DYNAMIC ECONOMETRIC MODELS


© 2012 Nicolaus Copernicus University Press. All rights reserved.  
http://www.dem.umk.pl/dem 

DYNAMIC ECONOMETRIC MODELS 
Vol. 12 (2012) 73−87 

Submitted October 22, 2011  ISSN 
Accepted March 21, 2012 1234-3862 

Milda Maria Burzała* 

The Probability of Recession in Poland Based on the 
Hamilton Switching Model and the Logit Model 

A b s t r a c t. In the article dating method for the four phases of economic activity is presented. 
Comparison of probabilities of recession occurrence in Poland based on the Hamilton switching 
model and the logit model was conducted in the empirical research. The study shows the conver-
gence of indications based both on the proposed dating method and on the Hamilton model. In the 
presented version the Hamilton model adequately describes the probability of occurrence of two 
decline phases. The logit model allows to obtain satisfactory results for the division on four phas-
es of economic activity. However, in the domain of the Polish economy, more research is needed 
in recognising the symptomatic properties of various macroeconomic indicators. The interest rate 
spread, used successfully in advanced marked economies, continues to alter its characteristics 
under Polish economic conditions and is currently not the best possible indicator forecasting a 
recession.   

K e y w o r d s: switching model, logit model, dating of economic activity phases, probability of 
recession. 

J E L Classification: E32, E37. 

Introduction 
 Analysts often emphasise that many financial and economic indicators tend 
to behave differently during growth and decline. Therefore, it is a well-
grounded assumption that the parameters of the models describing the for-
mation of such values change. The switching models allow to test such an as-
sumption. If the turning points, otherwise known as the moments of switching 
between periods of diversified behaviour of the variables, are known, then the 
segment model for the quantitative variable or the probability model for the 
selected variants of the qualitative variable, is estimated. If researchers cannot 

                                                 
* Correspondence to: Milda Burzala, Department of Econometrics, Faculty of Informatics and 

Electronic Economy, Poznan University of Economics, ul. Towarowa 53, 61-896 Poznań, Poland, 
e-mail: m.burzala@ue.poznan.pl 



Milda Maria Burzała 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 73–87 

74 

agree upon a single method for establishing such turning points, then the Mar-
kov-switching model, as proposed by Hamilton, can be used. In the case of eco-
nomic activity, the moments of switching depend, among others, on the accept-
ed method of decomposition of time series of selected macroeconomic ratios. 

In research on the American market, data provided by the NBER concern-
ing the turning points for growth and decline phases in the U.S. economy is 
used as the point of reference. Yet in many countries there is no established 
system to indicate the beginning and end of a recession. That is why it is 
worthwhile to analyse the convergence of indications resulting from various 
methods. 

Zarnovitz and Ozyildirim (2006) discuss the influence of the accepted 
method of decomposition on the variability of the course of a U.S. growth cy-
cle. They compare the dating of turning points based on cycles of levels, trend 
deviations and smoothed growth cycle. The results obtained with the use of the 
PAT method are very similar to the results obtained on the basis of the Hodrick-
Prescott filtering method, local linear trend as well as band-pass filtering 
method1. In Poland, a comparative analysis of business cycles obtained using 
different methods, made under various assumptions, was described, among oth-
ers, by Skrzypczyńska (2011) and Burzała (after revives, in press).  

This article presents a comparison of the indications for a recession phase 
based on the two models, with known and unknown switching points. Section 1 
presents the dating method of economic activity phases, which allows to deter-
mine the moments of switching between the phases of high and low economic 
activity. Sections 2 and 3 describe the models (Hamilton’s switching model and 
the logit model, respectively) which are used to estimate the probability of 
a recession. The research results and the comparison of indications based on the 
accepted method of dating of phases are described in section 4. Section 5 is 
a summary of the research results. 

