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DYNAMIC ECONOMETRIC MODELS 
Vol. 9 – Nicolaus Copernicus University – Toruń – 2009 

Jarosław Krajewski 
Nicolaus Copernicus University in Toruń 

Estimating and Forecasting GDP in Poland  
with Dynamic Factor Model† 

A b s t r a c t. Presented paper concerns the dynamic factor models theory and application in the 
econometric analysis of GDP in Poland. DFMs are used for construction of the economic indica-
tors and in forecasting, in analyses of the monetary policy and international business cycles. In the 
article we compare the forecast accuracy of DFMs with the forecast accuracy of 2 competitive 
models: AR model and symptomatic model. We have used 41 quarterly time series from the 
Polish economy. The results are encouraging. The DFM outperforms other models. The best fitted 
to empirical data was model with 3 factors. 

K e y w o r d s: Dynamic factor models, principal components analysis, GDP. 

1. Introduction 
 In recent years, dynamic factor models have become popular in empirical 
macroeconomics. They are believed to have been pioneered by Geweke (1977) 
and Sims & Sargent (1977) who applied this type of models to the analysis of 
small sets of variables. DFMs have a very wide field of applications. Such mod-
els are widely used for forecasting, constructing leading indicators of business 
climate, monetary policy analysis or the analysis of international business cy-
cles.  
 The purpose of the article is to estimate a dynamic factor model of GDP in 
Poland in 1997 – 2008.  
 The second part of the article presents a concept of a dynamic factor model. 
In the third part the approach to estimating model parameters as well as com-
mon factors are discussed. The methods of specifying the number of factors in 

                                                 
† Scientific work is financing by European Social Fund and national budget of Poland in The 

Integrated Regional Operational Programme framework, Measure 2.6 "Regional Innovation Strat-
egies and transfer of knowledge" Kujawsko-Pomorskie voievodship own project "Candidate for 
doctor's degree grants 2008/2009 – IROP". 



Jarosław Krajewski 

 
140 

the model are also presented. The data used in the study and the empirical re-
sults have been described in the fourth part. Final part summarises the whole 
study. 

2. Dynamic Factor Model  
 The concept of factor models bases on the assumption that the behavior of 
most macroeconomic variables may be well described using a small number of 
unobserved common factors. These factors are often interpreted as the driving 
forces in the economy. The particular variables may be then expressed as linear 
combination of up-to-twenty common factors which usually make it possible to 
explain a major part of variability of those variables (Kotłowski, 2008).  
    Let ty  stand for a variable and let tX express the vector of N variables con-
taining information that can be useful in modeling and forecasting the future 
values of ty . In the dynamic factor model we assume that all variables itx  con-
tained in vector tX  may be expressed as a linear combination of current and 
lagged unobserved factors itf .  

,,...,1for ,)( NieLx itiit =+= tfλ  (1) 

where ]',...,,[ 21 trtt fff=tf stands for vector r of unobserved common factors at 

moment t, q
iqiiii LLL λλλλλ ++++= ...)( 2

210  represent a lag polynomials and 

ite  express an idiosyncratic errors for variable itx  (see Stock, Watson, 1998). 

 In turn, ty  may be noted as the function of current and lagged common 

factors contained in vector tf  and the past values of variable ty , with the fol-
lowing formula 

.)()( tttt eyLLy ++= γβ f  (2) 

 The model described with equations (1) and (2) is a dynamic factor model. 

