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Modeling regime shifts
Markov switching models
Hidden Markov Model Applications
Fahad Javed Malik
Dept of Economics and Finance & Dept of Mathematical Sciences
Institute of Business Administration, Karachi
This thesis is submitted for the degree of
Bachelor of Science
Economics and Mathematics
Dr. Adnan Haider
Dr. Mohammad Nishat
Academic Program Coordinator Thesis Supervisor
Assistant Professor Professor and Associate Dean
Institute of Business Administration, Karachi. Institute of Business Administration, Karachi.
June 2016
Centre for Business and Economics
Research
Dr. Adnan Haider Dr. Mohammad Nishat
This thesis is dedicated to my dear parents.
Declaration
I hereby declare that except where specific reference is made to the work of others, the contents
of this dissertation are original and have not been submitted in whole or in part for consideration
for any other degree or qualification in this, or any other university. This dissertation is my o w n
work and contains nothing which is the outcome of work done in collaboration with others,
except as specified in the text and Acknowledgments. This dissertation contains fewer than
18,000 words including appendices, bibliography, footnotes, tables and equations and has fewer
than 15 figures.
Fahad Javed Malik
June 2016
Acknowledgements
Attaining this opportunity, I would like to thank all those individuals without whom my educa-
tion over the preceding years and particularly this thesis would not have been what they have
been.
First of all my thanks go to Professor Dr Muhammad Nishat for bringing the topic of this
thesis to my attention, for all the helpful advice throughout the time I worked on it. Moreover, I
am most grateful to him for enabling me to spend a Summer Semester in University of Oslo
and BI Norwegian Business School a year ago, to attend a research program in Norway.
Prof. Dr Muhammad Nishat patience, guidance and motivation as supervisor of this thesis,
for which I would like to thank very much indeed. Moreover, I am also heavily indebted to
Prof. Dr Adnan Haider for all the helpful advice and encouragement which he have given me
over the years. I would like to express my sincere gratitude to supervisors, for their insightful
comments and support and for giving me the opportunity to complete my thesis.
My special thanks for most helpful and influential discussions on the subjects of this thesis
go to Prof Dr. Javed Iqbal and Prof. Dr Muhammad Sheraz. For introducing me into mathemat-
ical research and all the helpful things that I learned from the very interesting work together
with them I am most grateful to Prof. Dr Nasir Touheed.
Moreover, I gratefully acknowledge the privilege of having been supported financially by
Institute of Business Administration Karachi via Merit Scholarship throughout my studies.
Furthermore, a special thank you is due to all my friends who made life and, in particular,
university life much more enjoyable. Last but not at all least I would like to thank especially
my family for simply everything.
Abstract
Regime shifts is an important aspect of financial economics, since some series have a tendency
to fluctuate abruptly, such that the new trend stays for a long period. The Markov switching
model captures volatility persistence, time varying correlations, skewness and kurtosis. The
thesis considers discrete time space models using Hidden Markov models to study regime
switching properties of the time series. The Hamilton (1988) model is referred, following the
assumption that current regime is only dependent on the regime one time ago; then incorporated
the extensions in regime switching models. The Hidden Markov models capture the non-
linear patterns, considering stochastic volatility, after the identification of the regimes, the
probability denoting the changes in regimes is also estimated. Since the financial time series
exhibits conditional variance that is not handled by traditional regression models, therefore
autoregressive conditionally heteroscedastic model techniques are used. The regime switching
technique extends GARCH model taking into account the variance structure considering
economic states are different in the long term such that the shifts is governed by Markov
chain process.
The thesis describes Markov chain model, highlighting the State Space method and Kalman
Filter. Then the Finite Markov mixture distribution is outlined, with emphasis on extensions of
Markov switching models. Later studied the structural break in the variance of growth rate of
GDP for the economies; USA, Pakistan, India and Japan. Further paper tries to capture linkages
between national stock markets of six different economies taking into account the degree of
dependence among markets. The real interest rate of Pakistan structural breaks were assessed,
working on volatility clustering, interest rate trend is itself unrelated representing a mixture of
distribution.
Table of contents
List of figures xiii
List of tables xv
Nomenclature xvii
1 Introduction 1
1.1 Markovchains................................... 1
1.2 Estimation.................................... 2
1.3 Smoothing.................................... 5
1.4 Time-varying transition probabilities . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 State Space Methods and Kalman Filter . . . . . . . . . . . . . . . . . . . . 8
1.6 HiddenMarkovModels............................. 9
1.7 Hidden Markov Model Parameter Estimation . . . . . . . . . . . . . . . . . . 11
2 Markov Switching Models 17
2.1 Finite Markov Mixture Distributions . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Statistical Modeling Based on Finite Markov Mixture Distributions . . . . . 18
2.2.1 The Basic Markov Switching Model . . . . . . . . . . . . . . . . . . 18
2.2.2 The Markov Switching Regression Model . . . . . . . . . . . . . . 19
2.3 Nonlinear Time Series Analysis Based on Markov Switching Models . . . . . 19
2.4 The Markov Switching Autoregressive Model . . . . . . . . . . . . . . . . 20
2.4.1 ModelDefinition............................ 20
2.5 Markov Switching Dynamic Regression Models . . . . . . . . . . . . . . . . . 21
2.6 Markov Switching Conditional Heteroscedasticity . . . . . . . . . . . . . . . 22
2.6.1 Switching ARCH Models . . . . . . . . . . . . . . . . . . . . . . . 23
List of figures
3.1 Real GDP growth rate GARCH(1,1) Model: Conditional Variance . . . . . . 36