1. Dating of Economic Activity Phases 
 In a time of growth-based market economies, it is difficult to clearly deter-
mine which of the observed changes have resulted from long-term economic 
growth and which stem from economic fluctuations. Despite the many research 
studies and upgraded decomposition methods, the division between ‘trend’ and 
‘cycle’ has always been accepted as a convention and can be considered some-
what artificial. Therefore, it would seem to make sense to employ an approach 
based on an analysis of the growth rates of a time series with the seasonal and 
random fluctuations removed, and with no decomposition into trend and eco-

                                                 
1 PAT (Phase Average Trend) is a 10-step procedure as described by Boschan, Ebanks (1978). 

It was applied by the NBER to a large number of indicators with generally satisfactory results in 
terms of timing and conformity to aggregate growth cycles. 



The Probability of Recession in Poland…  

DYNAMIC ECONOMETRIC MODELS 12 (2012) 73–87 

75 

nomic fluctuations. Thus, the research is focused on economic activity in gen-
eral, rather than on the course of a business cycle. 
In the empirical studies presented in this paper, the measure of economic activi-
ty were the annual indices of total industrial output sold PR_IRt – as recorded 
on a monthly basis between January 1993 and March 2011 – with seasonal and 
random fluctuations removed. The selection of output indices mostly resulted 
from a larger frequency of quotations than from the GNP. 
This approach is compatible with the generally accepted growth-based defini-
tion of the business cycle as commonly assumed in empirical analyses (Mintz, 
1972). The dating rules for economic activity phases, as used in this paper, 
make use of short- and long-term changes in the output indices (Burzała, 2005). 
The annual index PR_IRt measures the change in a value PRt with respect to 
the corresponding period of the preceding year PRt-12 and constitutes the meas-
ure of changes ‘within a long period of time’. The monthly index PR_IMt 
measures the change in a value PRt with respect to the preceding month PRt-1  
and is a measure of changes ‘within a short period of time’. These indices repre-
sent the following respective dependencies: 

.100_,100_
112

⋅=⋅=
−− t

t
t

t

t
t PR

PR
IMPR

PR
PR

IRPR  (1) 

The first index (PR_IRt) is a reference series, as has already been mentioned. 
The proposal of dividing the set of all observations of the reference series into 
separable subsets (economic activity phases) has been based upon tests which 
were to determine whether the short-term growth rate implies long-term chang-
es. Depending on whether a given index is above or below 100 (which corre-
sponds to a positive or negative growth rate), an observation is classified as 
characteristic of a given phase of the economy. 

This approach uses the rules of symbolic taxonomy, in which a phase (state) 
is described through a conjunction of values of selected indicators (Gatnar, 
1998). 
These rules allow to distinguish four phases of the economy (Burzała, 2005 
a,b): 

a) 100_ ≥
t

IMPR  and, simultaneously, 100_ ≥
t

IRPR  – implied growth, 
which denotes high economic activity (conventionally called the prosperity 
phase and marked with the code W_IM); 

b) 100_ ≥tIMPR  and 100_ <tIRPR  – non-implied growth, which de-
notes a rise of the monthly index and most often occurs following a reces-
sion (code W_NIM); 

c) 100_ <tIMPR   and  100_ <tIRPR  – implied decline, which denotes 
an economic recession (code S_IM); 



Milda Maria Burzała 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 73–87 

76 

d) 100_ <tIMPR  and  100_ ≥tIRPR  – non-implied decline, which de-
notes a dropping monthly index – which is usually a pre-recession phase 
(code S_NIM). 

The advantage of such a division is that two transitory phases have been identi-
fied, i.e. non-implied decline, which in itself is a warning against adverse 
change in business conditions, and non-implied growth, which indicates an im-
proving environment. The growth and decline phases discerned in this paper 
may be combined for a less detailed differentiation (prosperity vs. recession). 

