Aneta Wlodarczyk

Marcin Zawada

English

Research Papers in Economics

© Copyright by The Nicolaus Copernicus University Scientific Publishing House
DYNAMIC ECONOMETRIC MODELS 
Vol. 8 – Nicolaus Copernicus University – Toruń – 2008  
Aneta Włodarczyk, Marcin Zawada 
Czestochowa University of Technology 
Markov-Switching Models for the Prices of Electric  
Energy on the Energy Stock Market in Poland 
1. Introduction 
  GARCH models, characterised by autoregressive dependencies in the condi-
tional variance equation, are used in modelling financial time series on account 
of the variance grouping phenomenon they demonstrate. In turn, Markov-
switching models, through random process switching to different regimes, allow 
the differentiation of periods corresponding to different levels of volatility of 
the endogenous variable (Stawicki, 2004). Hamilton and Susmel (1994) pro-
posed the integration of these two approaches by introducing a random regime 
switching in the conditional variance equation of the ARCH model (Markov-
switching ARCH, SWARCH).
 A generalised version of the SWARCH model is 
the GARCH model with a regime switching (MS-GARCH), which makes it 
possible to provide a more detailed description of the dynamics of the variance 
process which differs across the regimes (Frömmel, 2004). 
In this paper we present theoretically different specifications of Markov-
switching models, which take into consideration the autoregressive dependen-
cies both in the conditional mean and in the conditional variance of process. In 
the empirical section of this paper the Markov-switching models for daily prices 
of electric energy in Poland are estimated and tested.  
2. General Overview of the Markov-Switching Model 
Hamilton (1990) proposed a form of the Markov-switching model MS(N)-
AR(p) which describes the changes of the mean and the variance of autoregres-
sive economic process across the regimes: 
t p t s p t s t s s t y y y c y
t t t t ε φ φ φ + + + + + = − − − , 2 , 2 1 , 1 ... , (1) © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Aneta Włodarczyk, Marcin Zawada 
 
172 
nowhere:  t s t u
t σ ε =  for ut ~ IID(0,1),  t s  - denotes a first-order homogeneous 
Markov chain with N states and transition probabilities matrix  [ ]
N N ij p P
× = , 
t i,s   ,φ
t s c - parameters depend on the state variable st . 
Hamilton (1994) formulates the following relation for the conditional distribu-
tion of the state variable st: 
() ()
()
( ) ( )
() () ∑
=
− −
− −
−
−
Φ = ⋅ Φ =
Φ = ⋅ Φ =
=
Φ
Φ =
= Φ = N
i
t t t t t
t t t t t
t t
t t t
t t
i s P i s y f
j s P j s y f
y f
j s y f
j s P
1
1 1
1 1
1
1
,
, ,
, (2) 
where the form of the density function of conditional distribution of the variable 
yt depends on the density function g(
.) of the postulated distribution of the inno-
vations (Doman, 2005):
1 
() ()
( )
() ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛ −
= Φ = − j
j y
g
j
j s y f
t
t t
j
t
t t t σ
μ
σ
1
, 1 , (3) 
The generalisation of the Markov-switching model of the form (1) consists in 
including an additional exogenous variable into a model. In such a case the ma-
trix form of model is following (Kaufmann, 2000): 
t s t t t X y ε β + = ,        t ε ~ IID( )
t S σ , 0 , (4) 
where:  t X = (1, yt-1,..., yt-p, xt,..., xt-q) – vector of observable variables, which 
may contain lagged both endogenous and exogenous variables, 
) ,..., , , ,..., , , ( , , 1 , 0 , , 2 , 1 t t t t t t t t s q s s s p s s s
T
s c ψ ψ ψ φ φ φ β =  – parameters vector, which 
depends on the state variable st . 
3. GARCH Structure for Markov-Switching Models 
The econometric literature provides the descriptions of various possible specifi-
cations of the conditional variance equation for the MS-GARCH model, which 
generalize the GARCH (p, q) model by allowing for regimes with different 
volatility levels  (Gray, 1996; Klassen, 2002): 
∑ ∑
=
− −
=
+ + =
p
i
i t s i t
q
i
s i s t h h
t t t
1
,
2
1
1
, β ε α ω , (5) 
                                                 
1 μt(j) and σt(j) are the conditional mean and the conditional standard deviation of 
the financial process respectively, dependent on the regime j, in which the process is 
found at a moment t. One can include the autoregressive scheme in the conditional 
mean equation depending on the properties of the modelled series. In empirical research 
it is frequently assumed that innovations have the following distributions: normal, Stu-
dent’s t-distribution, GED, skew Student’s t-distribution. © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Markov-Switching Models for the Prices of Electric Energy...  173
where:  t h – the variance of random term conditional on observable information 
Φt-1 and on the regime path  t s ~, 
t s ω – the intercept depends on the state variable 
st, 
t s i, α  – the ARCH parameters depend on the state variable st, 
t s i, β  – the 
GARCH parameters depend on the state variable st. 
However, the estimation of the conditional variance parameter ht-1 in equation 
(5) causes problems of numerical nature, owing to the occurrence of path-
dependence which illustrates the entire history of process switching to particular 
regimes (Cai 1994 and Hamilton, Susmel 1994). This problem will be presented 
for the MS(2)-GARCH (1,1) model.
  
