied prior adjustments of Part A, if any.

C19 = B1 adjusted for trading-day variation (if requested).

D9: D9 = D1 − D7S + I Note that D1 is adjusted for any trading-day and user supplied prior adjustments. This is also true of C19. However D1 is also adjusted for extreme irregular values whereas C19 is not. Thus D8 and D9 are not always identical. The table entries for D9 are the values that are not identical to D8. This highlights the observations that have extreme values for easy reference by the reader.

D10: To the D8 data, with extreme vlues replaced by the D9 entries, a 3x5 MA (a 7-term MA) is applied to the data for each month separately. This step is similar to D5 where a 3x3 MA is used.

MA(D8) = S (final seasonal factors)

Also in this step seasonal factor forecasts are calculated for the next 12 months according to the formula:

S_{t+12} = S_t + \frac{1}{2}(S_t - S_{t-12}) where t is the month of the last year of the sample.

Note that BLS does not use this method of projecting the seasonal factors for the next year. Instead, the last year’s worth of factors are simply repeated.

D11: (C19 or B1) - D10 = T + I

This is the final seasonally adjusted series.

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