In this paper we propose the prognosis of the unemployment rate in the European Union through the Box-Jenkins method and the TRAMO/SEATS method as well as the detection of the method which proves to provide the best results. The monthly unemployment rate in the European Union is affected by seasonal variations of deterministic and stochastic nature. The prognosis through the Box-Jenkins nature supposes the separate consideration of seasonal variations, according to their specific nature. The stochastic seasonal variations are modelled and prognosticated simultaneously with the other components of the time series, based on the generating stochastic process. The prognosis of the monthly unemployment rate in the European Union through the TRAMO/SEATS methods is done by aggregating the individual prognoses of the components of the time series, obtained according to the stochastic processes models that generate them.seasonal variations, stochastic process, moving average, prognosis, performance indicators of the prognosis

Balan Christiana Brigitte

Gagea Mariana,

English

Research Papers in Economics

848 
 
PROGNOSIS OF MONTHLY UNEMPLOYMENT RATE IN THE 
EUROPEAN UNION THROUGH METHODS BASED ON 
ECONOMETRIC MODELS  
Gagea Mariana  
Alexandru Ioan Cuza University of Iasi, Faculty of Economics and Business Administration, Bd-ul 
Carol I, no. 22, Building B, B315, Ia￿i, post code 700505, mariana.gagea@uaic.ro, Tel. +40 232 201 584 
Balan Christiana Brigitte 
Alexandru Ioan Cuza University of Iasi, Faculty of Economics and Business Administration, Bd-ul 
Carol I, no. 22, Building B, B315, Ia￿i, post code 700505, itte7@yahoo.com , Tel. +40 232 201 584 
 
