2. Three different chain-linking methods
Austrian GDP in million €
Date
nominal (at prices of the current quarter)
at average prices of the current year
at average prices of the previous year
blue = over-the-year method green = quarterly-overlap method red = annual-overlap method

3. Time series properties
Over-the-year method: Breaks in the series (compared to the previous quarter) occur every quarter. Does not represent any time series in a narrower statistical sense.
Annual-overlap method: Breaks in the series (compared to the previous quarter) occur every first quarter of a year. Represents a time series in a narrower statistical sense only within a year.
Quarterly-overlap method: No breaks in the series (compared to the previous quarter) occur. Represents a time series in a narrower statistical sense.

4. Vertical additivity property
Over-the-year method: approximatly vertically additive even away from the reference period
Annual-overlap method: fully vertically additive
Quarterly-overlap method: not additive, especially away from the reference period
⇨ Splitting-up annual discrepancies over quarters by a method generating time series in a narrower sense (proportional Denton procedure, spline functions) does not interfere with the time series properties of the benchmarked series.
Note: Quarterly-overlap method + pro-rata distribution of annual discrepancies ≙ annual overlap. According to the IMF‘s Quarterly National Accounts Manual (2001, p. 84): ‘Because of the step problem, the pro-rata distribution technique is not acceptable.‘

5. Seasonal and working day adjustment
For working day adjustment Eurostat explicitly recommends regression techniques.
Outlier detection procedures for preparing time series for a following seasonal adjustment are based on time series analysis.
For seasonal adjustment an extrapolation of the series beyond the time series horizon is necessary to apply filter techniques for the recent observations (wich are in the focus of interest). All extrapolation methods rely on time series properties.
For TRAMO-SEATS the seasonal component is extracted by factorization of the time series model. For X-12 instead, mathematical filters are applied (making this procedure slightly less dependent on neat time series properties.

6. Application of TRAMO-SEATS
Over-the-year

7. Annual-overlap

8. Quarterly-overlap

9. Differences in seasonal factors

10. Quarter-on-quarter growth rates

11. Austrian chained GDP HP-1600 filtered

12. Co-movement of cyclical components
Series
Coherence
Average Spectrum
Mean Delay
Cross-correlation
2 Y-8 Y r0
rmax
tmax (1)
GDPao
0.98
0.41
-0.01
0.96
GDPqo
0.88
0.42
0.1
0.92
Note: The + (-) sign refers to a lead (lag) with respect to the reference series

13. Bry – Boschan turning points
Peak
Trough
# of extra
cycles
Reference Series
Q1-1992
Q1-1993
Q1-1994
Q1-1995
Q1-1996
Q2-1997
Q2-2000
Q4-2003
GDPao -2 - -1
GDPqo
Note: The - (+) sign refers to a lead (lag) with respect to the reference series

14. Conclusions
Quarterly national accounts data are most of the time processed further. This goes for all kinds of modelling, whether for adjustment of working days, detection of outliers, seasonal adjustment or macro-economic purposes. Therefore, these data should show well defined time series properties.
Different chaining techniques can lead to different output of those models giving different or wrong information about economic developments.

15. Topics of further interest
Which benchmarking procedure interferes least with original time series properties?
Differences in model outputs if external information of outliers and the model selection process is used?
Regressing the output of different chaining methods on each other and observing the models‘ structure and their calculated residuals.