CYCLICAL INDICATORS SYSTEM REVISION
S.G. National accounts
National Institute of Statistics
March 2001
1. INTRODUCTION
The first modifications of the System of Cyclical Indicators (SCI), which the INE has presented since 1994, appear in this note, see INE (1994).The concurrence of changes in the main sources, the modification of the dynamic structure of the basic indicators and the availability of new computational procedures have meant an extensive revision of SCI. The first results are presented subsequently, detailing the methodology employed (section 2), the basic empirical results (section 3) and their dissemination scheme (section 4).
During 2001 and 2002 subsequent enlargements were incorporated, such as a new dissemination index which revises the one previously published. (see INE (1995)) and a probability indicator of the turning points depending, especially this last one, of the computer developments of the BUSY project ( Tools and Practices For Business Cycle
Analysis in National Statistical Institutes of EU).
2. STATISTICAL METHODOLOGY
The methodology exposed subsequently is an extension of the dynamic case of the known multi variant analysis factorial model, see Peña and Box (1987) and Peña and
Poncela (1999). This methodology permits the identification of common patterns like the determination of the variance structure (cyclical classification). In addition, it can be used for the design of synthetic indices. A detailed treatment is offered in Melis (1985),
Cristóbal et al . (1996), Cristóbal and Quilis (1998) and Abad et al . (2000).
The basic principle of the methodology may be expressed via the following example.
Given z
2,t
an advanced indicator of z
1,t
with a variance of 1 month. Then z
2,t-1
is a coincident indicator of z
1,t
and the relation which links the two series could be expressed in the following manner:
1

[1]

 z z
2
1 , t
, t

1







1
2

 f t





1 , t
2 , t

 where f t
is the common factor that links both series and which explains a large part of its variability.

I
are the corresponding loads and

I
are the idiosyncratic factors that represent the variations of z
I
which cannot be attributed to the systems common pattern. In this way, once the variance structure existing between the indicators is identified (in the example, the variance of one month between 1995 z
2,t
and z
1,t
), a static factorial model is estimated for the series vector adequately put into place.
Naturally, the usual measures of adjustment and diagnosis ease of this class of models are applicable, due to which there is a potent instrument for quantifying the factorial hypothesis available.
Consequently, in this class of models there exist two equations: one expresses the existing dynamic between the temporary series vectors and another which represents the factorial structure. The first is:
[2] W t

H ( B ) Z t con H ( B )
 diag ( B d 1
..
B dk
) where Z t
is the vector k of observed series, d
I
are the corresponding variances and W t is the vector of series that therefore develop coincidentally. The expression acts as a transition equation reflecting the dynamic structure of the system.
To complete the model a second equation is needed which relates W t
with its common unobservable factors.
[3] W t

PF t
  t where F t
and E t
are vectors that represent, respectively, the common r of the system
(r<k) and the idiosyncratic k factors . The specification is completed by assuming orthogonality between F t
and E t
and imposing a diagonal structure on the idiosyncratic components:
[4] E t
~ iid N ( 0 ,

E
)

E
 diag (

E 1
..

Ek
)
The empirical identification of this model requires, as essential steps, the determination of the variance structure d=[d
1
..d
k
] and the estimate of the corresponding load matrix P .
With the object of carrying out this empirical identification procedure the following algorithm is proposed.
1. Extension of the observed series vector
The original vector Z t
is enlarged including its variance values Z t-j
with j=0..M. The corresponding vector of elements k(M+1) is:
[5] Y
M , t







Z
Z
Z t t

1
...
t

M






2

The determination of the maximum value of the delay M may be carried out via a priori considerations either via some penalisation criteria such as the AIDC or the BIC. The choice of an appropriate value for M requires weighting two contrary considerations: parsimony (to avoid the overparemeterisation of the model and facilitate its interpretation) and resolution (to correctly identify the variance structure).
The experience accumulated with this method suggests that selecting M=12 or M=4 for monthly or quarterly series respectively, is a reasonable choice. Similarly, if the value of k is high the dimension of Y
M,t
becomes unmanageable for which its composition is carried out by considering j=0..M in a discontinuous fashion. For example, with M=12:
[6] Y
M , t








Z
Z
Z
Z
Z t t

3 t

6 t

9 t

12







2. Estimate of static factorial model
A static factorial model is identified and estimated for the enlarged vector Y
M,t
via the analysis of the autovalues and autovectors of the correlation matrix of the said vector.
Note that while M is a sufficiently large value, the said matrix encapsulates all the dynamic of the original vector Z t
in that it contains all the crossed correlation matrices of the said vector up to the delay M :
[7]

Y
( 0 )








 
Z
Z
Z
(
(
...
0
1
( M
)
)
)

Z

Z

Z
( 1 )
( 0 )
(
...
M

1 )
...
...
...
...