 
Figure 1. The evolution of conditional variance in the GARCH model with a regime 
path-dependence 
Source: Gray S. F., Modeling the conditional distribution of interest rates as a regime – switching 
process, Journal of Financial Economics 3/1996, p. 35. 
Each conditional variance (concerning the model with path-dependence) de-
pends not only on the current regime, but also on the entire history of the proc-
ess which controls the switches between different volatility regimes, which is 
reflected by the branching nodes of the above tree.  
In the subsequent specification, the conditional variance was made dependent 
only on the current regime st and not the entire path  1
~
− t s , through having intro-
duced an expected value operator into the expression (5) (Gray, 1996): 
] [ 1 2
2
1 − − − + + = t t s t s s t h E h
t t t β ε α ω , (6) 
where: 
2
2 2 1 1 2 1 1 2
2
2 2 1
1 1
2
1 2 1
2
2 1 2
2
1 1
} ) 2 ( ) 1 ( { ) ( ) 2 (
) ( ) 1 ( ]) [ ( ] [
μ μ μ
μ
⋅ Φ = + ⋅ Φ = − + ⋅ Φ = +
+ + ⋅ Φ = = Φ − Φ =
− − − − − − −
− − − − − − − −
t t t t t t t
t t t t t t t t
s P s P h s P
h s P y E y E h
 (7) 
Figure 2 illustrates the evolution of conditional variance in the MS-GARCH 
model without a regime path-dependence. © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Aneta Włodarczyk, Marcin Zawada 
 
174 
 
Figure 2. The evolution of conditional variance in the GARCH model without a regime 
path-dependence 
Source: Gray S. F., Modeling the conditional distribution of interest rates as a regime – switching 
process, Journal of Financial Economics 3/1996, p. 36. 
In any given period, as a consequence of the conditional variances (ht/i for 
i=1,2) integration, a conditional variance (ht) is constructed relative to the set of 
observed data (Φt-1), not regimes, as it occurred in the previous specification. 
The described dependency is illustrated by the tree nodes which combine into  
a shared node in each subsequent step. The final specification of the conditional 
variance of the GARCH (1,1) model with a volatile regime can be presented in 
the form of the following equation:
2 
] [ 1 1
2
1 t t t s t s s t s h E h
t t t − − − + + = β ε α ω , (8) 
where: 
. } ) 2 ( ) 1 ( {
) ( ) 2 ( ) ( ) 1 (
2
2 1 1 1 1 1
1 2
2
2 1 1 1 1
2
1 1 1 1
μ μ
μ μ
⋅ Φ = + ⋅ Φ = −
− + ⋅ Φ = + + ⋅ Φ = =
− − − −
− − − − − − −
t t t t
t t t t t t t
s P s P
h s P h s P h
 (9) 
The problem of the occurrence of path-dependence in the conditional variance 
equation is of vital importance when computing multi-periods-ahead forecasts. 
4. Numerical Sample 
The empirical study has been conducted on daily spot quotations of electric 
energy on the Polish Energy Stock Market in the period between January 02, 
2004 and December 31, 2006. The form of Markov-switching model proposed 
by Hamilton and Susmel (1994) was estimated: 
t 1 t t u y y + ⋅ + = − ϕ γ , (10) 
                                                 
2 The final specification of conditional variance for the GARCH model with a re-
gime –switching takes advantage of filtered probabilities, while in the previous specifi-
cation the ex-ante probabilities were applied (Klassen, 2002). © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Markov-Switching Models for the Prices of Electric Energy...  175
whereas for the innovation ut the SWARCH – L(3,2) specification was used: 
⎩
⎨
⎧
>
≤
=
⋅ ⋅ + + + =
⋅ =
⋅ =
−
− − − −
                            