In this paper we propose the prognosis of the unemployment rate in the European Union through the Box-
Jenkins method and the TRAMO/SEATS method as well as the detection of the method which proves to 
provide the best results. The monthly unemployment rate in the European Union is affected by seasonal 
variations of deterministic and stochastic nature. The prognosis through the Box-Jenkins nature supposes 
the  separate  consideration  of  seasonal  variations,  according  to  their  specific  nature.  The  stochastic 
seasonal variations are modelled and prognosticated simultaneously with the other components of the time 
series, based on the generating stochastic process. The prognosis of the monthly unemployment rate in the 
European Union through the TRAMO/SEATS methods is done by aggregating the individual prognoses of 
the components of the time series, obtained according to the stochastic processes models that generate 
them.  
Key words: seasonal variations, stochastic process, moving average, prognosis, performance indicators of 
the prognosis  
1. Introduction 
There exists a great variety of analysis and prognosis methods as far as seasonal time series are concerned. 
These methods can be grouped according to various criteria, such as: their funding principle; the way 
seasonal variations are treated, either individually or globally, etc. The Box-Jenkins and TRAMO/SEATS 
methods are both based on ARIMA stochastic processes and models. The two methods are different as far 
as the approach of the seasonal component is concerned. The Box-Jenkins method consists in identifying, 
estimating and validating the stochastic process model that generates the time series as a whole, as well as 
modelling and the simultaneous prognosis of the components of the series, according to this model. 
As opposed to the Box-Jenkins method, which it actually incorporates, TRAMO/SEATS is a method of 
time  series  decomposition,  involving  the  individual  treatment  of  components.  The  application  of  this 
method supposes finding the stochastic process that generates the analysed series and decomposing it in 
stochastic  processes  that  generate  the  time  series  components.  The  prognosis  of  the  values  of  this 
phenomenon is achieved by aggregating the individual prognoses of the components of the series. 
The  monthly  unemployment  rate  in  the  European  Union,  considered  as  an  average  level  of  the 
phenomenon in the27 member states, is analysed for the period  between January 2000 – November 2007. 
The data was extracted from the Eurostat portal. Data processing is done with SPSS, Demetra and JMuTi 
programmes. 
The phenomenon under study has an accentuated seasonal character, manifested regularly in time. The 
presence of seasonal variations can be verified through different methods, such as: the linear chronograph, 
the Fisher test, the autocorrelation function. 
We wish to make a comparison between the prognosis of the monthly unemployment rate in the European 
Union according to the Box-Jenkins and the TRAMO/SEATS methods, in order to facilitate the choice of 
the method providing the best results. The comparison will take into account the advantages, the limits and 
the results thus obtained. 849 
2. Method description 
a) The Box-Jenkins method 
The Box-Jenkins method, proposed in 1970 by the authors who gave its name, is a short-term prognosis 
method. The Box-Jenkins method algorithm comprises the following steps: identification of the stochastic 
process  model  that  generates  the  analysed  time  series,  estimation  of  the  model,  model  testing  and 
prognosis.  The  first  step  of  the  identification  stage  of  the  process  generating  the  time  series  is  the 
stationarity  analysis.  A  temporal  series  is  considered  stationary  if  the  generating  stochastic  process  is 
stationary. The identification of the stochastic process which generates the time series under analysis is 
based on the autocorrelation and partial autocorrelation functions (see 1, pages 47-82). 
The basic stochastic models used in the econometric analysis of time series describe the series behaviour 
through linear combinations of white noises, linear combinations of past values or combinations of past 
values  and  white  noises:  moving  average  models  (MA),  autoregressive  models  (AR)  and  mixed 
autoregressive and moving average models (ARMA).  
The estimation of the parameters of the stochastic process model is made using methods such as: the 
smallest least squares method, which can be easily applied in order to estimate autoregressive models; the 
method of maximum likelihood; the Yule-Walker algorithm. The validation of he estimated model requires 
the verification of the parameters meaning as well as error analysis (see 2, pp. 717-728; 3, pp. 225-247; 5, 
pp. 348-351). 
b) The TRAMO/SEATS method 
The TRAMO/SEATS method was developed by Agustin Maravall and Victor Gomez (1996), its starting 
point being the method proposed by Burman (1980). This method consists of estimating through OLS of 
the ‘hidden’ components in the time series (“signal extraction”), on the basis of ARIMA stochastic process 
which generates the series. Three main components of the time series are taken into account: the trend-
cycle component, the seasonal component and the irregular component. 
The TRAMO method (Time series Regression with ARIMA noise, Missing values and Outliers) consists 
of the estimation and prognosis of the regression model with ARIMA errors, called reg-ARIMA, in the 
possible presence of missing values, outliers of various types and other deterministic special effects. The 
application of the automatic procedure of the TRAMO method consists of the following steps [Demetra 2.0 
User  Manual,  pp.  18-58]:  identification  of  the  aggregation  scheme  of  the  time  series  components; 
identification  and  estimation  of  the  reg-ARIMA  model  on  the  grounds  of  the  method  of  maximum 
likelihood; testing and correction (wherever necessary) of the calendar effects: the working days effect, the 
leap year effect, the Easter effect, the legal holidays effect; testing and correction of the outlier points of 
the series, that can be: extreme values, bearing the AO symbols, transitory shocks, bearing the TC symbols, 
or level changes, bearing the LS symbols; optimum interpolation of missing values of the time series; 
extension of the time series for the following two years through prognosis, on the basis of the estimated 
ARIMA  model;  determination  of  correction  factors  for  the  total,  calendar  and  outlier  effects  and 
calculation of the pre-adjusted or ‘linearized’ temporal series values. (see 4, pp. 18-58).  
The SEATS method (Signal Extraction in ARIMA Time Series) is a method of decomposition of temporal 
series in components, according to the ARIMA modes. The stochastic components estimated through this 
method are added deterministic effects (the calendar and outlier effects) eliminated through the TRAMO 
method, the final components being thus obtained: the corrected series of seasonal variations, the tendency, 
the  seasonal  component,  the  irregular  component.  The  SEATS  method  offers  the  prognosis  of  these 
components and of the uncorrected values of the seasonal variations for the following two years. 
3. Presentation of results 
a) The prognosis of the monthly unemployment rate in the European Union through the Box-
Jenkins method 
The components of the unemployment rate in the European Union as well as their aggregation scheme are 
identified  using  the  Fisher  test  and  the  band  test  (see  3,  pp.  11-14;  21-22). The  analysed  time  series 850 
presents significant trend and seasonal variations, consequently the series is non-stationary. The adequate 
aggregation scheme of the components is thus the additive one. 
In order to identify the adequate stationarity techniques the Franses test is applied for testing the seasonal 
unit root, with the JMulTi programme (see 3, pp. 172-175). The model that includes a constant term, 
seasonal  binary  variables  and  trend  is  chosen.  The  estimated  and  tested  regression  model  presents 
statistically  significant  coefficients  of  the  seasonal  binary  variables  and  has  no  seasonal  roots  of  unit 
module.  Consequently,  the  unemployment  rate  in  the  European  Union  presents  deterministic  seasonal 
variations which are eliminated through the moving averages method (see 5, pp. 444-447). 
The simple autocorrelation function (figure 1) and the partial autocorrelation function (figure 2) for the 
unemployment rate in the European Union are then analysed. Their deterministic seasonal variations have 
been extracted. 
 