Z

(
Z

Z
( M )
M
...
( 0

)
1 )





This matrix is of Toeplitz type (symmetrical by blocks and as a whole) and condenses all the dynamic information from the vector of the observed series Z t
.
Usually, for practical reasons and also with the object of facilitating the interpretation of the model, a unifactorial model: r=1 is considered. Nevertheless, the algorithm is applicable to the general case r>1 without any problem.
[8] Y
M , t

Pf t
  t
The estimate of the loads vector P is carried out by calculating the autovector associated with the greatest autovalue of the contemporary correlations matrix of Y
M,t
.
3. Simplification of the load matrix
Habitually the dimension of P is very high, for which its interpretation is not easy.
Therefore its is convenient to plan some simplification criteria that makes the most important aspects contained in P stand out. where

I,j
the load of the variable z
I
in the delay j . In the first place, only those values that are greater, in absolute value than a certain threshold or “critical value“ will be
3

considered.
 c
. As the distribution of the loads in the sample is not easy to derive, it is recommended that you select a value for
 c
proportional to n -1/2 , where n is the number of observations. Consequently:
[9]
 * i , j
 
0
 i , j en
 los
|
 i , j demás
|
  c casos
The identification of the index of variance d
I
from each series is obtained by calculating the corresponding maximum:
[10]
 * * i , j



 0
 * i , j
 en
 * i , j los
 max j
(| demás
 * i , j
|) casos
The value of subindex j for which the series I is maximised constitutes the estimate of the corresponding delay d
I
.
4. Generation of model
Once the matrix of load P has been simplified and the variance structure d=[d
1
..d
k
] is known the complete model defined above may be formed:
[2] W t

H ( B ) Z t con H ( B )
 diag ( B d 1
..
B dk
)
[3] W t

Pf t
  t
5. Determination of the reference series
One of the advantages of this technique is that it offers a purely empirical way of establishing which is the reference series. Therefore it is worth remembering that the common factor summarises all the dynamic information of the series considered as it has been estimated considering the successive variances of Z t
(operating on the extended vector Y
M,t
and consequently, the loads

I,j
specify the link of each series in each variance with the factor. In this way, we can identify as a reference series those that are found to be more related with the common factor obtained. Formally:
[11] z h , t es la serie de referencia
 h
 max i
(|
 i , j
|)
Naturally the choice of the reference series may be done a priori in which case this stage is omitted.
6. Cyclical classification
The reference series z h,t
is maximally identified with the common factor f t
in a delay d
(0
 d

M). For each variable I
 h considers the delay j corresponding to its simplified non-null load

I,j
and its characteristic variance (de
I
=d-j) is calculated. In this way we have the following classification criteria:
[12] z i , t es
 



 a c r
 



 respecto a z h , t




 de i
 
L
L
 de i de i

L

L



4

where L is a fixed a priori parameter. Naturally, the classification may be more detailed including if possible, very advanced series and/or very delayed series. The case <I> appears when

I,j
= 0
 j, that is, when the series considered is not identified in any way with the dominant pattern and consequently, nor is it with the reference series z h,t
.
The following example is the proposed method: The following are considered k=4,
M=2, r=1 and
 c
=0.2. The factorial model for the enlarged vector
2,t
is:
[13]















 z z z

 z

 z
 z
 z z z z z z
2
2
4
3
4
3 ,
1 , t
2
3 ,
4
.....
1 , t
, t t
, t
.....
1 , t
, t
, t
, t
, t t

1

1

1

1

2

2 t

2

2























































0
0 .
.
91
72 



0
0
.
.
43

11 

.....
0 .
73



0
0
.
.
98

52

 f
0 .
09

.....