3.2 Markov switching heteroskedasticity model Variance: India . . . . . . . . . . . 37
3.3 Markov switching heteroskedasticity model Variance: Pakistan . . . . . . . . 38
3.4 Markov switching heteroskedasticity model Variance: USA . . . . . . . . . . 38
3.5 Markov switching heteroskedasticity model Variance: Japan . . . . . . . . . 39
4.1 Markov switching model: Filtered probability India . . . . . . . . . . . . . . . 51
4.2 Markov switching model: Filtered probability Germany . . . . . . . . . . . . 52
4.3 Markov switching model: Filtered probability UK . . . . . . . . . . . . . . . 52
4.4 Markov switching model: Filtered probability Pakistan . . . . . . . . . . . . 53
4.5 Markov switching model: Filtered probability USA . . . . . . . . . . . . . . 53
4.6 Markov switching model: Filtered probability China . . . . . . . . . . . . . 54
5.1 Markov switching model Filtered probability Pakistan Real Interest Rate . . . 63
5.2 Markov switching model: Pakistan Real Interest Rate Actual versus Fitted Plot 64
List of tables
3.1 DescriptiveStatistics ............................... 31
3.2 Real GDP growth rate: GARCH(1,1) Model Estimation . . . . . . . . . . . . 34
3.3 Diagnostic Tests: GARCH(1,1) Model Estimation . . . . . . . . . . . . . . . 35
3.4 Real GDP growth rate: Markov Switching Heteroskedasticity Model estimation 36
4.1 Descriptive Statistics: Stock Market Return . . . . . . . . . . . . . . . . . . 46
4.2 Correlation of Stock Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Stock return: Markov switching heteroscedasticity model . . . . . . . . . . . 48
4.4 Estimated Probability of remaining in the same State . . . . . . . . . . . . . 49
4.5
Concordance Correlation Coefficient between probabilities of High volatility state
49
4.6
Concordance Correlation Coefficient between probabilities of Low volatility state
50
5.1 Estimates: Real Interest Rate Linear Model and Markov Switching Model . . . 61
5.2 Estimates: Real Interest Rate Transition Probabilities and Expected Duration 62
A.1 Chapter 3: Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . 73
A.2 Chapter 5: Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . 73
A.3 Chapter 5: Real Interest Rate Unit Root Test . . . . . . . . . . . . . . . . . . 74
Nomenclature
Acronyms / Abbreviations
BiNom Binomial Distribution
EGARCH Exponential GARCH
TGARCH Threshold GARCH
AR Autoregressive
ARCH Autoregressive Conditional Heteroskedasticity
ARMA Autoregressive–Moving-Average
BSE Bombay Stock Exchange Index
CAPM Capital Asset Pricing Model
SSE Shanghai Stock Exchange
EM Expectation–maximization
FTSE Financial Times Stock Exchange Index
GARCH Generalized Autoregressive Conditional Heteroskedasticity
GARCH M GARCH-in-mean
HMM Hidden Markov models
IID Independent and identically distributed
JB Jarque Bera statistic
xviii Nomenclature
KSE Karachi Stock Exchange Index
LB Ljung-Box statistic
MCMC Markov chain Monte Carlo
MLE Maximum-Likelihood Estimation
MSAR Markov Switching Autoregressive
MSM Markov Switching Model
NYSE New York Stock Exchange Index
RSV Regime Switching Stochastic Volatility
SSM State Space Method
SWARCH Markov Switching Autoregressive Conditional Heteroskedasticity
VAR Vector Autoregression
Chapter 1
Introduction
The canonical a probabilistic model for temporal or sequential data is what’s called a Markov
model and Markov models are named after Andrey Markov who was a Russian mathematician
who did some work in stochastic processes in the late 1800 and early nineteen hundreds. The
idea behind a Markov model in exploits a very deep intrinsic fact about the real world; that
is the future is independent of the past given the present. In easier terms, if you know the
exact state of the world right now and you want to utilize that knowledge to predict the future
then knowing something about the past of the world 1 second ago will not be helping you in
predicting the future because you know everything that you want to know is already encoded in
the current state of the world.
1.1 Markov chains
Markov chain is a stochastic process defined as set of states
{St,t=0,1, ...}
using finite integer
values denoted by i,j. The probability of any future state value
St+1
is equivalent to j, therefore
the conditional probability distribution of any future state value
St+1
, given the past states
S0,S1, ..., St−1
and the present state
St
is only dependent on the present state and independent of
the past states.
P{St+1=j|St=it,St−1=it−1,S1=i1,S0=i0}=P{St+1=j|St=it}=pi j (1.1)
pi j
is the transition probability from i to j, that is the probability of the next state will be j with
immediately preceding state is i. Assuming there are N states, thena transition matrix can be
2Introduction
used to describe all the transitions.
M=
p11 p12 . . . p1N
p21 p22 . . . p2N
. . . . . .
. . . . . .
pN1pN2. . . pN N
(1.2)
The transition probability
pi j
is non-negative, every state including the current state must transit
into some state such that: N
∑
j=1
pi j =1,i=1,2, ..., N(1.3)
The above matrix signifies one-step transition probabilities. The one step case can be extended
to n-step transitions which is a result of multiple one-step transitions. A two-step transition
probability can described as
PSt+2=j|St=i
that is the summation of transition probabilities
from state i to all states and then similarly from all the states into state j, that is:
N
∑
k=1
P{St+2=j|St+1=k}P{St+1=k|St=i}
In general, the n-step transition probability matrix can be written:
P{St+n=j|St=i}=pn
i j (1.4)
The multi-step transition probabilities can be calculated using the Chapman-Kolmogorov
equation:
pm+n
i j =
N
∑
k=1
pn
ik pm
k j,i,j=1,2, ...N(1.5)
1.2 Estimation
The Markov chain process can be estimated through filtering by taking the joint conditional
probabilities of the future states as a joint conditional probabilities function of current states
and the transition probabilities. It is expressed as filtering because the conditional probabilities
of current states as input, that go through the procedure of filtering or system of dynamic
transformation that is transition probability matrix to derive the output in the form of conditional
1.2 Estimation 3
probabilities of the future states. Suppose, there is a simple two state Markov chain:
yt=µ1S1+µ2S2+εt(1.6)
where
S1=1
in state 1 and 0 otherwise,
S2=1
in state 2 and 0 otherwise, and the white noise
residual is
εt
. The joint probability of
yt
and
St
that transits over time is need to be calculated.
