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

The annual-overlap method The quarterly-overlap method

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 2004-01 55316.88 55883.83 54749.82 2004-02 58206.69 58323.31 57223.87 2004-03 60165.13 60055.64 59066.67 2004-04 62129.84 61555.77 60727.75 2005-01 57257.14 57433.55 56310.32 2005-02 60614.11 60743.55 59604.40 2005-03 62513.27 62528.67 61398.34 2005-04 64718.30 64397.05 63324.27 2006-01 59452.31   58996.74 2006-02 63595.43 62623.76 2006-03 65441.57 64489.25 2006-04 67899.57 66494.52 blue = over-the-year method green = quarterly-overlap method red = annual-overlap method

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.

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.‘

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

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

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

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, …)

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

Annual-overlap method

Quarterly-overlap method

Denton benchmarked quarterly-overlap method

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)

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

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

Austrian chained GDP HP-1600 filtered

Log differencies of HP-1600-transformed series

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.

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

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)

Capacity utilisation in the Austrian industry

Baxter-King 32-8-filtered series

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

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.