0
0
0
.
.
.
61
82
64




0 .
12  t
  t
The application [13] of the aforementioned simplification algorithm gives the following results:
[14]















 z z z

 z

 z
 z
 z z z z z z
1 , t
1 , t
1 , t
, t
, t
.....
2
3
4

1

1

1
.....
2
3 ,
4
2
3 ,
4
, t t
, t
, t
, t

1

2

2 t
 t
2

2














































0 .
91
0
0
0
.....
0








0
0
0
.
98



.....



 f
0
0
0
0
.
64





 t
  t
5

Consequently, it is worth establishing the following:

The series z
1
behaves procyclically and is identified with the common factor in a contemporary way.

For its part, the series z
2
is procyclical but its identification with the factor occurs with a delay of one period.

The series z
3
is anticyclic and is linked with the common factor with a variance of two periods.

Finally, z
4
is not identified with the common factor that is linked with the remaining series, for which it can be considered as anticyclic.
As has already been mentioned, one of the advantages of this focus is that it allows selecting one series of reference by virtue of the empirical characteristics of the data, without making a selection a priori . In the previous example , the series z
2
could act as a reference because it is the most identified with the factor which describes the common cycle of the vector series. As a consequence, since z
1,t
, z
2,t-1
and z
3,t-2
are coincident between themselves, it can be established that z
1
and z
3
are, respectively, delayed and advanced indicators of z
2
:
[15]



 z z z
1 , t
2
3
, t
, t

1

2



 son
 c
 

 z z
1 , t
3 , t

 r a

 z z
2 , t
2 , t


In this way, combining the information supplied by the loads with that supplied by the variances, a complete cyclical classification can be carried out like the one illustrated in the following graph:
Graph 1: Cyclical classification by means of factorial analysis
Carga
PROCICLICAS
ACICLICAS 0
ADELANTADAS
<a>
COINCIDENTES
<c>
ANTICICLICAS
RETRASADAS
<r>
0
Desfase relativo
6

Finally, from the groups of advanced series (Z
<a>
), current (Z
<c>
) and delayed (Z
<r>
) synthetic advanced indices can be formed (as t
), coincidence (c t
) and delayed (r t
) as linear combinations based on the structure of loads and taking into account the relative variance structure existing within each group: a t

A ( B ) Z
 a

, t
[16] c t

C ( B ) Z
 c

, t r t

R ( B ) Z
 r

, t
Said indices are one convex and normalised combination of the series which form each dynamic category, taking as a base the simplified load matrix. It must be noted that the groups are not commonly located in one single point, but rather on a temporal band.
Usually, the variance which begins at zero and is three months separate on each side from the one corresponding to the following group: 0, 3, 6 and 9 is selected as the centre of each group. Later each side is collapsed to said points retaining the internal variance structure.
7

3. BASIC EMPIRICAL RESULTS
The methodology set out in the previous section will be applied to 133 Industrial
Production Indices (IPRI), elaborated monthly by the INE. The sample period starts in
January 1975.
In general the IPRI series are characterised by an amalgam of tendency, seasonal and irregular elements that make the estimate of its cyclical profile difficult. Therefore, in this job the said series have been filtered via the combination of a band pass filter
(interannual rate) and a low pass smoother (the AR(2) filter described in Melis, 1991).
The result of a clear estimate of the preview rhythm of the temporal series object of study.
The resolution of gaps selected is M=12 which has been shown as a reasonable choice in the various trials carried out with this technique.
In graph 1 the first 37 auto values of the correlations matrix for Y t
are shown which, as has already been commented, encompasses the dynamic of the 133 IPI considered. The elevated distance between the first and the second auto value as well as the presence of a morrena ( scree ) phenomenon as of the second, suggests that the appropriate number of factors is one (r=1).I
Graph 2: Matrix autovalues

And
100
80
60
40
20
0
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
Once the appropriate number of factors has been determined the matrix of corresponding loads may be estimated, specifying the variance structure. A significance has not been observed a prioiri for the individual loads bearing in mind the unknown practice of its distribution. The dimension of the Toeplitz type matrix.

