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7/25/2019 Xu Jia.doc

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Momentum and Market States:

A Regime Switching Approach

Thesis Proposal

By Jia Xu

Department o !inance and Accounting

Business School

"ational #ni$ersity o Singapore

This $ersion:

!e% &''(

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A%stract

By using an alternative measure of market state, I will re-examine the relationship

between market state and momentum as well as long-run reversal profits. The measure

classifies excess market returns into two regimes using a arkov-switching model. There

are at least two advantages to this approach. !irst, this measure of market states is more

natural as we let the data to classify market states. "econd, in this study we can account

for the possibility of regime switches, the market state prior to as well as during the

holding period. I will investigate the effects of both the prior and the contemporaneous

market state on the momentum and long-run reversal profits.

)* )ntroduction

1.1 #b$ectives and %otential &ontribution of the "tudy

"ince 'eBondt and Thaler (1)*+ and egadeesh and Titman (1)), extensive research

has been done on the short-run momentum and long-run reversal. The anomalies have

been found in both /" and non-/" markets. 0ttempts using risk-based models to account

for these anomalies have largely been unsuccessful. "everal behavioral theories were

developed to $ointly explain both the momentum and contrarian phenomena. 'aniel,

irshleifer, and "ubrahmanyam ('", 1))* and ong and "tein (", 1))) each

constructed a behavioral model to explain these anomalies. %articularly, they link the

investor overreaction to new information with momentum profits. oreover, the degree

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of overreaction is believed to be higher following market gains as well as the decreasing

investor risk aversion.

!ollowing these models, &ooper et al (233 has tested the overreaction theories of short-

run momentum and long-run reversal in the cross section of stock returns. They find that

the momentum and long-run reversal profits depend on the state of the market prior to the

beginning of the momentum holding periods. They call attention for later research to take

regime switches into account.

4iven the critical effects of market states, it is natural to pay attention to the criterion for

identifying them. In &ooper et al (233, they define the two market states based on

whether the lagged market returns are positive (/% market or negative ('#56 market.

The lags they used are one-year, two-year and three-year. owever, in this study, I

propose an alternative measure of market states, which classifies excess market returns

into two regimes using a arkov-switching model. There are at least two advantages to

my approach. !irst, this measure of market states is more natural as we let the data to

classify market states. "econd, in this study we can account for the possibility of regime

switches, the market state prior to as well as during the holding period. "ome empirical

exploration can be conducted.

This study may have two potential contributions7 !irstly, based on the two market states I

identify, I will re-examine the relationship between market states and the momentum and

long-run reversal profits. Besides, in &ooper et al (233, there is an open 8uestion that


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there seems no initial momentum before the long-run reversal following the '#56

states. In my paper, it would be interesting to see if the phenomenon persists.

"econdly, since momentum profits can also be influenced by the market state during the

holding periods, the study will investigate the effects of both the prior and the

contemporaneous market state on the momentum and long-run reversal profits. I would

test which effect is more important to momentum and long-run reversal profits. The

results may lead to some interesting discussions about behavioral models.

1.2 #rgani9ation of the thesis

The thesis proposal is organi9ed as the following7 "ection II reviews the literature on

momentum and contrarian studies, behavioral models and market states, and regime

switching models and its application. "ection III details the research design, methodology

and data. "ection I: discusses further study beyond this thesis.

))* +iterature Re$iew

&*,*, Momentum Studies

The momentum effect, first elaborated by egadeesh and Titman (1)), is still a ma$or

pu99le in finance literature up to today. They show that a simple strategy, buy high sell

low, based on previous to 12 months stock returns will generate about 12; return for

the following year. It has drawn great attention thereafter. 0mong a lot other research,


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egadeesh and Titman (2331 give further evidence on the momentum effect, which

cannot be explained by data mining. !orner and arhuenda (233 confirmed that

momentum is significant. =ouwenhost (1))* and &han, ameed and Tong (2333 tested

and confirmed the momentum anomaly in the international e8uity market.

