Residual Seasonality: A Comparison of X13 and CAMPLET

Abstract

We compare residual seasonality properties for the series of U.S. real GDP. We download the raw series and the seasonally adjusted version, which is produced by X13ARIMA-SEATS, from FRED, the database of the St. Louis Federal Reserve bank, and calculate CAMPLET seasonal adjustments. In the second round, we seasonally adjust the X13 and CAMPLET seasonal adjustments again. We show graphs of unadjusted and first-round seasonal adjustments and compare unadjusted and all six seasonally adjusted series on the basis a selection of seasonality measures including our own Measure of Seasonality for the last eight years of the sample. Our empirical analyses confirm the strength of CAMPLET in seasonal adjustment. First-round and second-round seasonal adjustments of X13 and CAMPLET are similar. In addition, we do not find evidence of residual seasonality in U.S. real GDP.

Appendix: QS and Measure of Seasonality

4.1.1 $\boldsymbol{Q}\boldsymbol{S}$ Statistic

The QS test⁵ is a variant of the Ljung–Box test computed on seasonal lags, where we only consider positive autocorrelations $\hat{\gamma }$. More exactly,

$$\begin{aligned} \text {QS} = n(n+2) \sum _{i=1}^k \frac{[\max (0,\hat{\gamma }_{i-l})]^2}{n- i \cdot l}, \end{aligned}$$

where n is the number of observations (after differencing to achieve stationarity) and $k=2$, so only the first and second seasonal lags are considered. Thus, the test checks the correlation between the actual observation and the observations lagged by one and two years. Note that $l=4$ in the case of quarterly data. Under the null hypothesis, which states that the data are independently distributed, the statistics follows a $\chi (k)$ distribution. No attempt to rigorously establish an asymptotic theory for QS is published (Bell et al. 2022).

Measure of Seasonality (MofS)

The object of the MofS is to calculate the seasonality in a time series or in a stretch of the series. The length must be at least 4 years plus one observation, to allow calculation of growth between 2 corresponding periods that are 4 years apart. The MofS is explained for series of four years of quarterly data; for series with other periodicity the procedure is similar.

The MofS calculates the average value of each group of corresponding periods (all first quarters, all second quarters, etc.). This average value is called CPA, for Corresponding Periods’ Average. Also the overall average value is calculated. The differences between each of the four CPAs and the overall average value are taken to be seasonalities. To express seasonality as percentages, these differences (D) are divided by the average (A) of 4 CPAs, and the result is multiplied by 100: Percent seasonality $= [D/A] \times 100$.

Before this seasonality can be measured the CPAs must be cleaned of two non-seasonal influences:

• Average growth has more impact on later than on earlier CPAs.

• Aberrant values have an unseasonal impact on CPAs.

In the next two steps, the 4 CPAs are corrected for the average growth and for the impact of aberrant values. After that we will discuss how the MofS deals with a seasonal pattern that is changing during the selected length of the series.

Average Growth

The selected stretch must have a length of at least four years plus one observation. The non-seasonal development, to be called ‘growth,’ between the first and the seventeenth observation is calculated as $(\text {Obs.} \#17 - \text {Obs.} \# 1)/16$. This is seen as the average growth during this stretch of the series for four years. The impact of growth on each CPA depends on its distance from the center. We calculate the 4 CPAs without non-seasonal growth: CPA 1: plus 1.5 x growth; CPA 2: plus 0.5 x growth; CPA 3: minus 0.5 x growth; and CPA 4: minus 1.5 x growth. This correction for growth has no impact on the combined average of 4 CPAs.

Aberrant Values

Aberrant values must not be mixed up with seasonality, and seasonality is calculated by comparing each CPA to the average of four CPAs. In the standard setup, an outlier that causes a difference of one CPA from the average of four CPAs by 5% or more is replaced by this average (the operator can change this percentage). A divergence of 5% between a CPA and the collection of 4 CPAs indicates an outlier of over 25%. The impact of an aberrant value of say 25% on the seasonality can be seen as follows (selected CPA length 4 years). The impact on the (four years) CPA: 25%/4 = 6.25%; the impact on the collection of 4 CPAs: 6.25%/4 = 1.56%; the difference of impact on CPA and on collection: 6.25% − 1.56% = 4.69%. And the impact on seasonality: 6.25% − 1.56% = +4.69% for the quarter that includes the aberrancy and −1.56% for the other 3 quarters.

The MofS first calculates all CPAs and all collections of 4 CPAs in the selected stretch of the series, without replacing aberrant values. After this has been done, each CPA is compared to the average of the collection it belongs to. If a CPA differs from the collection by at least 4.69% (rounded to 5%), the CPA is replaced by the value of the collection, thus reducing the impact of the outlier on the seasonality to 1/16 of its value.

Aberrant Growth

Because the growth has been calculated as the difference between two observations at the beginning and at the end of the selected stretch of the series, one must be aware that, if one or both of these two observations are aberrant, this will have impact on the growth figure and on the calculated seasonality. This is one more reason to compare the seasonality for more than one section of the time series. It must be left to the judgement of the operator to discard an outcome between aberrant observations.

Changing Seasonality

Seasonality can also change in the course of time. Then, during the first years after the change, the MofS would present an average of seasonality before and after the change, which is unwanted. For the MofS, we adopt the CAMPLET approach. At the first occurrence of the change it is seen as an aberrant value, and the concerned observation is replaced by the average of corresponding periods. When a similar deviation occurs in the same quarter of the following year, it is concluded that the seasonality has changed. Now the average of corresponding periods is replaced by the deviating value, thus at once adopting the changed seasonality in the year after the change took place.