2. Hamilton’s Switching Model 
 The assumption in this case is that the behaviour of certain macroeconomic 
indicators changes as a result of changed economic activity. However, a phase 
of economic activity is not directly observable, and thus it is difficult to estab-
lish which phase an economy is in at a given moment. In the terminology of 
switching models, a phase of activity is one of the possible states referred to as 
regimes. In the original Hamilton model (1989), it was assumed that there were 
two possible regimes, s (s = 1, 2), corresponding to the condition of an econo-
my (prosperity vs. recession). In each regime, indicator values are generated by 
two separate and independent processes. In the Hamilton model, these were the 
conditional average processes, AR(4), for quarterly GNP changes. Thus, in 
a general case: 

,)(

...)()(
21 2211

tsptp

ststst

pt

ttt

y

yyy

εµφ

µφµφµ

+−+

+−+−=−

−

−−

−

−−  (2)  

where: εt ~N(0,σ
 2), µs denotes the expected value under the s regime. 

 At present, models with switches resulting from the changing variance σ2, 
or the average and a variance are estimated particularly with regard to financial 
markets (Doman, Doman, 2009). Such a model, entailing two regimes, was 
used in the research studies. Consequently: 

).1,0(~

,/.../1

,
2222

11 1

iid

h

h

t

sptpstt

ttst

ptt

t

ξ

σεασεα

ξσε

−− −−
+++=

=

 (3) 

The series of random variables si in the subsequent moments in time t (t=1,..., 
T) has the Markov property, i.e. its value at the time moment t+1, i.e. st+1, de-
pends only on the regime at the t moment, rather than on all the preceding re-
gimes, which is formally formulated as: 

.)(,...),( 111 ijttttt pisjsPksisjsP ======= −−+  (4) 



The Probability of Recession in Poland…  

DYNAMIC ECONOMETRIC MODELS 12 (2012) 73–87 

77 

The probabilities pij denote the probability of an economy’s switching from 
regime i into regime j. These are unknown parameters that are estimated, where: 

).22(

),21(

),12(

),11(

122

121

112

111

===

===

===

===

+

+

+

+

tt

tt

tt

tt

ssPp

ssPp

ssPp

ssPp

 (5) 

Based on conditional probabilities (4 and 5), it is possible to determine the un-
conditional probability P(st = i) of an economy remaining at the t moment un-
der an ith regime. In the case of two regimes, the following is obtained: 

.
2

1
)2(,

2
1

)1(
2211

11

2211

22

pp
p

sP
pp

p
sP tt −−

−
==

−−
−

==   (6) 

The inverse of probabilities P(st =i) is interpreted as the expected time of re-
sumption of regime i: 

,
)(

1
)(

isP
imt

t =
=  (7) 

whereas the expected time of remaining under regime i is provided by means of 
the dependency: 

.
1

1
1

ijii
i pp

d =
−

=  (8) 

The parameters of the model (pij, µs φi, σs
2) are estimated using the maximum 

likelihood method2. If at the t moment the process was under the st =j regime, 
then the conditional probability density function of the explained variable yt can 
be represented as ),( 1−Ψ= ttt jsyf , where 1−Ψt  denotes the history of the 
process until the t-1 moment. Even knowledge of the model’s parameters does 
not allow to establish which regime a given economy is under at the t moment. 
Any suppositions on the actual regime may only be made by means of a condi-
tional probability: 

,
)(),(

)(),(
)(

2

1 11

11

∑ = −−
−−

Ψ=⋅Ψ=

Ψ=⋅Ψ=
=Ψ=

i ttttt

ttttt
tt

isPisyf

jsPjsyf
jsP  (9)  

                                                 
 2 The switching models were estimated using the TSM programme. The relevant likelihood 

function is presented as part of the description of the programme (Davidson, 2011). 



Milda Maria Burzała 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 73–87 

78 

where: 

∑ −−− Ψ==Ψ=
2

1 111
).()( ttijtt isPpjsP  (10) 

The maximised likelihood function is as follows: 

∑ ∑
= =

−− Ψ=⋅Ψ==
T

t j
ttttt jsPjsyfL

1

2

1
11 ).(),(log  (11) 

It is not easy to estimate the model’s parameters. Numerical problems result 
from the occurrence of local extrema of the logarithmic likelihood function. 
This is why normally two, up to three, regimes under which a process may be 
are distinguished. 

