Internal Coherence in Seasonally Adjusted Chain Laspeyres Indices:
an Application to the Italian Hourly Labour Cost Indicators
Anna Ciammola, Donatella Tuzi
ISTAT - Istituto Nazionale di Statistica
via Tuscolana 1778 - 00173 Rome Italy
[email protected], [email protected]
Keywords: Coherence, seasonal adjustment, chain Laspeyres index, hourly labour cost index.
1
Introduction
In line with the ESS (European Statistical System) Quality Definition, output quality is assessed
in terms of several components. One of them refers to coherence and comparability. Leaving out comparability, that is itself an aspect of coherence across domains, coherence may refer to time (coherence
between sub-annual and annual statistics), to other domains (coherence with National Accounts, with
mirror statistics or with other surveys) and to the fulfillment of internal arithmetic and accounting
identities (internal coherence). The latter constraint is almost always fulfilled in the production of
unadjusted data, while for seasonally adjusted data the outcome depends on both the approach used
and the number of series treated. Time series disseminated by National Statistics Institutes (NSIs)
are generally part of systems of indicators classified by levels whose totals can be produced through an
aggregation process (sum, weighted average, ...). In this context, seasonal adjustment of such systems
could be carried out through two possible strategies: on the one hand, the independent treatment of
each series, the aggregate and its components (direct approach); on the other hand, the treatment of
the component series (i.e. the lowest available breakdown) and then their aggregation according to the
same rules utilized for unadjusted data (indirect approach). The question of whether adjustment should
be direct or indirect is still an open question and neither theoretical nor empirical evidence uniformly
favours one approach over the other. In accordance with the ESS guidelines on seasonal adjustment,
the direct approach could be preferred for transparency and accuracy, especially when component series
show similar seasonal patterns, while the indirect approach could be preferred when component series
show seasonal patterns differing in a significant way. Moreover, when the direct approach is followed,
another issue, strictly related to dissemination practices, should be taken into account: when only
few series are seasonally adjusted, incoherence may easily emerge between components and aggregate
(especially whether period-on-period (p-on-p) growth rates are compared) confusing less expert users.
On the contrary, incoherence may be less evident when many seasonally adjusted series are aggregated.
The former case applies to the hourly Labour Cost Index (LCI), a set of indicators required by
the Commission Regulation N.450/2003 (hereafter called LCI-R). In the LCI system the sub-levels are
represented by NACE sections and by the total labour cost components. Since these components are
affected by different exogenous shocks (changes in regulations, collective agreement renewals, ...) the
direct approach often results in a lack of internal coherence. The indirect approach represents a good
alternative, but the main drawback is that some indices (namely the total indices) are computed as
chain Laspeyres indices, for which the additivity is lost. This represents a crucial issue in seasonal
adjustment as the seasonally adjusted aggregates for the labour components are chain indices. Therefore they cannot be aggregated in accordance with the weighting system used to compute unadjusted
indices, that is indices not chained yet. In fact, aggregation precedes the chain linking in the compilation of unadjusted data, whereas aggregation follows the chain linking (that excludes additivity) when
seasonally adjusted data are considered.
This paper focuses on this issue and, considering the Italian LCI indicators, it shows how internal
coherence is a quality component that cannot be left out when seasonal adjustment is performed,
proposing a method to deal with the aggregation of seasonally adjusted chain indices. The work is
1
organized as follows: the next section describes the Italian LCI indicators and the problems arisen
from the direct approach for the seasonal adjustment of aggregated series; the third section deals with
chain Laspeyres indices and derives the updating coefficients to be applied to the original weighting
system in order to aggregate chain linked indices; the fourth section presents the indirect approach,
assesses the quality of the seasonal adjustment and compares the results with the direct approach; the
fifth section concludes.
2
The Italian Labour Cost Indicators
This paper focuses on the case of the quarterly Italian Labour Cost Index (LCI), described in
Ciammola et al. (2009), produced by the Italian Oros Survey (Congia et al., 2008) to satisfy the LCI-R.
According to it member states are required to produce comparable indices of wages, other labour costs
and, as a synthesis, total labour cost as relevant measures for an understanding of the inflationary
process and the dynamics of the labour market at European level. The Labour Cost Index is also
included in the list of the Principal European Indicators (PEEIs), a set of infra-annual macroeconomic
key statistics for the EU/Eurozone.
The LCI indices are required by economic activities and sections B to S, as defined by the NACE
Rev. 2 nomenclature, have to be covered. Together with unadjusted series, working-day and seasonally
