2. Whereas chain-linking of annuals in previous years prices is unambiguous, it is not at quarterly frequencies. Contrary to the US, annuals as well as quarters are calculated at average prices of the whole previous year in the EU. For the consecutive construction of quantity indexes, there exist three different methods:
The over-the-year method

3. The annual-overlap method
The quarterly-overlap method

4. 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

5. Time series properties
Over-the-year method: Breaks in the series (compared to the previous quarter) occur every quarter. Does not represent a 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.

6. Time consistency (additivity) property
Over-the-year method: approximately time consistent even away from the reference period
Annual-overlap method: fully time consistent
Quarterly-overlap method: not time consistent, 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 method. 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.‘

7. Differences between the sum of the quarters and annual data in Austrian GDP

8. Relative differences between the AO- and the Denton benchmarked QO-method

9. Distribution of annual chain-linking-differences by the AO and the B-QO-method

10. Consequences for time series modelling of QNA data chain-linked by different methods
Why time series modelling of QNA series?
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 (which 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).
For some kind of business cycle analysis (Beveridge-Nelson decomposition, unobserved components models, …)

11. Models suggested by TRAMO-SEATS
Over-the-year method

12. Annual-overlap method

13. Quarterly-overlap method

14. Denton benchmarked quarterly-overlap method

15. Overview over time series models
outlier detected
OTY
(0,1,1)(0,1,1)
none AO
AO 1993Q1 QO
TC 1993Q1
B-QO
(0,1,0)(0,1,1)

16. Quarter-to-quarter percentage changes seasonally and working day adjusted

17. BC extraction
Seasonally adjusted OTY, AO and B-QO series (by time-series modelling techniques) HP1600-filtered
Unadjusted series (not modelled) BK 32-8 filterd

18. Austrian chained GDP HP-1600 filtered

19. Log differencies of HP-1600-transformed series

20. 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) vis-à-vis the reference series.

21. Bry-Boschan BC turning points in HP-1600 filtered series

22. Business cycle turning points
1992
1993
1994
1995
1996
1997 1 2 3 4
Austria HP ▲ ▼ BK
HP Dyn Fact
BK Dyn Fact
Breuss (1984)
Hahn et (1992)
Artis et M. (2004)
Artis et K. (2004)
Euro area
Mönch et (2004)
Forni et (2000)
CEPR (2003)

23. Capacity utilisation in the Austrian industry

24. Baxter-King 32-8-filtered series

25. Bry-Boschan BC turning points for BK 32-8 filtered series

26. Conclusions
Different quarterly chain-linking methods generate different time series.
Their different time series properties can potentially interfere with modelling outliers, seasonalfactors, BC-components, …
This can lead to different results for analysis based on model pre-processed series.
The turning point detection process itself (not based on data pre-processed or pre-adjusted by timeseries models) seems to be rather robust to different quarterly chain-linking methods.