, 0 u ~    dla    0
0 u ~    dla    1
, u ~ u ~ u ~
                                                , υ u ~
                                            , u ~ u
1 - t
1 - t
1
2
1 t 1
2
2 t 2
2
1 t 1 0
2
t t
t
t
t t
t
s
d
d h
h
g
t
ξ α α α  (11) 
where state variable st follows a first-order homogeneous Markov chain with 
three regimes. 
A likelihood function on the basis of the observations of the variable yt for t = 1, 
2, ..., T can be formulated: 
() () ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
Φ = Φ = = ∑ ∑
=
− −
=
N
j
t t t t t
T
t
T j s P j s y f y y y L
1
1 1
1
2 1 ; ; , log ) ; ,..., , (
~
θ θ θ , (12) 
The estimation of the parameters of this model has been performed in the Ox 
package, using the program codes written by Hamilton. 
The volatility clustering of the electric energy prices is explained by the fact 
that the periods of high volatility are followed by periods of low volatility. It is 
worth to wonder if a period of high volatility will be followed by normal vola-
tility level and, similarly, a period of low volatility will be followed by normal 
volatility level. So, we used different specifications of Markov switching 
ARCH model for describing the polish electric energy volatility process. The 
results in Table 1 show that variances in particular states differ from each other 
for each estimated switching model. For example, the results obtained for the 
SWARCH(2,0) model indicate that the variance corresponding to the second 
regime is over five times higher than the variance which characterises the first 
regime. It is also worth to observe that the probabilities of remaining in the 
states of high-, and low-volatility are significantly large, which is reflected by 
the effect of variance grouping in series of returns of energy prices. Similar 
results were obtained for the SWARCH-L(3,2) model.  
In the case of the first specification of the switching model, in which the model-
ling of the financial leverage effect was left and a normal distribution of the 
random parameter was assumed, an atypical estimate of variance in normal-
volatility regime was obtained. Moreover, the estimates of switching probabili-
ties for the Markov chain indicate a lower stability of particular regimes in 
comparison to the remaining models. In the case of SWARCH(3,2) and 
SWARCH-L(3,2) models, the estimates of parameter vectors are significantly 
different from each other, what may be connected with some numerical prob-
lems (the Hessian matrix is not positive definite, so the conclusion may be that 
either local maximum has not been found or the estimates are up against boun-
dary condition). © Copyright by The Nicolaus Copernicus University Scientific Publishing House
 
176 
Table 1. E
 
const 
φ 
α0 
α1 
α2 
ξ 
ν 
g1 
g2 
g3 
p11 
p22 
p12 
p21 
p31 
p33 
Log lik
n 
T-statistics 
Figure 3. R
f
An additi
are the fil
a defined
ated in th
moment o
Estimates of pa
SWAR
0.104
-0.225
15.34
0.241
0.191
0.182
2.187
0.579
0.567
0.420
0.000
0.405
0.594
k -3314
10
in parentheses.
Returns and sm
for the SWAR
ional product
lter and smo
d probability 
he unobserve
of switching
Aneta W
arameters for t
RCH(3,2)  S
491258 
516631 
48751 
141784 
174416 
- 
- 
1 
257100 
73915 
932120 
702740 
067880 
000000 
554684 
445300 
4.7072 
095 
. 
moothed prob
RCH(2,0) mod
ts of the estim
othed probab
that the elec
d high or low
g the process
Włodarczyk, Ma
the SWARCH
SWARCH-L(3
-0.080703910
-0.14736107
3.4173519
0.0049463894
0.057990611
0.90538713
5.0749257
1 
4.0064729
13.553465
0.99240311
0.99388004
0.0075968939
0.00000000
0.0000000066
0.98016587
-3292.9939
1095 
babilities of ob
del for energy 
mation of M
bilities. Basi
ctric energy 
w volatility r
s from low v
arcin Zawada 
H model  
3,2) 
0  0.089
7 -0.222
13.21
4 
1 
5.56
0.91
0.80
9 
6 
btaining low- a
prices 
Markov switch
ing on them 
return at the
regime. One 
volatility reg
SWARCH(2,
988490      (0.70
255117      (-7.0
15834         (9.5
- 
- 
- 
- 
1 
615685     (7.84
- 
75262       (5.06
19476       (4.66
- 
- 
- 
- 
-3344.0826
1095 
and high-vola
hing model p
one may ind
e moment t w
may also es
gime to high
,0) 
0880937) 
0260506) 
5472570) 
415091) 
650989) 
635380) 
 