and working-day adjusted series must be delivered, but no instruction is given on the approach to be
used for the treatment of calendar and seasonal effects.
From a definitional point of view the main variables contributing to the calculation of the LCI
draws a two-way classified system of indicators that should satisfy two cross-sectional aggregation
constraints, one involving the sectorial aggregation, the other concerning the total labour cost as
aggregation of wages and other labor costs. Let S be the set of sectorial cross-sectional aggregation
constraints over the Labour cost components in each sector s, and C the set of Labour cost components
cross-sectional aggregation constraints over the sectors in each Labour cost component c. These set of
constraints can be written as follows (Dagum and Cholette, 2006):
YsC,t =
C−1
X
Ysc,t
(1)
Ysc,t
(2)
c=1
YSc,t =
S−1
X
s=1
where Ysc,t , YsC,t and YSc,t represent the interest variables observed at quarter t and referred, respectively, to the sectorial component (sc), to the sectorial total cost (sC) and to the total component (Sc).
Let us imagine the system organized in a two-way table whose rows and columns are, respectively, the
Nace sections and the labour cost components. In the system, for each quarter, there are S constraints
linking the components series to the “row” marginal totals YsC,t and C constraints linking the component series to the “column” marginal totals YCs,t . The respect of these constraints, that implies perfect
coherence in the system, is strictly dependent on the computational process the data are submitted to
in order to get a synthetic representation of the analyzed phenomena.
In accordance with LCI-R and Eurostat recommendations are hourly indices are compiled using
different formulas for components and for the aggregates. The sectorial indices of each of the three
labour cost variables are calculated as elementary indices:
b Isc,t
Ysc,t
H
à s,t !
=
Ysc,t
m b
t∈b
b Hs,t
(3)
where H the total hours worked in sector s at quarter t 1 . This index compares hourly level of each
component in quarter t with its average value calculated over the quarters of the fixed base year b.
1
In the notation Hs,t the subscript c is omitted as hours worked in sector s refer to every component.
2
Given relation (3), it can be demonstrated as the elementary index for the total labour cost
aggregate C can be obtained as a cross-sectional weighted mean of the elementary indices of its two
components, wages and other labour costs:
b IsC,t
=
C−1
X
b Isc,t b wsc
(4)
c=1
The “implicit” weights for the aggregation are defined as:
Ã
!
Y
b sc,t
m
C−1
t∈b
X
b Hs,t
! where
Ã
b wsc =
b wsc = 1.
Y
c=1
b sC,t
m
t∈b
b Hs,t
(5)
These weights, defined at section level and depending on the average values of the hourly indicators in the base year, are constant and guarantee the fulfillment of the coherence between the “row”
marginal totals (the elementary indices for the total labour cost) and their components.
As far as the aggregation over the economic sectors (S) is concerned, LCI-R requires the totals
to be calculated as Laspeyres indices based upon a weighting structure defined at section level and
referred to the previous year (a − 1) (year (a − 2) is also allowed). These indices are subsequently
converted to the fixed base b through a chain linking. The basic Laspeyres is:
S−1
X
a−1 LCISc,t
=
s=1
S−1
X
ysc,t
a−1 hs
(6)
a−1 ysc a−1 hs
s=1
where ysc,t is the hourly labour cost component c in sector s while a−1 hs are the corresponding hours
worked in the base year a − 1. The previous formula can can be easily transformed in the following:
a−1 LCISc,t =
S−1
X
a−1 Isc,t a−1 ωsc
(7)
s=1
where a−1 ωsc are the annual weights of each labour cost component c in sector s over the corresponding
aggregate in the reference period a−1 . It is worth stressing that a−1 ωsc 6=b wsc , for a − 1 = b.
The Laspeyres indices are finally chain linked according to the following expression:
b LCISc,t
= 1 · LSc,0,1 · LSc,1,2 · LSc,a−2,a−1 ·a−1 LCISc,t
(8)
where the annual links LSc,l,l+1 for year l to year l + 1 are defined as:
S−1
X
LSc,l,l+1 =
ysc,l+1 hsc,l
s=1
S−1
X
= m
t∈l+1
l LCISc,tl+1 a−1 Wsc
(9)
ysc,l hsc,l
s=1
The application of the chain linking technique has an important effect: the additivity property of
the basic Laspeyres shown by (7) is lost and, as a result, the total indice b LCISc,t cannot to be compiled
aggregating the sectorial elementary indices b Isc,t through the original weighting system. This issue
has a crucial implication in the choice of the approach to be used for the seasonal adjustment of the
aggregates: the weights used to compute the total unadjusted indices, that are indices not chained yet,
are inadequate to calculate indirectly the seasonally adjusted series, that must include the chaining.
3
Sectorial tlc (%) Incoherencies
Nace Rev. 2 sections - Total labour cost
B - Mining and Quarrying
0.3
9.0
C - Manufacturing
34.8
0.0
D - Electricity, Gas, Stream and Air Conditioning Supply
1.2
3.0
E - Water Supply; Sewerage, Waste Management, ...
1.5
5.0
F - Construction
9.8
16.0
G - Wholesale and Retail Trade, Repair of Vehicles
15.8
2.0
H - Transportation and Storage
9.0
6.0
I - Accomodation and Food Service Activities
4.3
0.0
J - Information and Communication
5.2
12.0
K - Financial and Insurance Activities
7.7
0.0
L - Real Estate Activities
0.4
0.0
M - Professional, Scientific and Technical Activities
4.3
7.0
N - Administrative and Support Service Activities
5.6
2.0
