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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, qiqiiii 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

t
itxNT

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

TN
kkVkIC

)
(7)

Jarosław Krajewski

142

,ln))(ln()( 22 ⎟⎟
⎠

⎞
⎜⎜
⎝

⎛
⎟
⎠
⎞

⎜
⎝
⎛ +

+= NTCNT
TN

kkVkIC
)

(8)

,
ln

))(ˆln()( 2
2

2 ⎟
⎟
⎠

⎞
⎜
⎜
⎝

⎛
+=

C
C

kkVkIC (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

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.