16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Lag Number
1.0
0.5
0.0
-0.5
-1.0
A
C
F
Lower Confidence
Limit
Upper Confidence Limit
Coefficient
 
Fig. 1. The autocorrelation function for the 
unemployment rate in the European Union 
out of which the deterministic seasonal 
variations have been extracted 
16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Lag Number
1.0
0.5
0.0
-0.5
-1.0
P
a
r
t
i
a
l
 
A
C
F
Lower Confidence
Limit
Upper Confidence Limit
Coefficient
 
Fig. 2. The partial autocorrelation function 
for the unemployment rate in the European 
Union out of which the deterministic 
seasonal variations have been extracted 
 
The autocorrelation functions of the analysed series indicate an autoregressive stochastic process, bearing 
the symbol AR(1), defined by the relation: 
t t t y y ε φ + = −1 1 . 
The  order  1  autoregressive  process  equation  with  constant  terms  estimated  through  the  method  of 
maximum  likelihood,  with  the  SPSS  programme.  The  estimated  equation  is: 
028 8 028 8 996 0 1 1 1 , ) , y ~ ( , ) y ~ ( y t t
*
t + − = + − = − − µ µ φ , with  t y ~  the value of the unemployment rate in 
the European Union in moment t, out of which the deterministic seasonal variations have been extracted. 
The parameters  1 φ  and  µ  of the model are statistically significant, the condition of rejection of the nil 
hypothesis being fulfilled:  ). , ( ) , Sig ( 05 0 00 0 = < = α  
According to the autocorrelation function and the Jarque-Bera test, the modelling errors define a stochastic 
process of the ‘white noise’ type. The autoregressive model AR(1) is validated. Based on this model the 
unemployment rate in the European Union is estimated for the period December 2007 – November 2008, 
according to the relation:  028 8 028 8 996 0 1 , ) , y ( , y T
*
T + − = + . Finally, to the values obtained one adds 
the deterministic seasonal variations, presented as seasonal coefficients:  j
*
T
) P (
T S y y + = + + 1 1 . 
The measurement of the performance of the unemployment rate in the European Union through the Box-
Jenkins method is made by means of the absolute prognosis errors average. Prognosis errors are calculated 851 
as  the  difference  between  the  observed  value  and  the  estimated  value,  according  to  the  relation: 
) P (
t t t y y e − = . The average of the absolute prognosis errors is:  079 0 1 .
T
e
) e ( M
T
t
t
=
￿
= = . 
b)  The  prognosis  of  the  unemployment  rate  in  the  European  Union  through  the 
TRAMO/SEATS method. 
The TRAMO/SEATS method is applied with the Demetra programme, automatic procedure. The output of 
the decomposition of the unemployment rate in the European Union through the T/S method is structured 
on three points and presents the results below.  
Information regarding the model used: 1) The terms of the series have been logarithmically transformed 
and the log-additive aggregation model was used; 2) Three outliers with significant statistic effect were 
identified, being presented in the inversed order of their importance : May 2006 (AO), January 2007 (LS), 
November  2005  (LS);  3)  The  model  of  the  stochastic  process  generating  the  time  series  is 
ARIMA(1,1,0)(0,1,0)12,  with  all  the  1 φ   parameters  corresponding  to  the  statistically  significant  non-
seasonal autoregressive part, according to the t-Student test. The estimation of the model parameters was 
made by means of the maximum likelihood method; 4) The decomposition in components was made based 
on the exact ARIMA model, estimated in the pre-adjustment stage with the TRAMO method. 
The equation of the reg-ARIMA model can be symbolically written in the following manner:  = t y 3 
outliers +  t x . The equation of the ARIMA(1,1,0)(0,1,0)12 mode of the regression errors  t x  is: 
t t t t x ) L , ( x ) L ( ε ∆∆ ε ∆∆ φ = + ⇔ = − 12 12 1 127 0 1 1 ,  
where:  -∆   represents  the  difference  operation,  resulted  from  the  relation  1 − − = t t t y y y ∆ ; 
12 12 − − = t t t y y y ∆  
- L - the delay operator, resulted from the relation  1 − = t t y Ly . 
Information regarding the validation criteria of the modelling (table 1): 1) The estimated value of the 
Ljung-Box statistic of order k=24 is Q24=15.55. The trust interval for Q24, guaranteed with a probability of 
95%, is [0; 35.2]. The empiric value is covered by the trust interval, so the estimated value of the Q24 
statistics is not statistically significant, the errors are not auto-correlated to the k=24 disparity; 2) The 
estimated value of the Jarque-Berra statistics is 65 . 0 = JB . The trust interval for JB, guaranteed with a 
probability of 95%, is [0; 5.99]. The calculated value of the JB statistics is included in the accepted interval 
of the nil hypothesis, the errors are normally distributed; 3) The identified outliers represent 3.61% of the 
total number of observations of the monthly unemployment rate in the European Union. The percentage is 
smaller than the maximum allowed limit of 5% (see 4, pp. 27-28). 
 