3. Logit Model 
 The dating method of economic activity phases, as proposed in Section 2, 
allows to date the time periods relating to a recession. Let us assume that this 
time the explained variable yt assumes the value of 1 at the t moment, if a reces-
sion occurs in an economy (S_IM); otherwise, the value is 0. The explained 
variable thus defined is a qualitative binary variable, usually modelled using the 
logit model. 
In the model with a qualitative explained variable, the theoretical probability Pt1  
of occurrence of the first option of the variable at the t moment is defined by the 
cumulative distribution function (β′xt), i.e.: 

),'(1 ttt FP ξ+= xβ  (12) 

where xt is a vector of explanatory variables, β is the parameters vector and ξt  
is a random disturbance. The type of distribution assumed for the ξt random 
variable generates the type of model under consideration. In the logit model, it 
is assumed that probability Pt1 is defined using the cumulative distribution func-
tion of a standardised logistic distribution ξt ∼ L(0, π

2/3). Hence: 

( )
.

1
1

1
'

'

21 t

t

t

t

e
dξ

e
e

P tξ
ξ

t xβ

xβ

−
∞−

−

−

+
=

+
= ∫  (13) 

It can be proved that the logit model’s random component is heteroscedastic 
(Jajuga, 1990). For this reason, to estimate the model’s parameters, the method 
with the utmost probability is most often used. The maximised logarithmic like-
lihood function is provided by means of the dependency: 

.lnln)
1

2

1
∑ ∑

= =

==
T

t j
tjtj

* PyLL )(( ββ  (14) 



The Probability of Recession in Poland…  

DYNAMIC ECONOMETRIC MODELS 12 (2012) 73–87 

79 

 As for the uses of models with the qualitative explained variable, the article 
by Estrella, Mishkin (1998) is of primary importance for measuring economic 
activity. These authors tested macroeconomic indicators that preceded the onset 
of an economic recession in the U.S. by using a binary probit model. As it 
turned out, the difference in interest rates applied to ten-year treasury notes and 
three-month treasury bills (the so-called spread) quite reliably indicates, with 
a one-year advance, an economic recession in the U.S. In the research studies 
carried out for the Polish economy, apart from the interest rate spread 
(SPREAD), the annual indices of money supply M1 (M1_IR), M2 (M2_IR), M3 
(M3_IR) and the WIG stock-exchange index (WIG_IR) were tested. 
The symptoms of a recession were chosen according to research studies carried 
out by this author in an unpublished PhD thesis (Burzała, 2005a). It should be 
emphasised that any given set of leading indicators is not effective for all coun-
tries. A different set of indicators is used by NBER or OECD researchers, and 
specialists from Germany or Japan use yet a different methodology. Under 
Polish economic conditions, the methods of testing the business cycle are still 
being improved and the list of quantitative indicators is being expanded. The 
idea of using the interest rate spread for forecasting economic activity is derived 
from the rational expectations hypothesis. In a market economy, a negative 
interest rate spread is one of the symptoms of a recession (short-term interest 
rates are higher than long-term interest rates)3. The growing interest in financial 
indicators stems from economists’ convictions that financial variables have an 
increasing influence on the real sphere of the economy, and thus the entire eco-
nomic policy. The meaning of a monetary policy that is handled appropriately 
results from the function that regulates both the money supply and lending in 
order to maintain employment, price stability and economic growth. The im-
portance of financial indicators was analysed by Atta-Mensah, Tkacz (1998), 
Estrella, Hardouvelis (1991), Stock, Watson (1993), Wright (2006), and Nyberg 
(2009). The purpose of this article is not to determine the best set of indicators, 
but to check the properties of some of these indicators. 

4. Research Results 
 In order to illustrate the recession indications based on the dating method as 
proposed in Section 1, the results obtained for the U.S. economy were compared 
with information from the NBER (Figure 1). 
 It turned out that the time periods shown for the recession appear to show 
high convergence, which may prove that the proposed dating method is correct. 
It is worth emphasising that in dating the cycles, NBER imposes additional 
restrictions for the duration of the decline and growth phase (a minimum of 6 
months) and for maintaining the growths and declines around the turning points 

                                                 
3 In the countries of Central and Eastern Europe (including Poland), the interest rate spread 

does not fulfil the given assumptions during an economic transformation.  