It can be derived in two steps. First, the conditional probability of being in state
st
, given
information available at time t-1 that is PSt=st|yt−1.
PSt=st|yt−1=PSt=st|St−1=st−1×PSt−1=st−1|yt−1(1.7)
Second, the probability of being in state
st
, given information available at time t
PSt=st|yt
,
hence calculating the joint probability distribution of Stand ytthat is:
fyt,St=st|yt−1
=fyt|St=st,yt−1×PSt=st|yt−1
=fyt|St=st,yt−1×PSt=st|St−1=st−1×PSt−1=st−1|yt−1
(1.8)
Then the joint probability of ytand Stcan be calculated as:
PSt=st|yt=fyt,St=st|yt−1
fyt|yt−1(1.9)
Let
yt
be N state Markov chain process, with autoregression of order
r
and residual
εt
and is
a function of exogenous variable
xt
and its lags known as dynamic process of autoregression.
When
yt
is
r
order autoregression, then joint conditional probability of the current state and
r
previous states, given information including all lag values of rperiods before 0; therefore:
PSt=st,St−1=st−1, ..., St−r=st−r|Ωt−1(1.10)
is being considered, with
Ωt−1=yt−1,yt−2,..., yt−r,xt−1,xt−2, ..., xt−r
is the available infor-
mation at time t-1. The filtering procedure is being used again to determine the updated version
of probability of being in state
st
, based on information available at time t-1, by deriving joint
4Introduction
density of Ytand St:
fyt,St=st,St−1=st−1, ..., St−r−1=st−r−1|Ωt−1
=fyt|St=st,St−1=st−1, ..., St−r−1=st−r−1,Ωt−1
×pSt=st,St−1=st−1, ..., St−r−1=st−r−1|Ωt−1
=fyt|St=st,St−1=st−1, ..., St−r−1=st−r−1,Ωt−1
×PSt=st|St−1=st−1)×PSt−1=st−1, ..., St−r−1=st−r−1|Ωt−1
(1.11)
Then the probability density function of ytis calculated:
fyt|Ωt−1
=
N
∑
st=1
N
∑
st−1=1
...
N
∑
st−r=1
fyt,St=st,St−1=st−1, ..., St−r−1=st−r−1|Ωt−1(1.12)
It is not yet the filter output, hence calculating the non-serial correlation residual:
PSt=st,St−1=st−1, ..., St−r−1=st−r−1|Ωt
=fyt,St=st,St−1=st−1, ..., St−r−1=st−r−1|Ωt−1
fyt|Ωt−1
(1.13)
Hence the output of the filter is:
PSt=st,St−1=st−1, ..., St−r=st−r|Ωt
N
∑
st−r−1=1
PSt=st,St−1=st−1, ..., St−r−1=st−r−1|Ωt(1.14)
Above is the probability of the states at time t, and based on current information it can be
obtained as:
PSt=st|Ωt=
N
∑
st=1
N
∑
st−1=1
...
N
∑
st−r=1
PSt=st,St−1=st−1, ..., St−r=st−r|Ωt(1.15)
Similarly the likelihood function can be derived:
Lθ=
T
∑
t=1
fyt|Ωt−1;θ(1.16)
1.3 Smoothing 5
where
θ
is the vector of parameters. Techniques like Gibbs sampling and the EM algorithm can
be used to estimate log likelihood function however maximum likelihood is still a convenient
and most appropriate method. Based on the assumption of two state Markov chain process,
with a normally distributed residual then the maximum likelihood function can be extended as
follows:
Lθ=
T
∑
t=1
fyt|Ωt−1;θ
=
T
∑
t=1
2
∑
st=1
fyt|St=st,yt−1;θ×PSt=st|St−1=st−1
=
T
∑
t=1
2
∑
st=1
2
∑
st−1=1fyt|St=st,yt−1;θ×PSt=st|St−1=st−1
×PSt−1=st−1|yt−1
=
T
∑
t=11
√2πσε
exp−yt−µ12
2σ2
ε×p11 ×P
tL 1+p21 ×P
tL 2
+1
√2πσε
exp−yt−µ12
2σ2
ε×p21 ×P
tL 1+p22 ×P
tL 2
(1.17)
where P
tL 1=PSt−1=1|yt−1and P
tL 2=PSt−1=2|yt−1.
1.3 Smoothing
Smoothing is a process of inferring about the present states, using future information. It is
the probability of the states at time t given current information available, therefore revising
PSt=st|Ωt
. Therefore,
PSt=st|ΩT
replacing
Ωt
by
ΩT
in the probability. In case there
are no lags in
yt
, then smoothing is carried out in two steps; otherwise three steps and including
one approximation when there are lags in yt.
6Introduction
For simplification setting St=stto St:
PSt−r, ..., St,St+1|ΩT
=PSt−r+1, ..., St,St+1|ΩT×PSt−r|St−r+1, ..., St,St+1,ΩT
=PSt−r+1, ..., St,St+1|ΩT×PSt−r|St−r+1, ..., St,St+1,Ωt
=PSt−r+1, ..., St,St+1|ΩT×PSt−r, ..., St+1,Ωt
PSt−r+1, ..., St,St+1|ΩT
=PSt−r+1, ..., St,St+1|ΩT×PSt−r, ..., St,Ωt×PSt+1|St
PSt−r+1, ..., St,St+1|Ωt
(1.18)
The second inequality is equivalent: PSt−r|St−r+1, ..., St,St+1,ΩT
=PSt−r|St−r+1, ..., St,St+1,Ωtiff
fyt+1,ΩTt |St−r,St−r+1, ...St,St+1,Ωt
=fyt+1,ΩTt |St−r+1, ...St,St+1,Ωt(1.19)
Because ΩT l =ΩT−Ωthence it follows:
PSt−r|St−r+1, ...St,St+1,ΩT=PSt−r|St−r+1,...St,St+1,Ωt,ΩT t
=fyt+1,St−rΩTt |St−r+1, ...St,St+1,Ωt
fyt+1,ΩTt |St−r+1, ...St,St+1,Ωt
=fyt+1,ΩTt |St−r, ...St,St+1,Ωt×PSt−r|St−r+1, ...St,St+1,Ωt
fyt+1,ΩTt |St−r+1, ...St,St+1,Ωt
(1.20)
Then summing it up over
St+1=1,2, ..., N
to get a smoothed states with residual with no serial
correlation is derived.