B-N labour cost components
Wages
0
Other costs
0
Total labour cost
0
Incoherencies (%)
23.1
0.0
7.7
12.8
41.0
5.1
15.4
0.0
30.8
0.0
0.0
17.9
5.1
0
0
0
Legend: tlc - total labour cost (the sectoral tlcs are expressed as % of the B-N tlc).
Table 1: Incoherencies in the LCI system of time series in terms of q-on-q growth rates
(2000Q2 - 2009Q4).
At the moment the Italian LCI time series start from first quarter 2000 for sections B to N and
their aggregation, while time series for sections O to S (that have been delivered since half 2009 due
to a past derogation to the LCI-R) start only from first quarter 2009. Given their shortness, the latter
are not treated and only the former are subject to the adjustment for seasonality.
Given the complexity of the LCI system of time series, at a first stage the seasonal adjustment of
these indicators, performed through the ARIMA model-based procedure implemented in Tramo-Seats
(riferimento), was based on the direct approach implying a series of advantages but also drawbacks
implicit in this technique. First of all, the treatment of the volatility affecting the LCI time series,
due the hours worked at the denominator, is significantly favoured by the independent adjustment
of the single time series. Furthermore, the direct approach guarantees the minimization of residual
seasonality on the adjusted time series for the aggregate and prevents them by spurious seasonality
due to a low quality of the ARIMA models estimated for the components. Nevertheless, this approach
does not control for the inconsistencies between the adjusted aggregate and its components.
Usually in the seasonally adjusted indices coherence is evaluated in terms of p-on-p growth rates:
the aggregate is inconsistent with its components when p-on-p growth rates are not included in the
range of the p-on-p growth rates calculated for the components. In the case of the quarterly LCI
indices, this problem is particularly noticeable in the row marginal totals, referred to the total labour
cost, because of both the number of composing variables (only two) and the fact that components are
affected by different exogenous interventions (changes in regulations, collective agreement renewals,
...). In order to reduce this inconvenience several expedients have been undertaken along the time.
Firstly, the ARIMA models for the sectorial totals are approximated to the ones chosen for the leading
variable in the aggregate (i.e. wages that represent about 2/3 of the total cost). Same strategy for the
most irregular time series of the other labour cost component whose ARIMA models, approximated to
those of the corresponding wages, implies a reduced number of inconsistencies but a higher propensity
to revision when new observations are available for the estimation.
Table 1 reports the weights of inconsistency situations in terms of q-on-q changes calculated
from the directly seasonally adjusted LCI series. The period ranging from I quarter 2000 to IV quarter
2009 is considered for a total of 40 observations (39 q-on-q changes). The table shows as none of the
labour cost component is affected by incoherencies on the sectorial aggregate: the large number of
4
components ranging from B to N (13) prevent discrepancies. A different situation distinguishes the
compliance of the sectorial coherence between the total labour cost and its 2 components: here the
number of inconsistencies is not negligible. In such a situation the adoption of an indirect approach
aimed at restoring coherence was seriously considered, requiring an updating of the original aggregation
weights to extend additivity to the chain linked indices. This is the aim of the next section.
3
An alternative approach to compute chain Laspeyres indices
As explained in the previous section, the aggregated indices refer both to the NACE sectorial
total cost and to the overall wages, other costs and total cost. The former are computed as elementary
indices, but it is shown that they can be computed as weighted average using “implicit” weights related
to the fixed base b. In order to synthesize the overall components of the labour cost, chain Laspeyres
indices are instead required.
In accordance with Eurostat recommendations, the following steps are followed to derive b LCISc,t :
1. compilation of the indices in previous year base (a − 1) for each sector and variable (the total
cost is treated like its components);
2. aggregation through a weighting scheme related to the base a − 1 in order to derive the overall
indices a−1 LCISc,t ;
3. calculation of the linking coefficients (links LSc,l,l+1 ) and their chaining;
4. application of the chain links to the indices of step 2 to express them in the first available year
base a = 0 (currently 2000);
5. splicing to express chain indices in the fixed base b (currently 2008).
When seasonal adjustment is carried out, elementary indices expressed in the fixed base b are
processed, so they cannot be aggregated through the weighting scheme used in step 2. This section,
proposing an alternative approach to compute chain indices b LCISc,t , shows that such drawback can
be avoided applying the chain linking technique to the annual weighting system. A vector-matrix
representation is used.
From section 2, let a−1 Ws be the weight of sector s for the generic component c (here omitted to
simplify notation), referred to the base year a − 1 and a−1 Is,ta the elementary index, t ∈ a. Arranging
both the annual weights and the sectorial quarterly time series of indices as matrices:


