Jan Zawadzki

Maria Szmuksta-Zawadzka

English

Research Papers in Economics

© Copyright by The Nicolaus Copernicus University Scientific Publishing House
DYNAMIC ECONOMETRIC MODELS 
Vol. 6 – Nicolaus Copernicus University – Toruń – 2004 
  
 
 
 
 
Maria Szmuksta-Zawadzka, Jan Zawadzki 
Technical University of Szczecin, Agricultural Academy of Szczecin 
 
 
 
On Hierarchic Models for Decade Data  
with Seasonal Fluctuations   
 
 
  Hierarchic predictive modelling and forecasting economical variables with 
12-month cycle seasonal fluctuations are discussed in Zawadzki (2003) and 
Szmuksta-Zawadza, Zawadzki (2002). As it has been proven, the essential ad-
vantage of these models is decreasing the number of estimated parameters. It 
allows for forecasting by using short time series limited in the extreme case to 
two cycles. 
  In the present paper hierarchic time-series models will be considered with 
decade fluctuations both for full series and series with unsystematic gaps. 
  A consequence of shortening the period from a month to a decade is an in-
crease in the number of hierarchic regular models. The number of permutations 
and permutations with repeated divisors of cycle fluctuations, which satisfy the 
following conditions: they are not less than 2 and not greater than m/2 and their 
product equals the length of the fluctuation cycle, equals 25. 
  This number is composed of seven two-level models, twelve three-level 
models and six four-level models. The models for periodic fluctuations or rela-
tively constant ones having 36-decade cycle together with the denotations of 
matrices corresponding to the hierarchy levels are given in Table 1.  
  The last column contains the numbers of the estimated parameters defining 
the seasonal fluctuations. The number ranges from 18 for two-factor models 
H218 and H182 to 6 for the four- factor models. We note for comparison that 
35 parameters should be estimated in classical models. Thus in the case of a 36-
decade cycle, advantages related to the increased number of degrees of freedom 
are entirely clear. Noncorrelation of the factors (hierarchy levels) for the con-
sidered model is an important property of the hierarchic models for full data. 
This property allows measuring the participation of variance of various factors 
in the explanation of the seasonal variance. In the case of the series with the © Copyright by The Nicolaus Copernicus University Scientific Publishing House
unsystematic gaps, e.g. when we have at least one observation for each sub-
period this property does not appear. 
For example a part of matrix X433 comprising the period of a year (36 decades 
for model H433) is presented below 
 
 KW1  KW2  KW3 MCKW1 MCKW2 DTD1  DTD2 
 1   0   01010  
 1   0   01001  
 1  0  0 1 0 -1 -1 
 1   0   00110  
 1   0   00101  
 1  0  0 0 1 -1 -1 
 1  0  0 -1 -1 1 0 
 1  0  0 -1 -1 0 1 
 1   0   0- 1- 1- 1- 1  
 0   1   01010  
 0   1   01001  
 0  1  0 1 0 -1 -1 
 0   1   00110  
 0   1   00101  
 0  1  0 0 1 -1 -1 
 0  1  0 -1 -1 1 0 
 0  1  0 -1 -1 0 1 
 0   1   0- 1- 1- 1- 1  
 0   0   11010  
X433 =  0   0   11001  
 0  0  1 1 0 -1 -1 
 0   0   10110  
 0   0   10101  
 0  0  1 0 1 -1 -1 
 0  0  1 -1 -1 1 0 
 0  0  1 -1 -1 0 1 
 0   0   1- 1- 1- 1- 1  
 - 1   - 1   - 11010  
 - 1   - 1   - 11001  
 -1  -1  -1 1 0 -1 -1 
 - 1   - 1   - 10110  
 - 1   - 1   - 10101  
 -1  -1  -1 0 1 -1 -1 
 -1  -1  -1 -1 -1 1 0 
 -1  -1  -1 -1 -1 0 1 
 -1  -1  -1 -1 -1 -1 -1 
 ,   ,   ,,,,,  
 ,   ,   ,,,,,  
 ,   ,   ,,,,,  
 
Source: Authors’ elaboration. 
 