PSt−r, ..., St|ΩT=
N
∑
St+1=1
PSt−r, ..., St,St+1|ΩT(1.21)
The above equation of smoothed states is derived when no lagged
yt
exist however when there
are lags then similar equation is set by summation:
PSt|ΩT−
N
∑
st=1
N
∑
st−1=1
...
N
∑
st−r=1
PSt,St−1, ..., St−r|ΩT(1.22)
1.4 Time-varying transition probabilities 7
1.4 Time-varying transition probabilities
The extension of Markov chain process is addition of time-varying transition probabilities that
makes the model more flexible and can be defined as:
P{St+1=j|St=it|Ωt+1}=pi j(t+1)(1.23)
And the transition probability matrix is given as:
P(t) =
p11(t)p12(t). . . p1N(t)
p21(t)p22(t). . . p2N(t)
. . . . . .
. . . . . .
pN1(t)pN2(t). . . pN N (t)
(1.24)
The time varying transition probabilities are used in binary models in the form of logit and
probit models. The exponential function and cumulative distribution function are symmetric
therefore deviation from the mean value will increase the probability. The time varying functions
are similar to smoothed transition models, however the advantage of time varying transition
probabilities is within the interval
0,1
. The logit function of transition probabilities is:
pi j(t) = 1
1+exp{−Ωtβ′
i j}(1.25)
where
β′
i j
is the coefficient vector of the set of endogenous and exogenous variables. In simpler
terms, −Ωtβ′
i j =ωi j0−γi j yt−1then we have:
pi j(t) = 1
1+exp{ωi j 0−γi jyt−1
(1.26)
The above equation has direct relation and implications, for instance symmetric responses are
only related to amount of deviation from central or equilibrium values no matter what is the
direction; however in asymmetric effect direction or sign of deviation is important.
8Introduction
1.5 State Space Methods and Kalman Filter
The state space models are used in dynamic time series models containing unobserved variables.
The simplest SSM is first order vector autoregressive model with:
yt=Γyt−1+wt(1.27)
where the states
yt,i
is determined from the past states
yt−1,j
,
j=1,2, ..., p
for
i=1,2, ..., p
and time
t=1,2, ..., n
. In order to ensure completeness,
wt
are
p×1
Gaussian white noise
vectors with covariance
Q
. The initial value of the state process is
y0
taken from variables being
normally distributed with
µ0
mean vector and covariance matrix. Though the state vector is not
observed but after some observed transformations, the equation can be written as:
zt=Atyt+vt(1.28)
Next, estimates are derived for particular point in time using Gaussian assumptions on unob-
served state vector.Based, on the initial conditions with mean
y0|0=µ0
and covariance
P
0|0=∑0
at time t=1,2, ..., n.
yt|t−1=Γyt−1|t−1(1.29)
P
t|t−1=ΓP
t−1|t−1Γ′+Q(1.30)
yt|t=yt|t−1+Ktzt−Atyt|t−1(1.31)
Then the Kalmn gain matrix is given as:
Kt=P
t|t−1A′
tAtP
t|t−1A′+R−1(1.32)
and the covariance matrix P
t|tis given as:
P
t|t=I−KtAtP
t|t−1(1.33)
The innovation vector of the basic state vector equation can be defined as:
vt=zt−Atyt|t−1(1.34)
1.6 Hidden Markov Models 9
and its covariance as
∑
t
=AtP
t|t01A′
t+R(1.35)
This is called Kalman smoother with initial conditions
yT|T
and
P
T|T
. Then the smoother
algorithm can be explained as:
yt−1|T=yt−1|t−1+Jt−1yt|T−yt|t−1(1.36)
P
t−1|T=P
t−1|t−1+Jt−1P
t|T−P
t|t−1J′
t−1(1.37)
Jt−1=P′
t−1|t−1+Γ′P
t|t−1−1(1.38)
Therefore during the filtering process of the smoothing algorithm, values are generated as
yt|t
and
P
t|t
. Assuming parameters are known, however if parameter are required to be determined;
it can be done by maximizing innovation form of likelihood function. The likelihood function
can be expressed as:
logL=−1
2
T
∑
t=1
log |∑
t
Θ| −1
2
T
∑
t=1
v′
tΘ−1
∑
tΘvtΘ(1.39)
where
Θ
is the vector of unknown parameters to emphasize upon the dependence of log
likelihood function on the parameters of the model.
1.6 Hidden Markov Models
The basic concepts of Hidden Markov Models will be related to state space models, the simplest
model involving specific number of counts is Poisson process. Variables are independently
and identically distributed, hence the variance is equivalent to mean. We can generate specific
number of counts by using two different Poisson processes with different means, then one of the
means are selected by using a random process known as emission process. Given the emission
process is a Markov chains, then serial dependence along with over dispersion will be found
in the observed counts. The word hidden itself signifies that states cannot be only estimated
using the observed data, there also exist hidden or latent data following a Markov chain process.
Therefore in Hidden Markov Model unobserved sequence of states following Markov chain
with finite state space, hence the probability of any observation at any time t is dependent upon
the current state.
10 Introduction
The Hidden Markov Model that is a mixture of densities over time, can also be used as an
alternative to kalman filter process which is factor analysis over time. The Kalman filter
involves linear measurements including linear evolution of states whereas Hidden Markov
Model represents a nonlinear evolution and measurement of one state at a time. In Hidden
Markov Mode, the Viterbi algorithm to estimate the most likely state sequence while the Baum-
Welch forward-backward algorithm is used to estimate the parameters following Expectation
Maximization. In order to derive HMM we include the following:
• Hidden states: The states of a system that can be described by a Markov process,
• Observable states: The states of the process that are visible, i.e. measurable,
•
Pi-Vector: Contains the probability that the model is in one of the hidden states at the
initial time,
•
State transition matrix: Contains the probability that a hidden state will evolve to another
state, given the previous state,
•
Emission probability matrix: Contains the probability that a particular measurable state
can be observed, provided that the model is in one of the hidden states.