· · · 0 ωS−1
0 ω1
0 ω2
 0 ω1
· · · 0 ωS−1 
0 ω2


 1 ω1
· · · 1 ωS−1 
1 ω2


 ···

···
···
···
(10)
a−1 Ω = (Ω(1) Ω(2) · · · Ω(S−1) ) = 

 b ω1

· · · b ωS−1 
b ω2

 ···

···
···
···
τ −1 ω1 τ −1 ω2 · · · τ −1 ωS−1
(the first two rows contains the same weights since the basic indices are expressed in base 1 for both
year 0 and 1) and

a−1 Ita
= (I(1) I(2)
0 I1,t0
 0 I1,t1

 1 I1,t2

···
· · · I(S−1) ) = 

 b I1,t
b+1


···
τ −1 I1,tτ
0 I2,t0
0 I2,t1
1 I2,t2
···
b I2,tb+1
···
τ −1 I2,tτ
5
···
···
···
···
···
···
···
0 IS−1,t0
0 IS−1,t1
1 IS−1,t2
···
I
b S−1,tb+1
···
I
τ −1 S−1,tτ










(11)
where ta is the t-th quarter of year a (t = 1, . . . , 4), τ is the current (last available) year and
P
S−1
s=1 a−1 ωs = 1. For notational simplicity, only complete years are considered and consequently
matrices a−1 Ω and a−1 Ita have dimension (τ + 1) × (S − 1) and 4(τ + 1) × (S − 1), respectively.
Let 10 = (1 1 1 1) be a vector of 4 ones. The time series (column vector) of the aggregated LCI
quarterly indices in the previous base (a − 1) can be easily calculated:
a−1 LCIta
=
S−1
X
diag(Ω(s) ⊗ 1)I(s)
(12)
s=1
where the operator diag(v) produces a diagonal matrix whose main diagonal contains the elements
of vector v = {v1 , v2 , · · · } and the symbol ⊗ represents the Kronecker product. From equation 9,
multiplying and dividing ys,l+1 by ys,l (the component c is again omitted), the annual links can be
derived from the aggregation of the annual average l ms,l+1 of the sectorial indices l Is,tl+1 or from the
annual average l mS,l+1 of the overall index l LCIS,tl+1 . In matrix notation these links are:
M = (M(1) M(1) · · · M(S) )