Negative components of the matrix mean that the corresponding parameters are 
equal to zero. © Copyright by The Nicolaus Copernicus University Scientific Publishing House
  The presented theoretical investigations will be illustrated by an empirical 
example concerning  the forming of the decade deposits on the current accounts 
in Bank C. Four full 36-decade cycles constituted the estimating period. Equa-
tions were estimated using full series as well as series with gaps. It was assumed 
in the considered example that unsystematic gaps appear in the three consecu-
tive decades of odd months. Due to the unsystematic type of gaps it was as-
sumed that the data were accessible for February and April in the first year, 
June and August in the second year and October and December in the last year. 
Since the estimating period was a four-year time (144 decades), 54 gaps appear 
in the data. 
  Using the estimated equations, the interpolative (for 54 missing data) and 
extrapolative forecasts (for 9 decades) were developed. Both types of the fore-
casts were empirically verified which were obtained using the models with gaps 
while the extrapolative forecasts were verified only in the case of models with-
out gaps. 
  Synthetic estimations of the stochastic structure parameters which are si-
multaneously the estimations of the predictive properties both for the case of 
models with and without gaps are given in Table 2. To distinguish between 
these types of the models, the models for the full data are denoted by P. 
  We note from the last column that the estimations of determination coeffi-
cients of equations without gaps are very similar. Maximum estimation of the 
parameter is for model P123 (0.9938), while the minimum for model P3322 
(0.9870). The difference is thus 0.68%. 
  Estimations of the standard deviations of the random components are more 
diversified. They are in the range from 462,400 PLN for model P123 to 562,400 
PLN for model P334. Three concentrations of the estimations can be seen. The 
first concentration is formed by models of estimations less than 420,120 PLN 
(eleven models). The second concentration is formed by models of the estima-
tions in the range 450,000-470,000 PLN (four models). Ten models having es-
timations greater than 497,000 PLN fall in the third concentration.  
  Composition of the group, especially extreme groups, is not casual. A spe-
cific feature of the models of the first group is that the last number in the deno-
tation of the model, being the divisor of the cycle length, is three or a multiplic-
ity of three whereas the models having the divisors of two or a multiplicity of 
two belong to the third group. 
  The above-defined principle is related to the occurrence of the three-decade 
cycle in the month resulting from the payments in or out of the accounts. 
  Estimations of the determination coefficients of equations estimated using 
the data with unsystematic gaps are given in the seventh column of Table 2. 
  The estimations are little diversified, similarly to the case of the full data. 
This is why it is important to measure the estimation of the predictive capabili-
ties using the estimations of the standard deviations of the random components. 
They are generally a few or sometimes even by more then ten percent more than 
 © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Table 2. Estimations of selected parameters of stochastic structure of hierarchic equa-
tions for decade data 
 
Models for full data  Models for nonsystematic gaps  No. 
Model R
2 Se DW  Model  R
2 Se DW 
1 P218  0.9934  418.95 1.41 H218 0.9936  472.22  1.58 
2 P312  0.9916  462.40 1.47 H312 0.9913  511.3  1.44 
3 P49  0.9932  414.00 1.41 H49 0.9935  452.23  1.64 
4 P66  0.9929  420.12 1.40 H66 0.9933  457.86  1.62 
5 P94  0.9892  522.36 2.04 H94 0.9914  522.15  1.96 
6 P123  0.9938  397.15 1.51 H123 0.9936  455.85  1.67 
7 P182  0.9907  497.53 1.87 H182 0.9924  524.49  1.87 
8 P229  0.9931  414.21 1.40 H229 0.9933  547.17  1.62 
9 P236  0.9929  418.12 1.39 H236 0.9931  459.07  1.62 
10 P263  0.9933 406.84  1.43  H263  0.9931  456.67  1.62 
11 P292  0.9900 500.30  1.80  H292  0.9911  525.05  1.9 
12 P326  0.9913 461.84  1.15  H326  0.9911  517.06  1.51 
13 P334  0.9870 562.40  1.74  H334  0.9893  565.72  1.79 
14 P343  0.9916 454.85  1.17  H343  0.9917  497.42  1.44 
15 P362  0.9894 535.35  1.55  H362  0.9898  555.35  1.74 
16 P433  0.9931 410.23  1.42  H433  0.9933  447.24  1.63 
17 P623  0.9929 417.24  1.40  H623  0.9932  452.93  1.61 
18 P632  0.9897 503.32  1.74  H632  0.9913  515.21  1.91 
19 P922  0.9892 520.48  2.04  H922  0.9914  518.93  1.97 
20 P2233  0.9931 410.45  1.41  H2233  0.9931  452.07  1.61 
21 P2323  0.9929 415.30  1.39  H2323  0.9930  454.24  1.61 
22 P2332  0.9897 500.53  1.72  H2332  0.9911  514.39  1.9 
23 P3223  0.9913 458.31  1.51  H3223  0.9917  495.58  1.43 
24 P3232  0.9891 536.76  1.15  H3232  0.9897  551.77  1.72 
25 P3322  0.9870 560.24  1.74  H3322  0.9893  562.39  1.78 
 