Milda Maria Burzała 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 73–87 

80 

(for a period of 5 months). The method proposed in this paper does not take 
such conditions into consideration in this version, yet it can easily be extend. 

0

0,2

0,4

0,6

0,8

1

19
59

M
01

19
62

M
01

19
65

M
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04

M
01

20
07

M
01

20
10

M
01

80

85

90

95

100

105

110

115

NBER_Rec recession PR_IR
 

Figure 1.  U.S. economy recession phases, January 1959 to March 2011, base on 
NBER data 

Fundamental research studies on Polish economic activity were based on statis-
tics from the International Monetary Fund (CEIC). Figure 2 shows the econom-
ic activity phases for Poland. The bar chart highlights the recession phases.  

0

0,2

0,4

0,6

0,8

1

19
92

M
01

19
93

M
01

19
94

M
01

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M
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M
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M
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M
01

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00

M
01

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20
09

M
01

20
10

M
01

20
11

M
01

80
85
90
95
100
105
110
115
120
125

S_IM  (recession) W_IM S_NIM W_NIM
 

Figure 2. Economic activity phases for Poland, January 1993 to March 2011, based on 
CEIC data 



The Probability of Recession in Poland…  

DYNAMIC ECONOMETRIC MODELS 12 (2012) 73–87 

81 

The regularities that occur are worth consideration: a recession phase and 
a prosperity phase are always preceded by a transitory phase, which signalises 
a deterioration or improvement of the economic conditions. The economic ac-
tivity phases determined as such were compared with the probability of occur-
rence of two regimes (prosperity and recession, by definition) based upon the 
Hamilton model. 
 The results Hamilton presented referred to the differences between the GNP 
logarithms. The proposed pattern, once directly applied to the output indices in 
the Polish economy, has not yielded satisfactory results. Therefore, it was pro-
posed that the explained variable under the Hamilton model be the logarithm of 
the annual output index PR_IRt (1), as recorded on a monthly basis. The param-
eters of the model as estimated are shown in Table 1. In the construction of a 
dynamic model describing changes in output, a conclusion was applied which is 
used in the dating of economic activity phases. The tests that were carried out 
have shown that the model which takes into account the first and twelfth lag 
(a counterpart of the short- and long-term changes) gives the best results. 

Table 1. Parameters of Hamilton’s switching model 
Parameters (standard error) Probability of transition 

φ1  = 0.952 (0.007) p 11  = 0.920 p 12  = 0.027 
φ12   = - 0.547 (0.032) p 21  =0.079 p 22  = 0.973 

 Expected regime i duration 
µ1 = 317.204 (6.548) d 1  = 12.577 d 22  = 36.873 
µ2 = 316.875 (6.545) Unconditional probability 
σ1  = 0.736 (0.052) P(s t  = 1) = 0.254 P(s t  =2) = 0.746 
σ2 = 0.175 (0.011) Expected time of regime i resumption 

R2 = 0.9829 mt(1) = 3.932 mt(2) = 1.341 
 

 Changes of output indices PR_IRt, phases of recession S_IM and the proba-
bility of the occurrence of regime 1 (Rg1_SmProbs) are shown in Fig. 3. Our 
analysis implies that regime 1 corresponds with decreased economic activity. 
The high probability of regime 1 covers not only the recession periods S_IM but 
also non-implied decline periods S_NIM. The non-implied decline phase usually 
precedes an economic recession. The period of June 2004 to May 2005 was one 
of significant turbulence in the Polish economy and was connected with the EU 
accession boom. The period’s prosperity was disturbed by a decline as well as 
by non-implied growth. None of the observations, however, has been consid-
ered a time of economic recession. Under the Hamilton model a high probabil-
ity of regime 1 occurring can be observed for this period. Assuming that regime 
1 was to appear, with the probability of its occurrence exceeding 0.5, then the 
accuracy of indications was evaluated. Higher accuracy rates were obtained 
with the assumption that regime 1 corresponds to a combination of two decline 
phases (indications for both decline phases proved to be 90% accurate, with 
a total accuracy of 97%; as compared to the accuracy for the recession phase 
equal to 94%, with a total accuracy of 81%). It is worth noting that both regimes 