0 m1,0
0 m2,0
 0 m1,1
0 m2,1

 1 m1,2
1 m2,2

·
·
·
···
=

 b m1,b+1 b m2,b+1


···
···
m
τ −1 1,τ τ −1 m2,τ
···
···
···
···
···
···
···
0 mS,0
0 mS,1
1 mS,2
···
b mS,b+1
···
τ −1 mS,τ





 = (Iτ +1 ⊗ ( 1 10 ))[a−1 Ita

4



a−1 LCIta ]
(13)
where Iτ +1 is an identity matrix of dimension (τ + 1). Note that the first row, {0 m1,0 , · · · ,0 ms,0 },
contains only ones. In order to express the indices in a unique fixed base b, the chaining technique has
to be applied. At this aim, the matrix CL of dimension (τ + 1) × S containing the chain links can be
computed2 :
 Qb−1

Qb−1
Qb−1
−1
−1
−1
a=0 a m1,a+1
a=0 a m2,a+1 · · ·
a=0 a mS,a+1


 Q

Q
Q
 b−1

b−1
b−1
−1
−1
−1
 a=0 a m1,a+1
a=0 a m2,a+1 · · ·
a=0 a mS,a+1 




 Qb−1

Qb−1
Qb−1
−1
−1
−1


 a=1 a m1,a+1
a=1 a m2,a+1 · · ·
a=1 a mS,a+1 
 (14)
CL = (CL(1) CL(2) · · · CL(S) ) = 
···
···
···
···