 
atility states 
parameters 
dicate with 
was gener-
stimate the 
h volatility © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Markov-Switching Models for the Prices of Electric Energy...  177
(the following dependency must be satisfied: ( ) 5 , 0 ; 1 > Φ = θ T t s P and 
() 5 , 0 ; 1 1 < Φ = + θ T t s P ) and vice versa. 
Figure 3 plots the estimated smoothed probabilities of being in the low and high 
volatility regime, as defined in (2). 
On the basis of the computed residuals it was possible to verify the hypotheses 
concerning the dynamic specification of particular models. The values of par-
ticular testing statistics are presented in Table 2.  
Table 2. Diagnostics for dynamic specification of Markov switching model 
Test  SWARCH(3,2)  SWARCH-L(3,2)  SWARCH(2,0) 
Ljung-Box test  311.9  331.9  313.5 
Jarque-Bera test  174.3  163.1  186.7 
ARCH effect test  84.6  85.5  94.1 
AIC -3325.7072  -3305.9939  -3350.0826 
SC  -3353.1972 -3338.4839 -3365.0781 
The results presented in Table 2 confirm that the residuals of given models do 
not have white noise properties, i.e. they are correlated and conditionally het-
eroscedastic, and their distributions significantly differ from the normal distri-
bution. 
The transition probability values are the starting point for determination of the 
expected duration of regime i (Hamilton, Susmel, 1994): 
ii p
i d
−
=
1
1
) ( , (13) 
The analysis of the information included in Table 1 leads to the finding that 
state 1 (low volatility level) is expected to last on average for 12 days, while 
state 2 (high volatility level) typically – for 5 days. 
4. Conclusions 
To conclude, attention ought to be paid to the following properties of the 
GARCH model with a regime-switching: 
-  the structure of the ARCH and GARCH model allows to take into considera-
tion the effect of conditional heteroscedasticity which differs significantly 
across the regimes. 
-  higher flexibility of the model with regard to the persistence of shocks in 
volatility level
3, 
                                                 
3 Not all the disturbances which occur on the market have permanent influence on 
the level of prices. Owing to a suitable construction of these models, the stability of this 
kind of disturbances may be reduced by means of switching to a regime with a lower-© Copyright by The Nicolaus Copernicus University Scientific Publishing House
Aneta Włodarczyk, Marcin Zawada 
 
178 
Analyzing the results of the empirical study which concerned the modeling of 
market prices of electric energy the following conclusions may be formulated. 
The specification of equation for both the conditional mean and conditional 
variance should be improved. The series of returns of electric energy prices are 
subject to periodic fluctuations due to the specific character of electric energy as 
a commodity traded on the stock exchange. The autoregressive structure of first 
order considered in models in question turned out to be insufficient for the de-
scription of the underlying dependencies. As a consequence, in further research 
the higher-order autocorrelation should be taken into account and the ARCH 
structure should be replaced by the GARCH structure with a volatile regime. 
References 
Cai, J. (1994), A Markov model of Unconditional Variance in ARCH, Journal of Busi-
ness and Economic Statistics 12. 
Doman, R. (2005), Forecasting the Polish Stock Market Volatility with Markov Switch-
ing GARCH Models, w Milo W., Wdowiński P. (ed.), Theory and Applications, 
Wydawnictwo Uniwersytetu Łódzkiego, Łódź. 
Frömmel, M. (2004), Modelling Exchange Rate Volatility in the Run-up to EMU using 
a Markov Switching GARCH Model, Universität Hannover, Discussion Paper 
No. 306. 
Gray, S. F. (1996) Modeling the Conditional Distribution of Interest Rates as a Regime 
– switching Process, Journal of Financial Economics 3. 
Hamilton, J. D. (1990), Analysis of Time Series Subject to Changes in Regime, Journal 
of Econometrics 45. 
Hamilton, J. D., Susmel R. (1994), Autoregressive Conditional Heteroskedasticity and 
Changes in Regime, Journal of Econometrics 64. 
Kaufmann, S. (2000), Measuring Business Cycles with a Dynamic Markov Switching 
Factor Model: an Assessment using Bayesian Simulation Methods, The Econome-
trics Journal 3. 
Klassen, F. (2002), Improving GARCH Volatility Forecasts with Regime-Switching 
GARCH, w Hamilton J. D., Raj B.(red.), Advances in Markov-Switching Models, 
Physica-Verlag, Heidelberg. 
Stawicki, J. (2004), Wykorzystanie łańcuchów Markowa w analizie rynku kapitałowe-
go, Wydawnictwo UMK, Toruń. 
                                                                                                                         
volatility level, which in turn results in a particular piece of information being quickly 
delivered “outside the market”. 

A Markov model of Unconditional Variance in ARCH,

Analysis of Time Series Subject to Changes in Regime,

Autoregressive Conditional Heteroskedasticity and Changes in Regime,

Improving GARCH Volatility Forecasts with Regime-Switching GARCH, w

Measuring Business Cycles with a Dynamic Markov Switching Factor Model: an Assessment using Bayesian Simulation Methods,

Modeling the Conditional Distribution of Interest Rates as a Regime – switching Process,

Modelling Exchange Rate Volatility in the Run-up to EMU using a Markov Switching GARCH Model, Universität Hannover, Discussion Paper No.

Wykorzystanie łańcuchów Markowa w analizie rynku kapitałowego, Wydawnictwo UMK, Toruń. volatility level, which in turn results in a particular piece of information being quickly delivered “outside the market”.

Markov-Switching Models for the Prices of Electric Energy on the Energy Stock Market in Poland

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Markov-Switching Models for the Prices of Electric Energy on the Energy Stock Market in Poland

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