3. Model Estimation and Selection of the Factor’s Number 
 One of the most widely used methods of parameters and factors estimation 
in a factor models is the method of principal components. Let us emphasise that 
both: the factor matrix and the coefficient matrix are unknown. Model (1) is 
thus equivalent to the model in the matrix form of  

,'1 eΛFHHX += −  (3) 

where matrix H  is any non-singular matrix of dimension rr × . It is necessary 
to carry out the appropriate normalisation of matrix H . Stock & Watson (1998) 



Estimating and Forecasting GDP in Poland … 

 
141

suggest that for this purpose condition in the form of rIΛΛ =)/( ' N  may be 
imposed on the parameters of model which would render matrix H orthonormal.  
 The estimation of matricies F and Λ using the method of principal compo-
nent consist in finding such estimates of matrices F̂  and Λ̂  that would mini-
mise the residual sum of squares in equation (3) as expressed with the following 
formula 

.)(1)(
1 1

2'∑∑
= =

−=
N

i

T

t
itx

NT
V ti FΛΛF,  (4) 

It is necessary, in the first step, to perform a minimisation of function (4) in 
respect to factor matrix F with the assumption that matrix Λ is known and fixed. 
Then we obtain estimate F̂ , as function Λ, which is  subsequently substituted in 
equation (4) for the true value of F. In the second step, we minimise function 
(4) in respect to matrix Λ with a normalisation condition ,)/( '

rIΛΛ =N  thus 
directly obtaining estimate .Λ̂  It should be emphasised that it is equivalent to 
maximisation of expression ].)([ '' ΛXXΛtr  

 Matrix Λ̂  is a matrix whose subsequent columns are eigenvectors of matrix  
XX ' multiplied by N  corresponding to highest eigenvalues of the same ma-

trix. In turn, the estimate of matrix F is expressed by the formula 

./)ˆ(ˆ NΛXF =  (5) 

Stock & Watson (1998) emphasise that if the number of variables is higher than 
the number of observations, i. e. N > T, then from the computational point of 
view it is easier to apply a procedure which determine estimate F~  by minimiz-
ing concentrated function (4) in respect to matrix F with the condition 

./'
rIFF =T  Matrix F~  will then contain eigenvectors of matrix XX '  corres-

ponding to r highest eigenvalues of this matrix and multiplied by T . In turn, 
the estimate of matrix Λ~  will assume the following form 

./)~(~ '' TXFΛ =  (6) 

 In practice, the number of factors necessary to represent the correlation 
among the variables is usually unknown. To determine the number of factors 
empirically a number of criteria were suggested. Bai and Ng (2002) have sug-
gested information criteria to be used to estimate the number of factors. 

,ln))(ln()(1 ⎟⎟
⎠

⎞
⎜⎜
⎝

⎛
⎟
⎠
⎞

⎜
⎝
⎛

+
⎟
⎠
⎞

⎜
⎝
⎛ +

+=
TN

NT
NT

TNkkVkIC
)

 (7) 



Jarosław Krajewski 

 
142 

,ln))(ln()( 2
2 ⎟⎟

⎠

⎞
⎜⎜
⎝

⎛
⎟
⎠
⎞

⎜
⎝
⎛ +

+= NTC
NT

TNkkVkIC
)

 (8) 

,ln))(ˆln()( 2

2

2 ⎟⎟
⎠

⎞
⎜⎜
⎝

⎛
+=

NT

NT

C
CkkVkIC  (9) 

where )(ˆ kV  is residual sum of squares from k – factors model 

and { }.min TNCNT =  

4. Description of Data and Empirical Results 
 The data used in the study are macroeconomic quarterly data describing 
Polish economy and encompass the period from first quarter of 1997 to third 
quarter of 2008 (47 observations). As explained variable we used polish GDP. 
All of data were taken from polish Central Statistical Office1. Before embarking 
on the work on factor model specification, the date had to be appropriately 
modified. In the first step variables were adjusted for the impact of seasonal 
fluctuations. Next, the series were transformed by taking logarithms and/or dif-
ferencing so that the transformed series were stationary (Green, 2003).  In the 
final step, all variables were standardized. In total, 41 time series were consider-
ing, representing of following macroeconomic categories: output & sales, con-
struction, domestic and foreign trade, prices and labour market, budgetary and 
monetary policy.  
 After preliminaries principal components analysis were used to estimate 
factors. Next Bai and Ng informational criteria were calculated to specify num-
ber of factors. Table 1 includes eigenvalues and values of information criteria. 
The first two criteria reach minimum for the number of factors equal to 3, while 
the third criteria assumes its lowest value for 10 factors. Due to the fact that two 
out of three criteria display the same value, we arbitrarily assume that the num-
ber of factors in the model is 3. The first three factors explain almost 82% of 
total variance of GDP. 
 Some econometricians maintain that factors estimated using the principal 
component method do not have an economic interpretation. However, in this 
paper try-out were done. For this reason, it is possible to carry out a regression 
of particular variables against each of the estimated factors and check which 
factor explains the behavior of a given variable to the greatest extent. The 
R-squared values on the regression of particular variables suggest that the first 
factor primarily affects the variability of labour market and foreign trade. The 
second factor determines the prices and incomes. The third factor to the greatest 
extent influences the values of sales.  

