（ 251 ）Note

A Note on the Stock Market Trend Analysis Using
Markov-Switching EGARCH Models
MITSUI, Hidetoshi*

1

Introduction

1）

The stock marketʼs fluctuations follow a continuous trend whether or not is still a topic with mixed
opinions. However, if they do, it may be possible to discern fluctuation trends, that is, identify bull
markets and bear markets, using a statistical model. Many methods have been applied to thisproblem. In particular, there are many prior studies based on the Markov-switching model, including
Maheu and McCurdy（2000）. Maheu and McCurdy（2000）suggested identifying trend changes byswitching means of time-series model according to Markov process state variables. This model uses
an extension of the Markov-switching ARCH model, suggested by Hamilton and Susmel（1994）andCai（1994）, for volatility variation. The Markov-switching model is frequently used to formularize
volatility. Other representative works include the Markov-switching GARCH model by Gray（1996）,
Klaassen（2002）and Haas et al.（2004）, and the Markov-switching stochastic volatility model by So
et al.（1998）.
Trend analysis on asset prices using Markov-switching models normally classify trends into two
regimes – bear markets and bull markets. The mean of the rate of variation in stock prices is eithera negative or positive value. When the value continues to be positive there is an upward trend（bullmarket）, and when the value continues to be negative there is a downward trend（bear market）.
The assumption is that the two states follow a Markov process. However, Maheu et al.（2009）statedthat investors distinguish more detailed trends including temporary trends, and suggested a 4-state
Markov-switching model identifying a bear negative, bear positive, bull negative and bull positive
growth. Therefore, this study suggests a 4-state Markov-switching EGARCH model, which combinesthe 4-state Markov-switching model and an Exponential GARCH（EGARCH）model.
The reason why an EGARCH model is used for formularizing volatility is because there is anasymmetry in stock market volatility. That is, volatility is higher the day after stock prices fall comparedto the day after stock prices rise. Furthermore, because volatility involves very high and very low
states that cannot be expressed with an EGARCH model, the model switches volatility variations
also. However, because it is not always the case that volatility switching occurs at the same time as＊

Associate Professor, College of Economics, Nihon University, E-mail: mitsui.hidetoshi@nihon-u.ac.jp− 111 −KEIZAI SHUSHI（ 252 ）

Vol. 83

No. ３

trends change（from bull to bear, or from bear to bull）, the model accommodates expected rate of
return and volatility switching separately, with two state variables that follow a Markov process. In
this note, this model shall be called a Markov-switching EGARCH（MSEGARCH）model.
Satoyoshi and Mitsui（2011）used the MS-EGARCH model to analyze the Nikkei 225 options
market and showed that the MS-EGARCH model is effective for identifying bull and bear markets of
the Nikkei Stock Average, which are underlying assets. Therefore, this study extends the
MSEGARCH model to analyze bull and bear markets with more detail. First, the asymmetry ofvolatility is studied to see if it differs in bull and bear markets. Next, we see whether the transition
probability of bull and bear markets are influenced by the rate of return on the previous term, inother words, whether a price growth in a bull market increases the probability of a continued bull
market, and whether a price fall in a bear market increases the probability of a continued bearmarket. The transition probabilities of state variables are considered time-varying transition
probabilities. According to the above, this study suggests a model that allows trend analysis using a
Markov-switching model.
The brief descriptions of the following chapters are as follows: Chapter 2 describes the 4-state
Markov-switching model, the 4-state Markov-switching EGARCH model, the asymmetry model andthe time-varying transition probability model. Chapter 3 explains the trend analysis model. Chapter 4
contains the conclusion.
2
2.1

Model

4-state Markov-Switching ModelIn this chapter, the author will explain briefly the 4-state Markov-switching model by Maheu et al.
（2009）. When Rt is the rate of return on asset price at time t, the process of Rt can be expressed asfollows:
Rt = μ + σzt, zt ∼ i.i.d., E［zt］= 0, V［zt］= 1,

（2.1）

μ = μ1Δ1t + μ2Δ2t + μ3Δ3t + μ4Δ4t,

（2.2）

σ = σ1Δ1t + σ2Δ2t + σ3Δ3t + σ4Δ4t,

（2.3）

pij = Pr
［Δt = j | Δt−1 = i］
, i, j = 1, 2, 3, 4.