A hidden Markov model is a standard Markov process augmented by a set of measurable states
and several probabilistic relations between those states and the hidden states. In developing
the algorithms for the model, we write the joint probability over hidden
yt
and observed
xt
states, and then use the Markov property to simplify it. This property allows us to assume that
all information about the history of the states is summarized by the value of the state at the
previous time step. The total probability is,
pyT,xT=py1
T
∏
t=2
pyt|yt−1
T
∏
t=2
pxt|yt(1.40)
where
yT
denotes
y1,y2, ..., yT
. Following three matrices are used in characterization of
HMM:
Ai,j=pyt=i|yt−1=j
Bt,i=pxt|yt=i
πt,i=pyt=i
(1.41)
1.7 Hidden Markov Model Parameter Estimation 11
These three equations define a HMM, hence referred as a set
λ={A,B,π}
; further we also
require following definitions:
γt,i=pyt=i|xT(1.42)
Ωt,i j =pyt=i,yt−1=j|xT(1.43)
αt,i=pyt=i|xt(1.44)
κt,i=pxt|xt−1(1.45)
βt,i=pxt+1, ..., XT|yt=i
pxt+1, ..., XT|xt(1.46)
1.7 Hidden Markov Model Parameter Estimation
The likelihood function over N iterations is used to determine the parameters:
Q=
N
∑
n=1ZdyTpyT|xT,nlogpy1
T
∏
t=2
pyt|yt−1
T
∏
t=1
pxn
t|yt
=∑
nZdy1py1|xT,nlog py1
+∑
n
T
∑
t=2Zdytdyt−1pyt,yt−1|xn
tlog pyt|yt−1
+∑
n
T
∑
t=1Zdytpyt|xn
tlog pxn
t|yt
=∑
n
∑
t
γn
1,ilogπ1,i+∑
n
T
∑
t=2
∑
i,j
Ωn
t,i j log Ai,j+∑
n
T
∑
t=1
∑
i
γn
t,ilogBt,i
(1.47)
Baum-Welch Algorithm (1970) is used to maximize the likelihood function simplified above,
since hidden states are known. The result will be equivalent to Expectation Maximization
Algorithm.The above equation will be solved by adding two constraints:
∑
i
π1,i=1 (1.48)
∑
i
Ai,j=1∀j(1.49)
12 Introduction
Finally, using Lagrange multiplier the updated objective function is:
L=Q−λπ∑
i
π1,i−1−∑
j
λjAi,j−1(1.50)
In order to maximize, the derivative of the objective function is calculated and equated to zero:
∂L
∂ π1,i
=∑
n
γn
1,i
π1,i−λπ=0⇒πnew
1,i=λπ
N
N
∑
n=1
γn
1,i(1.51)
Thus Lagrange coefficient is derived from constraint relation as:
πnew
1,i=1
N
N
∑
n=1
γn
1,i(1.52)
Similarly for matrix A:
∂L
∂Ai,j
=∑
n
T
∑
t=2
Ωn
t,i j
Ai,j−λj=0 (1.53)
Anew
i,j=λj
NT−1
N
∑
n=1
T
∑
t=2
Ωn
t,i j
=∑N
n=1∑T
t=2Ωn
t,i j
∑N
n=1∑T
t=2γn
t−1,j
(1.54)
where
Ωn
t,i j =γn
t−1,j
representing the ratio of expected number of transitions from state j to state
i, divided by number of transitions from state j.
Next, it is determination of emission probabilities that can expressed as:
pxt|yt=i;θ=Bt,iθ(1.55)
where θcan be termed as:
θnew =argmax
θ
N
∑
n=1
T
∑
t=1
∑
i
γn
t,ilogBt,iθ(1.56)
Here we are only concerned with discrete output, therefore parameters are probability masses
for k states of the output, xt:
Bk,i=pxt=k|yt=i(1.57)
1.7 Hidden Markov Model Parameter Estimation 13
Assuming time is independent, and must satisfy the constraint:
∑
k
Bk,i=1∀i(1.58)
Similarly, we again use Lagrange multiplier to show that:
Bnew
k,i=∑n,ts.t.sn
t=kγn
t,i
∑N
n=1∑T
t=1γn
t,i
(1.59)
The matrix B is the expected number of times the system is in state i, there is k that is divided
by the expected number of times the system is in state i; the maximization algorithm is now
completed.