                                                 
1 www.stat.gov.pl 



Estimating and Forecasting GDP in Poland … 

 
143

 The further stage of study BIC criterion was used to determine the number 
of GDP and factors delays. The BIC criterion indicated model only with current 
values of the first three factors. Model estimation results are presented in Ta-
ble 2. All coefficients are significant under 5% level. Factor model of GDP in 
Poland estimated in this way has R-squared over 70%. Real values of GDP and 
values based on the model are shown on the figure 1.    

Table 1. Selection of the number of factors in the model 
Number of 

factors Eigenvalues 
Contribution 
to variance 

Cumulative contribution to 
variance IC1 IC2 IC3 

1 76.949 0.721 0.721 -3.694 -3.491 -3.833 
2 6.249 0.059 0.779 -3.584 -3.177 -3.861 
3 4.278 0.040 0.819 -4.555 -3.945 -4.971 
4 2.921 0.027 0.846 -4.490 -3.677 -5.044 
5 2.460 0.023 0.870 -4.467 -3.451 -5.160 
6 1.976 0.019 0.888 -4.335 -3.115 -5.166 
7 1.551 0.015 0.903 -4.192 -2.769 -5.162 
8 1.429 0.013 0.916 -4.109 -2.482 -5.216 
9 1.282 0.012 0.928 -4.246 -2.417 -5.493 
10 1.066 0.010 0.938 -4.126 -2.093 -5.511 

  

 

Figure 1. Actual and fitted values of GDP in Poland in 1997 – 2008 

 Next stage of analysis was to check if lagging some of variables will influ-
ence the final result of estimation. This caused increasing the number of factors 
to four. The resulting model with four factors is presented in Table 3. It is not 

-3

-2

-1

0

1

2

3

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43

Actual Fitted



Jarosław Krajewski 

 
144 

hard to see that in four factors model R-squared is higher and true significance 
level of coefficients is lower.  

Table 2. Dynamic factor model of GDP in Poland in 1997-2008 

Dependent Variable: GDP 
 Coefficient Std. Error T - statistic P - value 

F1 -0.0362 0.0095 -3.7985 0.0005 
F2 0.0683 0.0335 2.0404 0.0478 
F3 0.3707 0.0405 9.1610 0.0000 

R-squared 0.7142 Akaike Criterion 1.7264 
Adjusted R-squared 0.7003 Schwarz Criterion 1.8481 

Durbin-Watson 2.1454 Hannan-Quinn Criterion 1.7715 

Table 3. Dynamic factor model of GDP in Poland in 1997-2008 – after changes in data 
set 

Dependent Variable: GDP 
 Coefficient Std. Error. T - statistic P - value 

F21 0.1722 0.0273 6.3160 0.0000 
F22 0.1686 0.0329 5.1320 0.0000 
F23 -0.2548 0.0361 -7.0669 0.0000 
F24 0.2241 0.0400 5.6090 0.0000 

R-squared 0.7909 Akaike Criterion 1.4858 
Adjusted R-squared 0.7748 Schwarz Criterion 1.6497 