（2.4）

Here, the constant term μ is the mean, zt is the error term, and the assumption is that there is no
autocorrelation in returns. i.i.d. expresses an independently and identically distributed. Δt is a statevariable that follows a first-order Markov process.（Δ1t,Δ2t,Δ3t,Δ4t）is（1, 0, 0, 0）when Δt = 1,（0, 1,
0, 0）when Δt = 2,（0, 0, 1, 0）when Δt = 3 and（0, 0, 0, 1）when Δt = 4. Also, the assumption is thatmean μ and volatility σ follow Δt and switch simultaneously. The following restrictions apply to μ.
μ1 < 0 · · · bear negative growth,
μ2 > 0 · · · bear positive growth,− 112 −

A Note on the Stock Market Trend Analysis Using Markov-Switching EGARCH Models（MITSUI）（ 253 ）

μ3 < 0 · · · bull negative growth,
μ4 > 0 · · · bull positive growth.
A bear positive growth is a temporary rise during a long-term downward trend. A bull negative
growth is a temporary fall during a long-term upward trend. The transition matrix of Δt is:

		

P＝

p11
p12

p21
p22

0
0

p41
0

0
p14

0
0

p33
p34

p43
p44

.

（2.5）

Here, p14 is the probability of a transition from bear to bull, and p41 is the probability of a transition
from bull to bear. The path from bear to bull and the path from bull to bear are each one path, making
it easier to identify between states for prediction. This model can be called the 4-state（1）model.
The model below is when mean μ and volatility σ transition independently. There are twovolatility states to avoid increasing the number of parameters.
Rt = μ + σzt, zt ∼ i.i.d., E［zt］= 0, V［zt］= 1,

（2.6）

μ = μ1Δ1t + μ2Δ2t + μ3Δ3t + μ4Δ4t,

（2.7）

σ = σ1Γ1t + σ2Γ2t, σ1 < σ2,（2.8）

pij = Pr［Δt = j | Δt−1 = i］
, i,j= 1, 2, 3, 4,

（2.9）

qkl = Pr［Γt = l | Γt−1 = k］
, k,l= 1, 2.

（2.10）

Here, Γt is the state variable that follows a Markov-chain, and it is independent.（Γ1t, Γ2t）is（1, 0）
when Γt = 1 and（0, 1）when Γt = 2. The transition matrix of Δt and Γt are each:

		

P＝

p11
p12

p21
p22

0
0

p41
0

0
p14

0
0

p33
p34

p43
p44

, Q＝

q11
q12

q21
q22

.

（2.11）

This model can be called the 4-state（2）model.
With only a bull and bear growth, the μ, pij and P of the 4-state（1）and 4-state（2）models are as
follows:
μ = μ1Δ1t + μ2Δ2t, μ1 < μ2,pij = Pr
［Δt = j | Δt−1 = i］
, i,j = 1, 2, P ＝

p11
p12

p21
p22

（2.12）.（2.13）As in Eq.（2.12）, the restriction of μ1 < μ2 will be applied. Depending on the asset price the− 113 −KEIZAI SHUSHI（ 254 ）

Vol. 83

No. ３

estimated results may or may not be μ1 < 0，μ2 > 0, but we shall call μ1 a bear growth and μ2 abull growth. These models can be called the 2-state（1）and 2-state（2）models.
2.2

4-state Markov-Switching EGARCH Model

Next is the 4-state Markov-switching EGARCH model, an extension of the 4-state Markov-switchingmodel of section 2.1.
Rt = μa + φaRt－1 +Vab,tzt, zt ~ i.i.d., E［zt］= 0, V［zt］= 1,ln（Vab,t）= ωb + βln（Vab,t－1）+ θ

Rt－1 －μa －φaRt－2
Vab,t－1

Rt－1 －μa － φaRt－2+γ（2.14）

Vab,t－1－ E［ | zt－1 | ］ ,（2.15）

μa = μ1Δ1t + μ2Δ2t + μ3Δ3t + μ4Δ4t,

（2.16）

φa = φ1（Δ1t +Δ2t）+ φ2（Δ3t +Δ4t）,（2.17）

ωb = ω1Γ1t + ω2Γ2t, ω1 < ω2,（2.18）

pij = Pr
［Δt = j | Δt−1 = i］
, i,j= 1, 2, 3, 4,

（2.19）

qkl = Pr［Γt = l | Γt−1 = k］
, k,l= 1, 2.

（2.20）

Here, coefficient φa of Rt − 1 is φ1 in a bear negative or bear positive growth, and φ2 in a bull negative
or bull positive growth. Vab,t expresses the conditional variance of Rt as Vab,t = V［Rt | Δt = a, Γt = b,
It − 1］, with the condition of information It − 1 = {Rt − 1, Rt − 2, · · · } until the term Δt = a, Γt = b, t−1. Δtand Γt are state variables that follow a Markov-chain, and they are independent. The transition
matrix of Δt and Γt is as follows:

		

P＝

p11
p12

p21
p22

0
0

p41
0

0
p14

0
0

p33
p34

p43
p44

, Q＝

q11
q12

q21
q22

.

