Alternative twosteps reconciliation practices Enrico Infante Eurostat Unit

Alternative two-steps reconciliation practices Enrico Infante Eurostat, Unit C. 2: National Accounts Production Università degli Studi di Napoli Federico II, dipartimento di scienze economiche e statistiche Germana Scepi Università degli Studi di Napoli Federico II, dipartimento di scienze economiche e statistiche NTTS 2017 Brussels, 14 -16 March 2017

The reconciliation issue • The High Frequency (HF) time series should add up to the Low Frequency (LF) time series temporal constraint • The sum of the component series should normally be equal to the main series contemporanues constraint • When in system(s) of time series both the temporal and the contemporaneus constraints are not satisfied, a reconciliation issue is faced • The final aim is to pass from preliminary estimates, which do not satisfy (completely or partially) both constraints, to estimates which are completely reconciled 2 Eurostat

From preliminary estimates. . . V 1 V 2 Z p 11 p 21 z 1 p 12 p 22 z 2 p 13 p 23 z 3 p 14 p 24 z 4 Y 1 Y 2 Tn Contemporaneous constraint Temporal constraint 3 Eurostat

. . . to reconciled estimates! V 1 V 2 Z r 11 r 21 z 1 r 12 r 22 z 2 r 13 r 23 z 3 r 14 r 24 z 4 Y 1 Y 2 Tn Contemporaneous constraint Temporal constraint 4 Eurostat

Reconciliation approaches Simultaneous approaches: contemporaneous and temporal constraints are solved at the same time: Ø Ø • • Multivariate Denton (or GRP) Multivariate optimal methods Di Fonzo-Marini (2011, 2015) QNA handbook (2013) Two-Steps approaches: the first step deals with the temporal constraint, while the second step solves the contemporaneous constraints: Ø Ø • • First step: univariate Denton (or GRP) Second step: quadratic distance measure to minimise Quenneville-Rancourt (2005) Di Fonzo-Marini (2011, 2015) 5 Eurostat

Two-steps approach All two-steps approaches follow the same philosophy: • First step: solve the temporal constraint (benchmarking) • Second step: solve the contemporaneous constraint without altering the temporal constraint (balancing) The second step is performed without considering the temporal profile of the series, as it is believed that doing it at the first step is enough 6 Eurostat

First step: benchmarking • The first step consists in (univariate) benchmarking each of the preliminary series of the system • In this way the temporal constraint is solved, and the discrepancies in the contemporaneous constraint are expected to be reduced • Both Quenneville-Rancourt (QR) and Di Fonzo-Marini (FM) methods propose the use of modified PFD Denton (or GRP) in the first step • Therefore, in the first step a method based on a minimisation of a loss function is used 7 Eurostat

But. . . • Other well-known (regression-based) benchmarking techniques could be used in the first step • • Chow-Lin Fernández Litterman Etc. • This way it would be possible to choose amongst different options for the first step 8 Eurostat

Second step: balancing • In the second step, a balancing approach is applied for each LF period • This is obtained via a constrained minimisation of a distance measure: QR FM 9 Eurostat

Nested approach When having more systems of time series which are nested, a possible way of proceeding is to reconcile the lower layers according the results obtained at higher layers It is clear that the results obtained in the lower layers will depend on the results obtained in the higher layers. However, the series in the lower layers are normally of lower quality as well, which would justify the use of a cascade approach First layer (1 digit) Second layer (2 digit) Third layer (3 digit) Cont. Constraint New Cont. Constraints 10 Eurostat

Reconciliation after SA • Reconciliation is often needed after seasonal adjustment has been performed using the direct approach • The results of the reconciliation process will depend on the level at which the indirect approach has been used • Testing for common seasonal patterns (Infante-Buono, 2015) could help the practitioner to decide at which level to directly adjust the series • The natural expectation is that directly adjusting series with common seasonal patterns would bring to results so that the discrepancies of the contemporaneous constraints are lower than when adjusting series with diverse seasonal patterns 11 Eurostat

Validation – Distance measures 1 • A first important validation tool is to calculate some measures of distance between the preliminary estimates and the reconciled figures could be computed • Mean and maximum Absolute Percentage Difference (Mean. APD and Max. APD) on levels and on growth rates: 12 Eurostat

Validation – Distance measures 2 • Mean and maximum Squared Percentage Difference (Mean. SPD and Max. SPD) on levels and on growth rates: • Concordance of growth rates: 13 Eurostat

Validation – Outliers • After the reconciliation is completed, it is expected that the outliers observed in the preliminary (related) series are exactly the same in the reconciled series • Focusing on the h extrapolated values, it would be possible to check the presence of possible outliers by applying SARIMA forecast intervals on both the preliminary and the reconciled series: The reconciliation results are considered validated in the case all the h extrapolated values are behaving in a consistent way with respect to the forecast interval on pt and rt 14 Eurostat

Case study • Simple small scale case study for the Quarterly Sector Accounts data D 7 N = Other current transfers D 74 = Current international cooperation D 75 = Miscellaneous current transfers D 76 = VAT and GNI based EU own resources • D 74, D 75 and D 76 are not available for many countries, thus only a preliminary estimate is available at quarterly levels; data are available at annual level • We applied different two-steps reconciliation methods: • QR, original and using Chow-Lin at the first step • FM, original and using Chow-Lin at the first step Eurostat 15

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Results • The results of the alternative two step approach using Chow -Lin seems to be very similar to the results achieved when using the modified Denton (original) Mean. SPD_GR FM original 4. 14 0. 76 FM CL 4. 17 1. 85 QR original 4. 31 1. 19 QR CL 4. 26 1. 77 22 Eurostat

Conclusions • As seen in the study presented, the method to be used as a first step could be chosen amongst all the existing benchmarking methods • Results could be similar by using different methods, but this might change on a case by case basis • Therefore the method to be used in the first step could be chosen • Regression-based approaches have specific statistical properties and a number of quality indicators to observe how the model is performing 23 Eurostat

Conclusions A tool for reconciliation of time series which supports two step approaches should in our view: • Have the possibility of selecting different options in the first step (Denton, Chow-Lin, Fernández, Litterman, etc. ) • Consent reconciliation of multiple systems at the same time A large case study application is ongoing 24 Eurostat