Durbin-Watson 2.2511 Hannan-Quinn Criterion 1.5463 

Table 4. Forecast errors 

 MAPE RMSE R-squared 
AR 90.8002 0.8116 0.1246 

DFM 1.7898 0.0160 0.7003 
DFM2 12.8691 0.1150 0.7748 

Causal Model 4.9267 0.0440 0.8694 

 In the last stage of our study we generated forecasts and forecast errors. The 
forecasting performance of the factor models was evaluated by comparing the 
accuracy of GDP forecasts obtained on the basis of the factor models with the 
accuracy of GDP forecasts derived from other competitive models. Two com-
petitive models were taken into consideration: an univariate autoregressive 
model and causal model with two variables. An univariate autoregressive model 
was adopted as the main benchmark model for evaluating the forecasting per-
formance of the factor models (see Marcellino, Stock, Watson, 2001). The BIC 
criterion indicated model AR(1). The causal model included industrial produc-
tion sales and average employment as explanatory variables. Forecasting mod-
els were estimated on a shorter sample (up to 4 quarter 2007). The forecast was 



Estimating and Forecasting GDP in Poland … 

 
145

produced on the one period ahead. First factor model had the best forecast accu-
racy.  Table 4 presents results.  

5. Summary 
 The principal component analysis reduced the number of explanatory va-
riables from 41 to 3 factors. The resulting dynamic factor model of GDP in 
Poland is satisfactory from the statistical point of view. 
 Changes in data set influenced the final result of model estimation. In this 
study it brought out increasing number of factors and improvement estimation 
performance. Unfortunately, it did not improve forecasting performance. 
 First dynamic factor model of GDP in Poland in 1997 – 2008 gave the best 
forecasting performance in comparison with three competitive models described 
above. 

References  
Bai, J., Ng, S. (2002), Determining the Number of Factors in Approximate Factor Models, Eco-

nometrica, 70, 191–221. 
Geweke, J. (1977), The Dynamic Factor Analysis of Economic Time Series, [in:] Aigner D. J., 

Goldberger A. S. (ed.), Latent Variables in Socio–Economic Models, Amsterdam, North 
Holland. 

Greene, W. H. (2003), Econometric Analysis, Pearson Education, New Jersey. 
Kotłowski, J. (2008), Forecasting Inflation With Dynamic Factor Model – the Case of Poland, 

Working Papers, 2-08, SGH, Warszawa.  
Marcellino, M., Stock, J. H., Watson, M. W. (2001), Macroeconomic Forecasting in the Euro 

Area: Country Specific versus Area–Wide Information, Working Paper, 201, Innocenzo 
Gasparini Institute for Economic Research. 

Sargent, T., Sims, C. (1977), Business Cycle Modelling Without Pretending to Have Too Much 
A-Priori Economic Theory, in Sims C.  (ed.), New Methods in Business Cycle Research, 
Minneapolis, Federal Reserve Bank of Minneapolis. 

Sims, C. A. (1980), Macroeconomics and Reality, Econometrica, 48, 1–48.  
Stock J., Watson, M. W. (1998), Diffusion Indexes, Working Paper, 6702, National Bureau of 

Economic Research. 

Zastosowanie dynamicznego modelu czynnikowego do modelowania 
i prognozowania PKB w Polsce 

Z a r y s  t r e ś c i. Referat traktuje o podstawach konstrukcji dynamicznych modeli czynniko-
wych i ich zastosowaniu empirycznym. DFM stosuje się do prognozowania, konstruowania 
głównych wskaźników koniunktury, analiz polityki monetarnej i badania międzynarodowych 
cykli koniunkturalnych. W referacie oszacowano dynamiczny model czynnikowy PKB w Polsce 
w latach 1997–2008, a także oceniono trafność uzyskanych na jego podstawie prognoz w porów-
naniu do modelu AR i modelu symptomatycznego. Zbiór danych wykorzystanych do badania 
zawiera 41 zmiennych makroekonomicznych. Najlepszym ze statystycznego punktu widzenia 
okazał się model z 3 czynnikami. 

S ł o w a  k l u c z o w e: dynamiczny model czynnikowy, metoda głównych składowych, PKB.



 