Milda Maria Burzała 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 73–87 

82 

differ insignificantly in terms of the expected value. The period of lower eco-
nomic activity is characterised by definitely higher variance. Both activity states 
are relatively durable (with the probability of remaining in the lower activity 
phase equal to 0.920, during a prosperity phase equal to 0.973), whereas the 
probabilities of transition between the states are small. The table also shows the 
unconditional probabilities which indicate a higher probability of prosperity 
occurring (0.746). Prosperity is also characterised by a longer expected duration 
period (c.37 months) and a shorter time of return from the state of lower eco-
nomic activity (c.1.3 months). 

0

0,2

0,4

0,6

0,8

1

19
92

M
01

19
93

M
01

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94

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01

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11

M
01

80

90

100

110

120

130

Rg1_SmProbs S_IM (recession) W_IM S_NIM W_NIM
 

Figure 3.  Hamilton model-based probability of regime 1, set against the background of 
the economic activity phases in Poland 

 The second research approach using the logit model assumed that the mo-
ments of switching to the recession phase are known. The qualitative explained 
variable yt assumes a value of 1 if the observation concerned an implied decline 
S_IM, otherwise it takes on the value of 0. The model was used to estimate the 
probability of a recession occurring on the basis of values assumed by selected 
leading indicators. They were represented in the model by annual indices of 
change – the exception being the interest rate spread as the difference in the 
interest rate of the ten-year treasury notes and the three-month treasury bills. 
Should the sample be balanced (the number of 1s being identical with the num-
ber of 0s for the explained variable), we assume that a recession is to occur if 
the probability of it occurring exceeds 0.5. For an unbalanced sample (which 
appeared in the research study), the limit value is usually assumed to be the 
share of 1s in the sample. Measures of fit and the accuracy of estimations from 
the logit models are shown in Table 2. The forecast of symptomatic variables 
(leading indicators) was determined based on the maximum value of McFad-
den’s R2 (determination coefficient) and Akaike’s information criterion. 



The Probability of Recession in Poland…  

DYNAMIC ECONOMETRIC MODELS 12 (2012) 73–87 

83 

Table 2. Logit models and their verification 

M
od

el
 

Va
ria

bl
e 

La
g 

Pa
ra

m
et

er
 e

st
i-

m
at

e 

P-
va

lu
e 

M
cF

ad
de

n 
 

R-
sq

ua
re

d 

Ak
ai

ke
 in

fo
rm

at
io

n 
cr

ite
rio

n 

Fr
eq

ue
nc

y 
of

 1
s 

(e
m

pi
ric

al)
 