Source: Authors’ elaboration. 
  
for the models without gaps. The least value of the standard deviation was ob-
tained for model H433 (447,240 PLN) and the highest for model H334 (565,720 
PLN). Division of the models into three concentrations, according to the stan-
dard deviations, is valid also for this model class. 
      The courses of the investigated variable (Y) and fitted values on the base of 
hierarchical model H433 are presented  in Fig.1. 
 
 © Copyright by The Nicolaus Copernicus University Scientific Publishing House
       Source: Authors’ elaboration. 
H433
Y
Fig. 1. Empirical and theoretical series
4000
8000
12000
16000
20000
0 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144
  
 
       As it has been earlier remarked using the estimated equations, inter- and 
extrapolative forecasts were calculated and the ex post analysis of their accu-
racy was carried out (for the models without gaps only the extrapolative fore-
casts were analysed). Mean relative errors of the inter- and extrapolative fore-
casts are given in Table 3. It follows from the third column that the estimations 
of mean relative errors of the interpolative forecasts were in the range from 4.61 
(H123) to 7.50% (H922). Estimations less than five percent were obtained 
jointly for ten models, from five to six percent for four models, and more than 
six percent for ten models. The models of the third group belonged to the last 
case. Model H312 was the only model of this group with error estimation less 
than five percent. 
  Very interesting results were obtained for the extrapolation forecasts. Error 
estimations of the forecasts with gaps are generally less than the forecast errors 
obtained using models without gaps. The average difference was 0.63%. The 
least estimation among the models with gaps was for model H433 (2.57%) and 
among models without gaps P326 (2.73%). The greatest estimations were for 
H94 (3.35%) and P182 (4.08%), respectively. 
For all the models with unsystematic gaps belonging to the first and second 
group the error did not exceed 3%. The only model of the third group with such 
an estimation was model H312. The limit for the models without gaps was   
a little higher, equal to 3.32%. The only model of the third group above the 
limit was model P312. Little higher effectiveness of the forecasts obtained using 
models with gaps can be explained by disturbance of the examined process in 
the periods where gaps appear. 
 
 © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Table 3. Estimations of mean relative errors of inter- and extrapolative forecasts (%) 
 
  Models with gaps  Models without gaps 
Model Interpolative 
forecast 
Extrapolative 
forecast 
Model Extrapolative 
forecast 
H218  5.95 2.98 P218 3.19 
H312 4.88  2.94  P312  3.03 
H49 5.54  2.69  P49  3.16 
H66 4.99  2.93  P66  3.16 
H94 7.48  3.35  P94  3.92 
H123  4.61 2.77 P123 3.13 
H182  6.59 2.83 P182 4.08 
H229  4.91 2.72 P229 3.06 
H236  4.73 2.83 P236 2.73 
H263  4.77 2.63 P263 3.05 
H292  6.27 3.32 P292 3.93 
H326  5.99  2.80 P326 2.73 
H334  7.42 3.25 P334 3.63 
H343  4.63  2.74 P343 2.82 
H362  6.40 3.18 P362 3.55 
H433  5.51 2.57 P433 3.08 
H623  4.94 2.89 P623 3.12 
H632  6.57 3.33 P632 3.95 
H922  7.50 3.32 P922 3.91 
H2233 4.81  2.59  P2233  2.98 
H2323 4.63  2.89  P2323  3.01 
H2332 6.40  3.28  P2332  3.34 
H3223  4.88  2.66 P3223 3.32 
H3232 6.57  3.10  P3232  3.4 
H3322 7.42  3.22  P3322  3.61 
 
Source: Authors’ elaboration. 
 
  
  It follows from the presented example that the hierarchic models with dec-
ade fluctuations can be successfully used for predictive modelling and forecast-
ing of economical processes with decade fluctuations. They require estimation 
of the less number of parameters than the classical models. This conclusion is 
related to both full series and series with unsystematic gaps.  
 