−1
−1
−1
m
m
·
·
·
m


b−1 1,b
b−1 2,b
b−1 S,b




1
1
···
1




m
m
·
·
·
m
b
1,b+1
b
2,b+1
b
S,b+1




···
···
···
···
Qτ −2
Qτ −2
Qτ −2
a=b a m2,a+1 · · ·
a=b a mS,a+1
a=b a m1,a+1
It is worth making two remarks. Firstly, the b + 2nd row is composed only by ones because it should
contain the chain links to be applied to indices b Is,tb+1 that, being already expressed in base b, need no
linking. Secondly, the chain links are computed in different ways depending on whether the original
base a precedes or follows the fixed base b.
Only for completeness, the final indices b LCIta can be easily obtained through the following
equation:
b LCIta
= diag(CL(S) ⊗ 1)
a−1 LCIta .
(15)
Since the sectorial indices are expressed in the fixed base b (this happens for seasonally adjusted
series), an alternative approach could be followed aimed at deriving b LCIta as weighted average of
b Is,ta . The deriving issue is that component indices are referred to base b and consequently the original
2
It should be noted that the first and the second row coincide since 0 ms,0 = 1.
6
weights a−1 Ω cannot be used (the latter are used to aggregate a Ita+1 ). A new weighting system, say
b Ω, has to be derived. It can be proved that this new weighting system, b Ω = (Ω̃(1) Ω̃(2) · · · Ω̃(S−1) ),
can be derived utilizing appropriately the chain links. In particular, the equation is:
Ω̃(s) = diag(CL(s) )−1 diag(CL(S) ) Ω(s)
(16)
and the computation of the final indices b LCIta is accordingly modified:
b LCIta
=
S−1
X
diag(Ω̃(S) ⊗ 1) Is .
(17)
s=1
When weights b Ω are used, attention must be paid to their properties. In fact it can be proved
P
that S−1
s=1 b ωs = 1 only if a = {b − 1, b, b + 1}. Moreover, when more levels of aggregation are dealt
with, weights Ω̃(s) are not additive and consequently they have to be defined for each level. In a sense,
the non-additivity of the chain indices is transferred to the updated weights. This feature is not treated
in this paper as it does not concern the LCI indices system.
4
Direct and indirect adjustment
The comparison between direct and indirect approach to seasonally adjust sectorial (or geographical) aggregated time series represents an important issue for NSIs, especially when data are
released to users. Nevertheless literature has not treated this problem with the same interest showed
for the temporal aggregation and disaggregation. From the methodological point of view the most
detailed treatment, carried out by Geweke (1978), dates back to about thirty years ago. The author
sets the seasonal adjustment of aggregated time series in a multivariate context showing that the multivariate approach is preferable to the univariate one (both direct and indirect) in terms of minimum
Mean Square Error (MSE) criterion for the final estimator and assuming the joint distribution of the
components is known.
Ghysel (1997) compares both the direct seasonal adjustment of the aggregated time series and
the indirect seasonal adjustment aggregating the seasonally adjusted components, minimizing the MSE
of the final estimator and formalizing the conditions for which one approach overcomes the other. The
analysis is carried out in the model-based context, but it is extended to the use of uniform filters (ad
hoc filters) and shows that the direct approach often gives better results than the indirect approach.
More recent developments focus on the model-based approach. Planas and Campolongo (2000)
have discussed the seasonal adjustment of aggregated time series comparing the univariate, direct and
indirect, and the multivariate methods. Their contribution is twofold. On the one hand, the variance
of the final estimator and the variance of the revision error of the concurrent estimator are considered
to select among the different approaches. On the other hand, focusing on the univariate context, two
models are proposed for the aggregated series: one is estimated directly on the series, the other derived
aggregating the models for the components series (results are achieved only for the latter).
Otranto and Triacca (2002) propose a criterion to measure the discrepancy between direct and
indirect approach: the autoregressive metric originally proposed by Piccolo (1989, 1990) to measure
the dissimilarity d2 between two ARIMA (AutoRegressive Integrated Moving Average) models, one for
the seasonally adjusted component of the aggregated series, the other obtained as aggregation of the
ARIMA models for the seasonally adjusted component of the elementary series.
Although the interest of the previous proposals, the treatment of many components and the
computational complexity reduce their use for the current production of aggregated seasonally adjusted
indicators. Consequently, several empirical criteria are used in the NSIs practice in order to choose, in
a univariate context, between direct and indirect approach: residual seasonality, smoothness, stability
and revision size (see Ladiray and Mazzi (2003) and Hood and Findley (2003)). However, it is worth
stressing that the issues related to the sectorial/geographical aggregation is not only the result of an
imperfect methodology, but it derives from the lack of an exact definition of both seasonality and
seasonal adjustment for aggregated time series. This is clearly pointed out in Maravall (2006) where,
through a theoretical example, it is shown how the indirect seasonal adjustment does not consider the
aggregate features.
7
Indices
MPD MAPD
Nace Rev. 2 sections - Total labour cost
B - Mining and Quarrying
0.00
0.24
C - Manufacturing
0.00
0.01
D - Electricity, Gas, Stream and Air Conditioning Supply -0.02
0.12
E - Water Supply; Sewerage, Waste Management, ...
0.00
0.05
F - Construction
0.00
0.17
G - Wholesale and Retail Trade, Repair of Vehicles
-0.01
0.05
H - Transportation and Storage
0.00
0.13
I - Accomodation and Food Service Activities
0.00
0.01
J - Information and Communication
-0.09
0.23
K - Financial and Insurance Activities
0.00
0.01
L - Real Estate Activities
0.00
0.01
M - Professional, Scientific and Technical Activities
0.00
0.08
N - Administrative and Support Service Activities
0.00
0.04
B-N labour cost components
Total labour cost
-0.03
0.14
Wages
-0.03
0.14
Other costs
-0.03
0.17
Q-on-Q changes
MPD MAPD
-0.02
-0.01
0.01
0.00
0.00
0.00
0.01
0.00
0.01
0.01
-0.01
0.00
-0.01
0.35
0.01
0.18
0.07
0.29
0.06
0.20
0.01
0.33
0.02
0.01
0.09
0.06
0.03
0.01
0.02
0.17
0.18
0.24
Table 2: Differences between direct and indirect seasonally adjusted LCI indices(2000Q2
- 2009Q4).
However, a criterion that generally is not considered in comparing several approaches is the