（2.21）

It is known that estimating parameters becomes difficult when a Markov-switching model andvolatility variation model are simply combined, due to the dependence of state variables. However,this study’s model makes estimation possible with maximum-likelihood estimation by using the ideafrom the Markov-switching GARCH model by Haas et al.（2004）. This model can be called the 4-state（2）-MSEG model.
With only bull and bear growth, the transition matrix P of μa, φa, pij, Δt of the above 4-state（2）-MSEG Model is as follows:− 114 −

A Note on the Stock Market Trend Analysis Using Markov-Switching EGARCH Models（MITSUI）（ 255 ）

μa = μ1Δ1t + μ2Δ2t, μ1 < μ2（2.22）

φa = φ1Δ1t + φ2Δ2t,

（2.23）
p11
p12

pij = Pr
［Δt = j | Δt−1 = i］
, i,j = 1, 2, P ＝

p21
p22

.

（2.24）

This model can called the 2-state（2）-MSEG Model.
2.3

Asymmetry Model

The present study focuses on the asymmetry model, whereby the asymmetry of volatility in thebullish sentiment differs from that in the bear sentiment. Here, the Eq.（2.15）is extended into thefollowing model, whereby the parameter θ, which captures the asymmetry of volatility, can also go
through switching.

ln（Vab,t）= ωb + βln（Vab,t－1）+ θa+γ

Rt－1 －μa －φaRt－2
Vab,t－1

Rt－1 －μa － φaRt－2－ E［ | zt－1 | ］ ,Vab,t－1

θa = θ1Δ1t + θ2Δ2t.

（2.25）（2.26）

This model can be called the “as” model for convenience.
2.4

Time-varying Transition Probability Model

This note also focuses on the time-varying transition probability model, whereby the probability oftransition between the bull and bear markets sentiment is influenced by the return of the previous
period.
p11, t = Pr
［Δt = 1 | Δt－1 = 1, Rt－1］=
p22, t = Pr［Δt = 2 | Δt－1 = 2, Rt－1］=

exp（λ0 + λ1Rt－1）
1 + exp（λ0 + λ1Rt－1）
exp（ξ0 + ξ1Rt－1）
1 + exp（ξ0 + ξ1Rt－1）,（2.27）,（2.28）

Here, if λ1 < 0, p11 becomes higher when Rt−1 < 0. That is, it becomes highly probable that adecrease in stock prices will maintain the bear market sentiment. In case of ξ1 > 0, p22 becomes
higher when Rt−1 > 0. Then, p12, t and p43, t can be expressed as follows:
p12,t = Pr［Δt = 2 | Δt－1 = 1, Rt－1］
=

exp
（ζ）

,
1 + exp（λ0 + λ1Rt－1）+ exp（ζ）− 115 −

（2.29）KEIZAI SHUSHI（ 256 ）

p43,t = Pr［Δt = 3 | Δt－1 = 4, Rt－1］=

Vol. 83

No. ３

exp
（υ）
,
1 + exp（ξ0 + ξ1Rt－1）+ exp（υ）（2.30）

Thus, it becomes highly probable that a rise in stock prices will maintain the bull market sentiment.
Hence, this model can be called the “tv” model for convenience.
Trend Analysis Model

3

When conducting a stock market trend analysis, if in the 4-state（2）-MSEG-tv the error term follows
t distribution, the following model can be derived from Eqs.（2.14）–（2.21）and Eqs.（2.27）–（2.30）:
Rt = μa + φaRt－1 +Vab,t zt, zt ~ i.i.d.t（0, 1, ν）

ln（Vab,t）= ωb + βln（Vab,t－1）+ θ

Rt－1 －μa －φaRt－2
Vab,t－1

Rt－1 －μa － φaRt－2+γ－ E［ | zt－1 | ］ ,Vab,t－1

μa = μ1Δ1t + μ2Δ2t + μ3Δ3t + μ4Δ4t,
φa = φ1（Δ1t +Δ2t）+ φ2（Δ3t +Δ4t）,
ωb = ω1Γ1t + ω2Γ2t, ω1 < ω2,
pij = Pr
［Δt = j | Δt−1 = i］
, i,j = 1, 2, 3, 4,
qkl = Pr［Γt = l | Γt−1 = k］
, k,l = 1, 2.