 © Copyright by The Nicolaus Copernicus University Scientific Publishing House
References 
 
Zawadzki, J. (ed.) (2003), Application of hierarchic models of time series for forcasting 
of economical variables with seasonal fluctuations (in Polish), Agriculture Uni-
versity of Szczecin.  
Szmuksta-Zawadzka, M., Zawadzki, J. (2002), Hierarchic models of time series with 
seasonal fluctuations. Construction. Estimation. Forecasting, in: Zeliaś, A., Spa-
tial and time modelling and forecasting of economical processes (in Polish), Eco-
nomic Academy of Cracow.  © Copyright by The Nicolaus Copernicus University Scientific Publishing House
 
Table 1. Specification of regular hierarchic model for decade data 
 
Model  Type of variation (first 
factor) 
Matrix  Type of variation 
(second factor) 
Matrix  Type of variation 
(third factor) 
Matrix  Type of variation 
(fourth factor) 
Matrix Number  of  esti-
mated parameters 
1  2  3  4  5 6 7 8 9  10 
H218  half-year in a year  PR  decade in a half-year  DPR      18 
H312  four-month period in  
a year 
CZM  decade in a four- 
month period 
DCZM     13 
H49  quarter in a year  KW  decade in a quarter  DKW      11 
H66 two-month   
period in a year 
DWM  decade in a two- 
month period 
DDWM     10 
H182  two-decade period in  
a year 
BIR  decade in a two- 
decade period 
DD     18 
H123  month in a year  MCR  decade in a month  DTD      13 
H94 four-decade   
period in a year 
CZD  decade in a four  
decade period 
DCZD     11 
H263  half-year in a year  PR  month in a half-year  MCPR  decade in a month  DTD    8 
H236  half-year in a year  PR  two month period in 
a half-year 
DWMPR  decade in a two-
month period 
DDWM    8 
H362  four-month period in  
a year 
CZM  two-decade period in 
a four-month period 
BICZM  decade in a two-
decade period 
DD    8 
H326  four-month period in  
a year 
CZM  two-month period in 
a four-month period 
DMCZM  decade in a two-
month period 
DDWM    8 
H433  quarter in a year  KW  month in a quarter  MCKW  decade in a month  DTD    7 
H343  four-month period in  
a year 
CZM  month in a four- 
month period 
MCCZM  decade in a month  DTD    7 
H334  four-month period in  
a year 
CZM  four-decade period in 
a four-month period 
CZDCZM  decade in a four-
decade period 
DCZD    7 
H623  two-month period in  
a year 
DWM  month in a two- 
month period 
MCDM  decade in a month  DTD   
 
8 © Copyright by The Nicolaus Copernicus University Scientific Publishing House
 
Table 1 continued 
   
          
1 2  3 4 5  6  7  8 9  10 
H632 two-month   
period in a year 
DWM  two-decade period in 
two-month period 
BISZ  decade in a two-
decade period 
DD 
 
  8 
H292  half-year in  
a year 
PR  two-decade period in 
a half-year 
BIPR  decade in a two-
decade period 
DD     
H229  half-year in  
a year 
PR  quarter in a half-year  KPR  decade in  
a quarter 
DKW    10 
H922    CZD  two-decade period in 
a four decade period 
BICZD  decade in a two-
decade period 
DD    10 
H2233  half-year in a year  PR  quarter in a half-year  KPR  month in a quarter  MCKW  decade in a month  DTD  6 
H2332  half-year in a year  PR  two-month  
period in a half-year 
DWMPR two-decade  period 
in a two-month 
 period 
BISZ  decade in a two-
decade period 
DD  6 
H2323  half-year in a year  PR  two-month  
period in a half-year 
DWMPR  month in a two- 
month period 
MCDW  decade in a month  DTD  6 
H3322  four-month period in  
a year 
CZM  four-decade period in 
a four-month period 
CZDCZM two-decade  period 
in a four-month 
period 
BICZD  decade in a two-
decade period 
DD  6 
H3232  four-month period in  
a year 
CZM  two-month period in 
a four-month period 
DMCZM two-decade  period 
in a two-month 
 period 
BISZ  decade in a two-
decade period 
DD  6 
H3223  four-month period in  
a year 
CZM  two-month period in 
a four-month period 
DMCZM  month in a two- 
month period 
MCDM  decade in a month  DTD  6 
 
Source: Authors’ elaboration. 
 

Application of hierarchic models of time series for forcasting of economical variables with seasonal fluctuations

On Herarchic Models for Decade Data with Seasonal Fluctuations

http://www.dem.umk.pl/dem/archiwa/v6/06_Zawadzak.pdf

On Herarchic Models for Decade Data with Seasonal Fluctuations

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Research Papers in Economics