1. 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
Qs test Ljung-Box (seasonal test)
NTTS 2017 Brussels, March 2017

2. 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

3. From preliminary estimates...V1 V2 Z
p11
p21 z1
p12
p22 z2
p13
p23 z3
p14
p24 z4 Y1 Y2 Tn
Contemporaneous
constraint
Temporal
constraint

4. ...to reconciled estimates!V1 V2 Z
r11
r21 z1
r12
r22 z2
r13
r23 z3
r14
r24 z4 Y1 Y2 Tn
Contemporaneous
constraint
Temporal
constraint

5. Reconciliation approaches
Simultaneous approaches: contemporaneous and temporal constraints are solved at the same time:
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:
First step: univariate Denton (or GRP)
Second step: quadratic distance measure to minimise
Quenneville-Rancourt (2005)
Di Fonzo-Marini (2011,2015)

6. 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

7. 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

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

9. 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

10. Nested approach
First layer
(1 digit)
Second layer
(2 digit)
Third layer
(3 digit)
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
Cont.
Constraint
New Cont.
Constraints

11. 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-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

12. 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 (MeanAPD and MaxAPD) on levels and on growth rates:

13. Validation – Distance measures 2
Mean and maximum Squared Percentage Difference (MeanSPD and MaxSPD) on levels and on growth rates:
Concordance of growth rates:

14. 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

15. Case study
Simple small scale case study for the Quarterly Sector Accounts data
D74, D75 and D76 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
D7N = Other current transfers
D74 = Current international cooperation
D75 = Miscellaneous current transfers
D76 = VAT and GNI based EU own resources

22. 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)
MeanSPD
MeanSPD_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

23. 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

24. A large case study application is ongoing
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