Ac
cu

ra
cy

, i
n 

to
ta

l 

Ac
cu

ra
cy

 o
f 1

s 

M1 SPREAD 12 -0.345 1.40E-03 0.122 78.36 0.105 76% 54% 
Const   -2.771 1.88E-11 

M1A 
SPREAD 12 -0.340 1.89E-02 

0.412 55.82 0.099 82% 85% PR_IR 1 -0.333 3.00E-04 
Const   30.710 6.00E-04 

M2 M1_IR 7 -0.105 4.40E-03 0.083 97.98 0.105 65% 44% 
Const   9.647 1.78E-02 

M2A 
M1_IR 7 -0.06 1.72E-01 

0.322 75.49 0.105 78% 88% PR_IR 1 -0.269 4.98E-05 
Const   32.07 1.00E-04 

M3 M2_IR 4 0.077 3.31E-02 0.046 102.44 0.103 63% 56% 
Const   -11 9.00E-03 

M3A 
M2_IR 4 0.023 6.23E-01 

0.309 77.24 0.103 80% 100% PR_IR 1 -0.267 4.12E-05 
Const   22.481 1.76E-02 

M4 M3_IR 4 0.069 6.08E-02 0.035 103.58 0.103 63% 63% 
Const   -10.08 1.85E-02 

M4A 
M3_IR 4 0.026 5.63E-01 

0.280 80.23 0.103 78% 100% PR_IR 1 -0.254 3.82E-05 
Const   20.914 1.91E-02 

M5 WIG_IR 2 -0.058 5.95E-06 0.312 79.91 0.083 80% 94% 
Const   2.981 4.20E-03 

M5A 
WIG_IR 2 -0.082 4.20E-03 

0.409 70.00 0.087 84% 94% PR_IR 1 0.017 8.71E-01 
Const   3.443 6.96E-01 

M6 

SPREAD 12 -0.354 1.65E-02 

0.462 53.59 0.099 88% 100% M3_IR 4 0.006 9.59E-01 
WIG_IR 2 -0.085 3.80E-03 
Const   3.811 7.85E-01 

Taking the individual indicators into account, the WIG stock-exchange index 
appears to have the highest symptomatic properties. Unfortunately, from a prac-
tical standpoint, a two-month forecast is too short. Due to the length of the fore-
cast period, the interest rate spread better fits the purpose. A higher accuracy of 
indications for the recession phase, given a four-month forecast, was obtained 
for annual changes in the index of money supply M3. In all cases the accuracy 
of indications was clearly improved if a lagging industrial output index was 



Milda Maria Burzała 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 73–87 

84 

added to the model. An attempt was also made in this research to combine the 
symptomatic properties of three leading indicators (i.e. index of money supply 
M3, the stock-exchange index and the interest rate spread). This latter model is 
characterised by the highest degree of measure of fit, the lowest Akaike’s crite-
rion value as well as the highest accuracy of indications within the sampling 
period. The probability of occurrence of a recession based on the M6 model is 
shown in Figure 4. 

0

0,2

0,4

0,6

0,8

1

19
92

M
01

19
92

M
12

19
93

M
11

19
94

M
10

19
95

M
09

19
96

M
08

19
97

M
07

19
98

M
06

19
99

M
05

20
00

M
04

20
01

M
03

20
02

M
02

20
03

M
01

20
03

M
12

20
04

M
11

20
05

M
10

20
06

M
09

20
07

M
08

20
08

M
07

20
09

M
06

20
10

M
05

80

90

100

110

120

130

logit M6 S_IM (recession) W_IM S_NIM W_NIM
 

Figure 4.  Probability of a recession based on the logit model M6 and the recession 
phases in Poland 

Summarising the results of the research – the duration of a recession was ana-
lysed based on the proposed dating method, the Hamilton model and one of the 
logit models – the M6 (Table 3).  

Table 3. Recession time periods 

Beginning of recession End of recession 

S_IM, 
S_NIM S_IM 

Hamilton 
Model 

Logit Model 
M6 

S_IM, 
S_NIM S_IM 

Hamilton 
Model 

Logit Model 
M6 

1998-06  1998-04  1998-12  1998-12  
2001-01 2001-05 2001-01 2001-02 2002-04 2002-04 2002-03 2002-04 
2004-06  2003-10  2005-01  2005-02  
2008-03 2008-09 2008-02 2008-08 2009-02 2009-02 2009-02 2009-09 

The classic approach was additionally adopted in the comparisons, which states 
that every phase should last at least half a year (Bry, Boschan, 1971). This al-
lowed us to exclude short periods of implied decline in the comparisons and 
means that the rise or drop of a measure must remain on both sides of the turn-
ing point for at least six months. For the Hamilton model this was the rise/drop 