After maximization step of the algorithm, quantities
Ωt,i j
and
γt,i
are computed. It is achieved
by computing αt,i,κt,βt,ias follow:
αt=pyt|xt=∑yt−1pxt|ytpyt|yt−1pyt−1|xt−1
pxt|xt−1(1.60)
αt,i=∑jBt,iAi,jαt−1,j
κt
(1.61)
Therefore the forward variable αis obtained:
αt=py1|x1=px1|y1py1
∑y1px1|y1py1(1.62)
α1,i=B1,iπ1,i
∑jB1,jπ1,j
(1.63)
Then for calculating κ:
κt=pxt|xt−1=∑
yt
∑
yt−1
pxt|ytpyt|yt−1pyt−1|xt−1(1.64)
κt=∑
i,j
Bt,iAi,jαt−1,j(1.65)
14 Introduction
and the initial value is:
κ1=px1
=∑
y1
px1|y1py1(1.66)
Now through backward recursions, we initiate as follows:
βt−1=pxt, ..., xT|yt−1
pxt, ..., xT|xt−1
=∑ytpxt, ..., xT,yt|yt−1
pxt|xt−1pxt+1,..., xT|xt
=∑ytpxt, ..., xT|ytpyt|yt−1
pxt|xt−1pxt+1,..., xT|xt
=∑ytpxt|ytpxt+1, ..., xT|ytpyt|yt−1
pxt|xt−1pxt+1,..., xT|xt
(1.67)
Then
βt−1,j=∑iBt,iβt,iAi,j
κt
(1.68)
Afterwards, this recursion is started, that is:
βT−1,j=pxT|yT−1
pxT|xT−1
=∑yTpxT|yTpyT|yT−1
κT
(1.69)
Similarly, for initializing:
βT−1,j=∑iBT,jAi,j
κT
(1.70)
βT,j=1 (1.71)
1.7 Hidden Markov Model Parameter Estimation 15
The recursions of γtand Ωtcan be calculated using the values of αt,κt,βt:
γt=pyt|xt
=pyt,xT
pxT
=pxt+1, ..., xT,yt|xtpxt
pxt+1, ..., xT|xtpxt
=pyt|xtpxt+1,..., xT|xt
pxt+1, ..., xT|xt
(1.72)
Hence γt,iis attained:
γt,i=αt,iβt,i(1.73)
And the final recursion:
Ωt=pyt,yt−1|xT
=pxt, ..., xT|ytpyt|yt−1pyt−1|xt−1
pxt|xt−1pxt+1,..., xT|xt−1
(1.74)
Finally, the expectation step is derived:
Ωt,i j =Bt,iβt,iAi,jαt−1,j
κt
(1.75)
If the convergence is achieved, then EM-algorithm is an alternate to M-step and E-step.
Chapter 2
Markov Switching Models
The chapter starts with the definition of a finite Markov mixture distribution, whose properties
are studied in some detail and introduces the basic Markov switching model and deals with its
extensions.
Finite mixture models are extended to deal with time series data that exhibit dependence over
time. Broadly speaking, this is achieved by substituting the discrete latent indicator
Si
introduced
as an allocation variable for finite mixture models by a hidden Markov chain. This leads to
a surprisingly rich class of nonlinear time series models that solve a variety of interesting
problems in applied time series analysis.
2.1 Finite Markov Mixture Distributions
Let
yt=1, ..., T
denote a time series of T univariate observations taking values in a sampling
space
y
which may be either discrete or continuous. As common in time series analysis,
yt=1, ..., Tis considered to be the realization of a stochastic process {Yt}T
t=1.
It is assumed that the probability distribution of the stochastic process
Yt
depends on the
realizations of a hidden discrete stochastic process
St
. The stochastic process
Yt
is directly
observable, whereas
St
is a latent random process that is observable only indirectly through
the effect it has on the realizations of
St
. A simple example is the hidden Markov chain model
Yt=µSt+εtwhere εtis a zero mean white noise process with variance σ2
18 Markov Switching Models
2.2 Statistical Modeling Based on Finite Markov Mixture
Distributions
Researchers have found Markov mixture models increasingly useful in applied time series
analysis.
2.2.1 The Basic Markov Switching Model
Assume that a time series
y1, ..., yT
is observed as a single realization of a stochastic process
Y1,..., YT
. In the basic Markov switching model the time series
y1, ..., yT
is assumed to be a
realization of a stochastic process
Yt
generated by a finite Markov mixture from a specific
distribution family:
Yt|St∼TθSt(2.1)
where
St
is a hidden K state Markov chain, where
Yt
satisfies the assumption; Conditional
on knowing
S= (S0, ..., ST)
, the random variables
Y1,..., YT
are stochastically independent.
For each
t≥1
, the distribution of
Yt
arises from one out of K distributions T
θ1, ...,
T
θK
,
depending on the state of St
Yt|St=k∼Tθk(2.2)
The basic Markov switching model found widespread applications in many practical areas
including bioinformatics, biology, economics, finance, hydrology, marketing, medicine, and
speech recognition. Various terminology became usual to denote models based on hidden
Markov chains. Because one may choose Markov mixtures of any discrete distribution, it is
possible to model many different types of discrete valued time series data, for example, binary
time series by
PYt=1|St=πSt(2.3)
time series of bounded counts by a Markov mixture of binomial distributions,
Yt|St∼BiNomnt,πSt(2.4)
or time series of unbounded counts by a Markov mixture of Poisson distributions,
Yt|St∼PµSt(2.5)
2.3 Nonlinear Time Series Analysis Based on Markov Switching Models 19
An important feature of applying Markov mixture models to discrete-valued time series is the
ease with which autocorrelation is introduced, and the properties of the marginal distribution
are easily analyzed. The basic Markov switching model has been generalized in several ways as
outlined in the following sections.
2.2.2 The Markov Switching Regression Model
An early attempt at introducing Markov switching models into econometrics in order to deal
with time series data that depends on exogenous variables is the switching regression model of
Goldfeld and Quandt (1973), which extends the switching regression model (Quandt, 1972)
described earlier in subsection 2.2.1. Whereas Quandt (1972) assumes that
St
is an i.i.d. random
sequence, Goldfeld and Quandt (1973) allow explicitly for dependence between the states by
modeling St as a two-state hidden Markov chain. The general Markov switching regression
mode is:
Yt=xtβSt+εt,εt∼N0,σ2
ε,St(2.6)
where
St
is a hidden Markov chain and
xt
is a row vector of explanatory variables including
the constant.For discrete-valued explanatory variables, the Markov switching regression model
will suffer from the same identifiability problems as the standard finite mixture of regression
models.
2.3 Nonlinear Time Series Analysis Based on Markov Switch-
ing Models
In practical time series analysis, an important aspect is properties of the marginal distribution of
Yt
as well as properties of the one-step ahead predictive density
pyt|yt−1,ϑ
, implied by the
chosen time series model. Typical stylized facts of the marginal distribution of practical time
series are asymmetry and nonnormality with rather fat tails, and autocorrelation not only in the
level Ytbut also in the squared process Y2
t.
It is well known that standard ARMA models (Box and Jenkins, 1970) often are not able to
capture stylized facts of practical time series. Some unrealistic features of ARMA models based
on normal errors are normality of the predictive as well as the marginal density, linearity of the
expectation EYt|yt−1,ϑand homoscedasticity of VarEYt|yt−1,ϑ.