Table 1. Synthesis of statistics used for the validation of decomposition by means of the TRAMO 
method 
Programme diagnosis. All tests being passed, the modelling of the analysed series through this variant of 
the TRAMO/SEATS method is accepted and there follows the next step: prognosis for the components of 
the series for the following two years. 852 
We wish to make a comparison between the prognosis of the monthly unemployment rate in the European 
Union  according  to  the  Box-Jenkins  and  the  TRAMO/SEATS  methods.  Due  to  the  fact  that  the  two 
methods  are  based  on  different  principles  and  ways  of  treating  the  seasonal  variations,  a  comparison 
between the results obtained with these two methods is quite difficult to achieve. The simplest solution 
seems to be a graphic comparison (figure 3), yet this method only provides intuitive information. The 
simple graphical comparison between the two prognoses does not say exactly which of the two methods 
has led to better results. 
Prognosis methods based on re-seasonalization do not allow the determination of the theoretical values of 
the  series,  necessary  for  the  comparison  with  the  empirical  values.  These  situations  prevent  the 
determination of prognosis errors, on which the performance indicators of the prognosis are actually built. 
The  prognosis  absolute  error  average,  which  equals  0.079,  was  calculated  for  the  prognosis  of  the 
unemployment  rate  in  the  European  Union  through  the  Box-Jenkins  method.  The  performance 
measurement of the prognosis obtained through the TRAMO/SEATS method, by means of this specific 
indicator, is not allowed.  
One possible solution in order to compare the results of the two prognosis methods is the following: the 
analysed time series is divided into two sub-series, the work series and the control series. The work series 
can be considered the unemployment rate in the European Union in the period January 2000 – November 
2006, while the control series is represented by the unemployment rate in the period December 2006 – 
November 2007. Modelling and prognosis are based on the data comprised in the work series, while the 
control series is used for the calculation of prognosis errors and the performance indicators, according to 
which the methods are compared and the best method is chosen. Following this procedure, the following 
results have been obtained: the average of the absolute prognosis errors for the Box-Jenkins method is 
0.669, while for the TRAMO/SEATS method the average is 0.295. The method that provided the best 
results in this particular situation is thus the TRAMO/SEATS method. 
 
MAY 2008
JUL 2007
SEP 2006
NOV 2005
JAN 2005
MAR 2004
MAY 2003
JUL 2002
SEP 2001
NOV 2000
JAN 2000
Date
10
9
8
7
6
5
Prognoza ratei
somajului din UE
prin metoda
TRAMO/SEATS
Prognoza ratei
somajului din UE
prin metoda Box-
Jenkins
Rata somajului
din UE
 
Fig. 3. Unemployment rate in the European Union initiated and prognosticated through the Box-
Jenkins and TRAMO/SEATS methods 
4. Conclusions 
The Box-Jenkins and TRAMO/SEATS methods have the advantage of using analysis instruments that 
ensure the basis for the statistical interface. The Box-Jenkins method is limited by the experience and 
subjectivity of the user in the identification stage of the stochastic process model. Such an example could 
be provided the modelling of the unemployment rate in the European Union based on the initial series in 
comparison  with  the  work  series.  In  the  latter  case  the  prognosis  errors  that  were  obtained  were 
substantially bigger then in the case of the initial series. As far as limiting user subjectivity is concerned, 
the  TRAMO/SEATS  method  provides  the  advantage  of  the  automatic  procedure  of  the  Demetra 
programme and the automatic guidance of the user when it comes to finding a solution to an identified 
problem. 
The direct numerical comparison of the results of prognosis methods based on different approaches of 
seasonal variations is possible neither individually nor globally. We are reserved in accepting the solution 
of prognosis methods’ comparison by splitting the initial series into working series and control series, even 853 
if this is one of the few solutions that exist. Therefore, the application of this method raises two important 
issues: data representativity for the two sub-series, and subjectivity in applying the methods on different 
time series: the initial and the working series. 
Bibliography 
1.  Alonso, J. H., Econometria de series temporales, Editorial Universitas, S.A., Madrid, 2000 
2.  Andrei, T., Statistic￿ ￿i Econometrie, Ed. Economic￿, Bucure￿ti, 2003 
3.  Bourbonnais, R., Terraza, M., Analyse des series temporelles. Aplication à l’economie et à la 
gestion, DUNOD, Paris, 2004 
4.  Demetra 2.0x User Manual, http://circa.europa.eu 
5.  Jaba, E., Statistica, 3
rd edition, Ed. Economic￿, Bucure￿ti, 2002 
 
 
 

PROGNOSIS OF MONTHLY UNEMPLOYMENT RATE IN THE EUROPEAN UNION THROUGH METHODS BASED ON ECONOMETRIC MODELS

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PROGNOSIS OF MONTHLY UNEMPLOYMENT RATE IN THE EUROPEAN UNION THROUGH METHODS BASED ON ECONOMETRIC MODELS

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