internal coherence of a system of seasonally adjusted indicators in terms of their p-on-p growth rates.
In fact incoherent growth rates represent an unpleasant result for dissemination practices. For the
Italian LCI indicators, in the comparison between direct and indirect seasonal adjustment particular
attention is paid to both coherence and stability.
In Table 2 the directly adjusted indices and related quarter on quarter changes are compared with
the indirect ones in terms of mean percentage differences (MPD) and mean percentage of the absolute
differences. The seasonally adjusted data produced by the two approaches do not differ significantly.
The larger values of the MAPD show as differences are not systematic in sign. The quarter on quarter
changes are more affected by the approach used for the seasonal adjustment then indices. The series
with the highest differences are those that showed lower performances in the direct approach.
Several diagnostics are considered to compare the quality of the two approaches: residual seasonality, smoothness and stability of seasonally adjusted data. As far as the residual seasonality is
concerned, due to the shortness of time series, the spectral diagnostics are not considered, while autocorrelations for seasonal lags of the stationary transformation of seasonally adjusted series are negative
and not significant. Concerning the smoothness, measures R1, R2, M ar(S) and M ar1(T C) used in
Ladiray and Mazzi (2003) do not show that one approach outperforms the other. More importance is
given to stability of the estimates, that occurs when the revisions to the past estimates are small if
new data are added into the estimation procedure. In order to provide a measure of stability of the
seasonally adjusted LCI indices two diagnostics have been computed, based on the sliding spans and
on the revisions history (USCensusBureau, 2009).
The sliding spans technique consists in the comparison of the seasonally adjusted data of a given
time series from overlapping spans through descriptive statistics. These statistics measure how the
seasonally adjusted data (and their infra-annual changes) vary when the span of data used for the
estimation is modified in a systematic way. Spans differ from each other for the starting and ending
date, that are shifted one year per time. The length used to define the spans is constant and depends
on the number of observations of the original series and the filters used for its seasonal adjustment
(in a model-based decomposition such filters depend on the seasonal moving average parameter). In
this example almost all the series have been analyzed using 4 spans, with the exception of one section
8
for which 3 spans have been imposed. The seasonal adjustment in each span is performed fixing the
parameters to the values singled out in the reg-Arima model used to estimate the original time series.
The most common sliding spans statistics are A% and M M % (for further details see Findley et al.
(1998) based on the comparison of the maximum and minimum seasonal adjusted data (and related
p-on-p changes) selected from comparable spans (at least two). A% refers to the seasonally adjusted
indices, while M M % considers the p-on-p changes.
In our exercise both the direct and the indirect approaches give good results in terms of stability,
since the threshold values are never reached. In general, the largest revisions occur in the quarters
where the highest seasonal peaks are observed.
A further instrument to assess stability of the seasonal adjustment process is based on the
evaluation of the revisions history diagnostics. This technique compares the revisions on the seasonally
adjusted data produced by the RegArima model estimated for the original time series with the adjusted
data from the same model with parameters re-estimated on a decreasing number of observations.
Several revision measures can be used for revision lag analysis. In this exercise we refer to the concurrent
target that is the estimate of the adjusted data for t when t < T that is the most recent observation
of the time series. The concurrent estimate gives a measure of the change in the seasonal adjustment
when more data are added in the time series.
In this exercise revisions have been calculated omitting 8 quarters from the initial time series
and gradually adding 1 observation for a new estimate. Revisions have been evaluated comparing the
seasonal adjustment of each quarter t belonging to the range of the omitted data with the estimates
obtained adding 1, ..., 4 observations (lags) to the truncated time series. Results confirm once again
the stability of the seasonally adjusted data produced by both the approaches. Nevertheless, a slight
superiority of the indirect approach emerges implying almost always smaller revisions. This effect
appears amplified on the q-on-q changes since they involve revisions on both t and t − 4.
5
Final remarks
In this paper the seasonal adjustment of the Italian LCI indicators has been considered stressing
the importance of internal coherence when seasonally adjusted components at different levels of aggregation are disseminated together. Working in a univariate context, direct and indirect approach have
been compared. The former is generally suggested, especially when model-based procedures are used
to decompose time series, since it better reflects the features of time series to be handled. In turn, the
latter allows the fulfillment of the aggregation constraints, but it could produce residual seasonality in
particular when serious non linearity problems occur for very disaggregated components.
The work has addressed two main issues. Firstly it shows how seasonally adjusted indices,
expressed in fixed base can be aggregated to produce seasonally adjusted chain linked indices. Using
a matrix representation, the problem has been tackled starting from the original weighting system,
utilized to aggregate indices expressed in the previous year base, and deriving a new weighting system
that allows to aggregate indices already linked and therefore not additive.
Secondly the direct and the indirect approaches are compared. Results are very similar and the
discrepancies between the two approaches are negligible. The indirect approach slightly outperforms
the direct one in terms of revision size of p-on-p growth rates. As a consequence the fulfillment of the
internal coherence becomes the crucial element to prefer the indirect seasonal adjustment.
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