This chapter discusses Markov switching models that constitute another very flexible class of
20 Markov Switching Models
nonlinear time series models and are able to capture many features of practical time series by
appropriate modifications of the basic Markov switching model introduced in subsection 2.2.1.
Section 2.4 deals with the Markov switching autoregressive model and Section 2.5 considers
the related Markov switching dynamic regression model and later highlights that Markov
switching models give rise to very flexible predictive distributions. Section 2.6 deals with
Markov switching conditional heteroscedasticity and switching ARCH models are introduced.
Later the section studies further extensions, namely hidden Markov chains with time-varying
transition probabilities and hidden Markov models for longitudinal data and multivariate time
series.
2.4 The Markov Switching Autoregressive Model
Markov mixture model introduces autocorrelation in the process
Yt
even for the basic Markov
switching model, where conditionally on knowing the states, the process
Yt
is uncorrelated. In
this section the Markov switching autoregressive model is introduced that deals with autocorre-
lation in a more flexible way than the basic Markov switching model.
2.4.1 Model Definition
The standard model to capture autocorrelation is the AR(p) model,
Yt−µ=δ1Yt−1−µ+... +δpYt−p−µ+εt(2.7)
where εt∼N0,σ2
εwhich is equivalent to model:
Yt=δ1Yt−1+... +δpYt−p+ζ+εt(2.8)
where
ζ=µ1−δ1−... −δp
. An important extension of the basic Markov switching model
is the Markov switching autoregressive (MSAR) model, where a hidden Markov chain is
introduced into model and through the work of Hamilton (1989) who allowed for a random
shift in the mean level of process (2.7) through a two-state hidden Markov chain:
Yt−µSt=δ1Yt−1−µSt−1+... +δpYt−p−µSt−p+εt(2.9)
2.5 Markov Switching Dynamic Regression Models 21
An important alternative to model (2.9) was suggested by McCulloch and Tsay (1994b), who
introduced the hidden Markov chain into (2.8) rather than into (2.7), by assuming that the
intercept is driven by the hidden Markov chain rather than the mean level:
Yt=δ1Yt−1+... +δpYt−p+ζSt+εt(2.10)
Although the parameterization (2.7) and (2.8) are equivalent for the standard AR model, a model
with a Markov switching intercept turns out to be different from a model with a Markov switch-
ing mean level. In (2.9), after a one-time change from
St−1
to
St̸=S+t−1
, an immediate mean
level shift from
µSt−1
to
µSt
occurs. Both models violates assumption;
Yt|St=k∼
T
θk
as the
one-step ahead predictive density
pyt|yt−1,St,ϑ
depends upon past values
yt−1
. For a model
with switching mean level it is evident from (2.9) that the predictive density pyt|yt−1,St,ϑ
depends not only on
St
but also on the past values
St−1, ..., St−p
of the hidden Markov chain
fulfilling only assumption;
pyt|yt−1,St,ϑ=pyt|yt−1,St, ..., St−p,ϑ
. On the other hand
for a model with switching intercept the predictive density
pyt|yt−1,St,ϑ
depends only
on
St
and such a process fulfills the stronger condition;
pyt|yt−1,St,ϑ=pyt|yt−1,St,ϑ
.
As a result, econometric inference for an MSAR model with switching intercept is not more
complicated than for the basic Markov switching model, whereas for an MSAR model with
switching mean inference on the hidden Markov chain St is far more involved.
The assumption that the autoregressive parameters switch between the two states implies differ-
ent dynamic patterns in the various states, and introduces asymmetry over time. Asymmetry
over time between the states is introduced also through the hidden Markov chain as different
persistence probabilities imply different state duration. Subsequently the notation MS(K)-AR(p)
is used occasionally to denote a Markov switching autoregressive model with K states and
autoregressive order p. A more subtle notation that also differentiates between homo- and
heteroscedastic variances, switching in the mean level or in the intercept as well as between
invariant and switching autoregressive parameters is introduced in Krolzig (1997).
2.5 Markov Switching Dynamic Regression Models
An important extension both of Markov switching autoregressive models and the Markov switch-
ing regression model, discussed in section 2.5, is the Markov switching dynamic regression
22 Markov Switching Models
model.
Yt−µSt−ztβ=δ1Yt−1−µSt−1−zt −1β+... +δpYt−p−µSt−p−zt−pβ+εt(2.11)
where the regression coefficient
β
is considered to be unaffected by
St
. In the following
dynamic regression model all parameters, including the regression coefficient
β
, are affected by
endogenous regime shifts following a hidden Markov chain:
Yt=δSt,1Yt−1+... +δSt,pYt−p+ztβSt+ζSt+εt(2.12)
For estimation it is useful to view this model as a Markov switching regression model as in
Subsection 2.2.1, without distinguishing between endogenous variables, exogenous variables,
and the intercept:
Yt=xtβSt+εt(2.13)
where
xt=yt−1...yt−pzt1...ztd 1
. In the mixed-effects Markov switching dynamic regression
model only certain elements of the parameter
βSt
in (2.13) actually depend on the state of the
hidden Markov chain, and others are state independent (McCulloch and Tsay, 1994b):
Yt=xf
tα+xr
tβSt+εt(2.14)
where
xf
t
are those columns of
xt
that correspond to the state-independent parameters
α
whereas
the columns of xr
tcorrespond to the state-dependent parameters.
2.6 Markov Switching Conditional Heteroscedasticity
Markov switching models are often used by researchers to account for specific features of
financial time series such as asymmetries, fat tails, and volatility clusters.
To deal with skewness and excess kurtosis in the unconditional distribution of daily stock returns
standard finite mixtures of normal distributions have been applied quite frequently (Fama, 1965;
Granger and Orr, 1972; Kon, 1984; Tucker, 1992). Such a modeling approach, however, is
appropriate for time series data only if the processes
Yt
and
Y2
t
do not exhibit autocorrelation;
Volatility clustering implies persistence of states of high volatility and leads to the rejection of
standard time series models in favor of time series models that allow the conditional variance
2.6 Markov Switching Conditional Heteroscedasticity 23
Varyt|yt−1,ϑ
to depend on the history
yt−1,yt−2
,... of the observed process such as the
autoregressive conditionally heteroscedastic (ARCH) model.