		

P＝

p11
p12

p21
p22

0
0

p41
0

0
p14

0
0

p33
p34

p43
p44

p11, t = Pr
［Δt = 1 | Δt－1 = 1, Rt－1］=
p22, t = Pr［Δt = 2 | Δt－1 = 2, Rt－1］=
p12,t = Pr［Δt = 2 | Δt－1 = 1, Rt－1］=
p43,t = Pr［Δt = 3 | Δt－1 = 4, Rt－1］
=

, Q＝

q21
q22

q11
q12

.

exp
（λ0 + λ1Rt－1）
1 + exp（λ0 + λ1Rt－1）
exp（ξ0 + ξ1Rt－1）
1 + exp（ξ0 + ξ1Rt－1）

,
,

exp
（ζ）

,
1 + exp（λ0 + λ1Rt－1）+ exp（ζ）
exp
（υ）

,
1 + exp（ξ0 + ξ1Rt－1）+ exp（υ）

Therefore, the parameter {μ1, μ2, μ3, μ4, φ1, φ2, ω1, ω2, β, θ, γ, ν, p22, p33, q11, q22, λ0, λ1, ξ0, ξ1,
ζ, υ} can be estimated using maximum likelihood estimation1）. Moreover, a variety of combinations
1）

For details, refer to Mitsui（2011），Mitsui（2012）.
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A Note on the Stock Market Trend Analysis Using Markov-Switching EGARCH Models（MITSUI）（ 257 ）

can be used using the 4-state（2）-MSEG-as model, 2-state（2）-MSEG-tv model, 2-state（2）-MSEG-asmodel, etc.
4

SummaryThis note suggests a 4-state Markov-switching EGARCH model, which is a combination of a 4-state
Markov-switching model and EGARCH model, and explains the use of an asymmetry model andtime-varying transition probability model in order to conduct trend analysis in more detail. Thesemodels can be used to conduct empirical analysis on actual stock index data such as the Nikkei Stock
Average, TOPIX, etc.
References［1］ Cai, J.（1994）, “A Markov Model of Switching-Regime ARCH,” Journal of Business & Economic Statistics, 12, pp.
309–316.
［2］ Gray, S. F.（1996）, “Modeling the Conditional Distribution of Interest Rates as a Regime-Switching Process,” Journalof Financial Economics, 42, pp. 27–62.
［3］ Haas, M., Mittnik, S. and Paolella, M. S.（2004）, “A New Approach to Markov-Switching GARCH Models,” Journalof Financial Econometrics, 2, pp. 493–530.
［4］ Hamilton, J. D. & Susmel, R.（1994）, “Autoregressive Conditional Heteroskedasticity and Changes in Regime,”
Journal of Econometrics, 64, pp. 307–333.
［5］ Henry, O. T.（2009）, “Regime Switching in the Relationship between Equity Returns and Short-Term Interest
Rates in the UK,” The Journal of Banking & Finance, 33, pp. 405–414.
［6］ Klaassen, F.（2002）, “Improving GARCH Volatility Forecasts with Regime-Switching GARCH,” Empirical
Economics, 27, pp. 363–394.
［7］ Lunde, A. and Timmermann, A.（2004）, “Duration Dependence in Stock Prices: An Analysis of Bull and Bear
Markets,” Journal of Business & Economic Statistics, 22, pp. 253–273.
［8］ Maheu, J. M. and McCurdy, T. H.（2000）, “Identifying Bull and Bear Markets in Stock Returns,” Journal of Business& Economic Statistics, 18, pp. 100–112.
［9］ Maheu, J. M., McCurdy, T. H. and Song, Y.（2009）, “Extracting Bull and Bear Markets from Stock Returns,” Working
Paper from University of Toronto, Department of Economics.
［10］Mitsui, H.（2011）, “A Note on the 4-state Markov-Switching EGARCH Model,” KEIZAI SHUSHI（The Nihon
University Economic Review）,Vol.81，No.3，pp. 165–169.（in Japanese）.
［11］────（2012）, “Application of Markov-Switching EGARCH Models to the Nikkei Stock Average Variation,”
KEIZAI SHUSHI（The Nihon University Economic Review）,Vol.82，No.1，pp. 79–84.
［12］Satoyoshi, K. and Mitsui, H.（2011）, An Empirical Study of Option Pricing under Bull and Bear Market
Conditions,” Journal of Business Research, Institute of Business Research, College of Economics, Nihon University,
No.33, pp. 63–87.（in Japanese）.
［13］So, M. K. P., Lam, K. & Li, W. K.（1998）, “A Stochastic Volatility Model with Markov Switching,” Journal of Business& Economic Statistics, 16, pp. 244–253.

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