Y
is
1729x1729, which supplies an average of the computational requirements of this class of technique.
The average point of the variance reticula (d=6) has been taken as a temporary reference forming five groups by virtue of the distance of each classification variance
I
.
In this way we have:

<ff>: very advanced series: of
I

[-6,-5]
8


<f>: advanced series: of
I

[-4,-2]

<c>: coincident series: of
I

[-1,1]

<b>: delayed series: of
I

[2,4]

<bb>: very delayed series: of
I

[5,6]
Bearing in mind the high dimension of the system modelled here the matrix of corresponding loads is not detailed. Rather what is offered in graph 3 are the five corresponding synthetic indices. The said indices are a convex and normalised combination of the IPI that integrates each dynamic category taking as a base the matrix of simplified loads. It has to be said that the groups are not usually located at just one point but rather in a temporary band. The variance that, starting at zero is finds separated a quarter by each side of that corresponding to the following group: 0, 3, 6 and 9 has been chosen as the centre of each group. Below each band collapses to the said points retaining the internal variance structure. In Cristóbal and Quilis (1996) has a detailed display.
9

15
10
5
0
-5
Graph 3: General CPI and SCI synthetic indicators
20
15
-10
92 93 94 95 96 97 98 99 00 01
M U Y A D ELA N TA D O IPI
10
5
0
-5
-10
92 93 94 95 96 97 98 99 00 01
A D ELA N TA D O IPI
20
15
10
5
0
-5
-10
-15
92 93 94 95 96 97 98 99 00 01
C O IN C ID EN TE IPI
20
15
10
5
20
15
10
5
0
-5
0
-5
-10
92 93 94 95 96 97 98 99 00 01
R ETR A SA D O IPI
-10
92 93 94 95 96 97 98 99 00 01
M U Y R ETR A SA D O IPI
Finally, it is worth asking is these dynamic properties are found to be linked to some structural characteristic of the IPI. As an illustration of this in table 1 the cyclical group obtained with that corresponding to the National Classification of Economic Activities
(NCEA-74) is compared.
10

Table 1: NAS cyclical classification and groups
NCEA-74 Division
Cyclical classification
<aa> <a> <c> <r> <rr>
1: Energy and water
2. Mining of ores. Chemical
3: Transformation of Metals
4. Other products
1
6
15
8
1
7
5
6
3
10
13
8
1
8
1
10
2
13
4
11
In the first place there does not exist a clear concentration pattern in the previous table in such a way that CPI from the different activity branches are not grouped in a very highlighted way in some of the five dynamic categories. Nevertheless, the indices corresponding to the basic industries and other products tend to be delayed while those transformed are predominantly coincident or very advanced. Given the large existing uncertainty over the dynamic phenomenon, it seems reasonable to use both classifications with the object of settling the short term diagnosis on a firmer basis given that each one of them contributes information on aspects not contemplated in the other classification.
4. DISSEMINATION OF THE RESULTS
Accompanying the Quarterly National Accounting data (QNA) the SCI results will be offered, in http://www.ine.es/daco/daco42/daco4214/btcmenu.htm
, whose analysis will be regularly carried out in the The Economic Moment report which is published in the
Quarterly Situational Bulletin of the INE.
11

REFERENCES
Abad, A., Cristóbal, A. and Quilis, E.M. (2000) “Economic fluctuations, turning points and cyclical classification”, Quarterly Situational Bulletin, no. 78, p. 135-171.
Cristóbal, A., Melis, F. and Quilis, E.M. (1996) “The industrial orders portfolio and their dynamic classification”, Fiscal Studies Institute, Work Papers no. 27/96.
Cristóbal, A. and Quilis, E.M. (1998) “Analysis of the economic cycle by means of dynamic indices models. Application to the Industrial Production Indices”,
Quarterly Situational Bulletin, no. 68, p. 73-93.
INE (1994) System of Cyclical Indicators of the Spanish Economy. Methodology and advanced, coincident and delayed synthetic indices , National Institute of
Statistics, Madrid, Spain.
INE (1995) “The elaboration of the employment distribution index”, Quarterly Short
Term Bulletin no. 58, p. 61-76.
Melis, M. (1985) “Characterisation and classification of economic series. Application to the industrial production system indices”, National Institute of Statistics, Internal
Document.
Melis, F. (1991) “The estimate of variation rhythm in economic series”, Statistics Spanish , no. 126, p. 7-56.
Peña, D. and Box, G.E.P (1987) “Identifying a simplifying structure in time series”,
Journal of the American Statistical Association , vol. 82, no. 399, p. 836-842.
Peña, D. and Poncela, P (1999) “Non stationary dynamic factor analysis”, Universidad
Carlos III, Internal Document.
12