2.6.1 Switching ARCH Models
A simple model to capture volatility clusters in financial time series is the ARCH model (Engle,
1982) which may be written as
Yt=σtεtεt∼N0,1
σ2
t=γt+α1Y2
t−1+... +αmY2
t−m
(2.15)
with γt=γ. Alternately,
Yt=√γthtεt
h2
t=1+α1
γt−1
Y2
t−1+... +αm
γt−m
Y2
t−m
(2.16)
The two parameterizations are equivalent if
γt=γ
, however they generate different processed if
γt
is time dependent. The switching ARCH model results by allowing time dependence of
γt
through a hidden K-state Markov chain
γt=γSt
. Such a switching parameter was introduced by
Hamilton and Susmel (1994) into parameterization (2.17):
Yt=√γsthtεt
h2
t=1+α1
γst−1
Y2
t−1+... +αm
γSt−m
Y2
t−m
(2.17)
whereas Cai (1994) introduced a two-state and Kaufmann and Fr
¨
uhwirthSchnatter (2002) a
K-state switching parameter into parameterization (2.16):
Yt=σtεt
σ2
t=γSt+α1Y2
t−1+... +αmY2
t−m
(2.18)
Gray (1996) introduced switching into all coefficients of the ARCH process, represented by
(2.16):
Yt=σtεt
σ2
t=γSt+αst,1Y2
t−1+... +αst,mY2
t−m
(2.19)
24 Markov Switching Models
The switching ARCH model may be combined with a Markov switching autoregressive model
for the mean equation that includes the same hidden Markov chain (Gray, 1996):
Yt=ζSt+δSt,1Yt−1+ut
ut=σtεt,εt∼N0,1
σ2
t=γSt+αSt,1u2
t−1+... +αSt,mu2
t−m
(2.20)
The switching ARCH model has been extended by including a leverage effect into the ARCH
specification (Hamilton and Susmel, 1994; Kaufmann and Fr
¨
uhwirth-Schnatter, 2002) to deal
with asymmetries in the marginal distribution:
Yt=σtεt,εt∼N0,1
σ2
t=γSt+α1y2
t−1+... +αmy2
t−m+ρdt−1y2
t−1
(2.21)
where dt=1 if yt≤0.dt=0 if yt>0 and ρ>0.
2.6.2 Switching GARCH Models
Francq et al. (2001) consider the following switching GARCH(m, n) model, where all coeffi-
cients are switching,
Yt=σtεt,εt∼N0,1
σ2
t=γSt+αSt,1y2
t−1+... +αSt,my2
t−m+δSt,1σ2
t−1+... +δSt,nσ2
t−n
(2.22)
By recursive substitution it becomes evident that the predictive density
pyt|yt−1,ϑ
depends
on the whole history of
St
. For the switching GARCH(1, 1) model, for instance, the variance of
the predictive density reads:
σ2
t=γSt+αSt,1y2
t−1+δSt,1γSt−1+αSt−1,1y2
t−2+δSt,1γSt−1γSt−2+αSt−2,1y2
t−3+...
Thus the model obeys only the weakest assumption that the observation density
pyt|yt−1,St,ϑ
depends on yt−1and all past states of St.
2.6.3 Time-Varying Transition Matrices
Whereas the transition matrix ζof the hidden process Stis time invariant under assumptions;
2.6 Markov Switching Conditional Heteroscedasticity 25
•St
is an irreducible, aperiodic Markov chain starting from its ergodic distribution
η=
η1,η2, ..., ηK:
PS0=k|ζ=ηk
The stochastic properties of
St
are sufficiently described by the (K
×
K) transition matrix
ζ
, where each element
ζjk
of
ζ
is equal to the transition probability from state j to state k:
ζjk =PSt=k|St−1=j,∀j,k∈ {1, ..., K}
•St
is a first-order homogeneous Markov chain with arbitrary transition matrix
ζ
, which
need not be irreducible or aperiodic, and starts from an arbitrary distribution
p0=
p0,1, ..., p0,Kthen:
p0,k=pS0=k
For a two-state Markov switching model, the transition probabilities ζSt−1,Stmay be reparame-
terized through a logit model in the following way:
ζSt−1,St=expκSt−1,1
1+expκSt−1,1,St̸=St−1
A univariate exogenous variable ztmay then be included:
ζSt−1,St=expκSt−1,1+ztκSt−1,2
1+expκSt−1,1+ztκSt−1,2,St̸=St−1(2.23)
Note that the transition probability
ζSt−1,St
not only depends on
zt
, but also on the state of
St−1
.
The logit transform could be substituted by another increasing function F(·),
ζSt−1,St=FκSt−1,1+ztκSt−1,2,St̸=St−1(2.24)
A model with time-varying transition matrices may be estimated through the EM algorithm
(Diebold et al., 1994) or through MCMC methods (Filardo and Gordon, 1998).
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Appendix A
Table A.1 Chapter 3: Correlation Analysis
India JAPAN USA Pakistan
India 1 -0.13006 0.145842 -0.63359
JAPAN 1 0.021093 0.154037
USA 1 -0.2651
Pakistan 1
Table A.2 Chapter 5: Descriptive Statistics
Real Interest Rate
Mean -0.406993
Median 0.571534
Maximum 12.126104
Minimum -30.046879
Std. Dev. 5.350612
Skewness -2.088641
Kurtosis 10.321620
Jarque-Bera 1850.411571
Probability 0.000000
74
Table A.3 Chapter 5: Real Interest Rate Unit Root Test
Unit Root Test t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -4.40837 0.00030
Phillips-Perron test statistic -5.55596 0.00000
*MacKinnon (1996) one-sided p-values.