The Probability of Recession in Poland…  

DYNAMIC ECONOMETRIC MODELS 12 (2012) 73–87 

85 

of the probability of recession above/below the limit value of 0.5. For the logit 
model, as a result of not balancing the statistic sample, the limit value was the 
empirical frequency of 1s occurring for the qualitative explained variable. In the 
proposed dating method it was the change in behaviour of the annual and 
monthly output indices. 
 All of the analysed dating methods show significant convergence in terms 
of dating the end of recession (a 1-month difference). The only exception is the 
logit model, which shows the end of the last recession to be later. However, the 
dating of the beginning of recession by the Hamilton model shows convergence 
with the results from the logit model in 2001 and the two phases of decline 
S_NIM, S_IM in 1998 and 2001. In the case of the last economic recession, the 
Hamilton model and the combined phases S_NIM and S_IM indicate the begin-
ning of recession in February/March 2008, while the logit model and the phase 
of implied decline S_IM indicate it as late as in August/September. Yule’s coef-
ficient of association, based on Chi-square statistics, which is a measure of in-
terdependence of qualitative variables measured in a nominal scale, was used to 
quantitatively estimate the similarity of indications (it has a value from -1 to 1). 
The indications of the Hamilton model demonstrate the highest level of associa-
tion with the indications of two phases of decline S_NIM and S_IM – 0.55, 
however, the logit model indications demonstrate the highest level of associa-
tion with the indications of implied decline S_IM – 0.42.  

5. Conclusions 
 The research conducted shows the convergence of indications based both on 
the proposed dating method and on the Hamilton model. In the presented ver-
sion the Hamilton model adequately describes the probability of occurrence of 
two decline phases. The dating of phases as proposed in Section 2 is simple in 
empirical applications and allows for a division into four phases of economic 
activity, which is beneficial as it distinguishes two transitory phases. 
The logit model allows to gain satisfactory results for a more detailed division 
of the recession phase S_IM, based on the value of selected leading indices. 
However, in the domain of the Polish economy, more research is needed in 
recognising the symptomatic properties of various macroeconomic indicators. 
The interest rate spread, used successfully in advanced marked economies, con-
tinues to alter its characteristics under Polish economic conditions and is cur-
rently not the best possible indicator forecasting a recession. 

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Burzała, M. (2005b), Kilka uwag na temat wyboru modelu dyskretnego do prognozowania ak-
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Burzała, M., (2011), Porównanie faz aktywności gospodarczej uzyskanych na podstawie filtru 
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of the Phases of Economic Activity Generated from Hodrick-Prescot Filter, Smoothed 
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usługach ekonomii (Mathematics in the Service of Economics), 65–77, Wydawnictwo 
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Prawdopodobieństwo kryzysu w Polsce z modelu prze-
łącznikowego Hamiltona i modelu logitowego 

Z a r y s  t r e ś c i. W wielu krajach brakuje wypracowanego systemu oznaczania początku 
i końca kryzysu. W proponowanej metodzie periodyzacji każda z czterech faz aktywności gospo-
darczej opisywana jest przez koniunkcję wartości rocznych i miesięcznych indeksów produkcji 
przemysłowej. Analitycy rynku zwracają szczególną uwagę na zróżnicowanie zachowania się 
większości wskaźników makroekonomicznych w czasie spadków i długookresowego wzrostu. W 
związku z tym uzasadnione jest założenie o zmieniających się parametrach modeli opisujących 
kształtowanie się tych wielkości. Realizację takiego założenia umożliwiają zarówno modele 
przełącznikowe jak i modele logitowe. W badaniach empirycznych przeprowadzono porównanie 
prawdopodobieństwa wystąpienia kryzysu z obu modeli Analiza wyników pokazuje duże podo-
bieństwo wskazań z zaproponowanej metody periodyzacji i modelu Hamiltona. Model Hamiltona 
w prezentowanej wersji dobrze opisuje prawdopodobieństwo wystąpienia dwóch faz spadko-
wych. Model logitowy pozwala na uzyskanie zadawalających rezultatów dla podziału bardziej 
szczegółowego. Na gruncie gospodarki polskiej należy jednak w dalszym ciągu prowadzić bada-
nia nad rozpoznaniem własności symptomatycznych różnych wskaźników makroekonomicznych. 

S ł o w a  k l u c z o w e: model przełącznikowy, model logitowy, periodyzacja faz aktywności 
gospodarczej, prawdopodobieństwo kryzysu.  




	Introduction
	1. Dating of Economic Activity Phases
	2. Hamilton’s Switching Model
	3. Logit Model
	4. Research Results
	5. Conclusions
	References