&urrent asset pricing models have difficulty in explaining momentum. "ome studies find

that it is not primarily driven by market risk. In !ama and !rench (1))>, they show that

their unconditional three-factor model cannot explain the momentum. 0nd 4rundy and

artin (2331 confirmed the result later on. &hordia and "hivakumar (2332 try to use a

conditional asset pricing model with lagged macroeconomic risk factors to forecast

momentum profits. But 4riffin et al (233 show that &hordia-"hivakumar model cannot

be applied internationally. oreover, &ooper et al (233 also find that macroeconomic

factors cannot explain momentum profits after simple methodological ad$ustments to take

account of microstructure concerns.

&*,*& -ontrarian Studies

'eBondt and Thaler (1)*+ suggest that over long period of time there are reverse

changes in stock returns7 specifically, the stocks that have lowest returns (the losers

during the previous three to five years (the formation period will do better during the

following three to five years (the test period than those that previously had the highest

returns (the winners. Based on the findings of ?ahnem and Tversky (1)*2 in the field

of cognitive psychology, 'eBondt and Thaler (1)*+ interpret their results as irrational

behavior on the part of investors. This will lead to excessive optimism toward good news

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and excessive pessimism over bad news. "uch a situation can cause the stock price

temporarily stray from its fundamental value. This potential violation of the @fficient

arket ypothesis is called as the overreaction effect. It remains one of the most

controversial topics in finance field. !orner and arhuenda (233 showed contrarian is

significant and not from data mining. "upporting evidence has also come from

ac'onald and %ower (1))1 and &ampbell and Aimmack (1)) in the /?, ai (1))+

from the !rench arket, 'a &osta (1))


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the precision of his private information rather than the public information and whose

confidence will change asymmetrically to confirming versus disconfirming pieces of

news. Thus, the arrival of confirming news tends to raise the overconfidence of the

investor, which may trigger further overreaction. "uch continuing overreaction will cause

momentum in security prices. But, in the long run, with further public information

arriving, the momentum is eventually reversed towards fundamental values. Thus, the

model may account for both short-run momentum and long-run reversal.

ong and "tein (1))) provide another unified account for under- and overreaction. In

their models, under the assumption of the gradual diffusing news about fundamentals,

there are two different groups of investors7 CnewswatchersD and Cmomentum tradersD.

The authors emphasi9e on the interaction of heterogeneous investors. The newswatchers

base their decisions only on their private information. 5hile the momentum traders, on

the contrary, only condition on the past price changes. They show that the newswatchers

tend to underreact to the private information at the beginning. 0nd the momentum traders

try to exploit the underreaction and, thus, create an excessive momentum profit which

inevitably leads to overreaction.

4ood theories must be potentially re$ectable by empirical tests. By extending the above

theories (mainly '" (1))* and " (1))) to the relationship between market states

and momentum, &ooper et al (233 tests the short-run momentum and long-run reversal

in the cross section of stock returns conditioned on /% and '#56 markets. They find

the results are, in general, consistent with the overreaction theories7 the monthly


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momentum profit does depend on the market state, and that the up-market momentum

will reverse in the long run.

But &ooper et al (233 leaves an open 8uestion7 they find significant long-run reversal in

the '#56 states without any initial momentum. They note that, CEtFhis finding indicates

that long-run reversal is not solely due to the corrections of prior momentum.D 'oes this

suggest the limits of overreaction theoriesG #r Cthere may be other factors causing long-

run reversal in generalD, as the authors mentionedG In my suspicion, it may simply

because of measurement error. By using another way of measuring and identifying the

market states, this study will examine whether the pu99le is still there.

&*. Regime/switching Model and Market State

In &ooper et al (233, they define two market states7 CH/% is when the lagged three-year

market return is non-negative, and H'#56 is when the three-year lagged market return

is negative.D 0lthough their results are robust using one- or two- year lagged market

states and risk-ad$ustments, the definition of market states is still rather sub$ective. 5hy

not use six months as the windowG 5hy not set the cutting return to be 3.+; instead of

3;G ... Juestions as these are unavoidable.

Besides, defining market states according to the sign of index returns may misclassify the

true state of the market. "uppose the index return over a given lag period, say > months

is positive. Thus, the market is considered to follow an /% state according to &oopers et.

al (233. owever, if the market is volatile, it is possible to have many down states

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within the >-month period. It is very likely that the holding period is actually following

a recent down market, instead of /% market. &onse8uently, it may be 8uite misleading to

classify that period as a bullish period. &onversely, a '#56 market according to their

classification may actually consist of many up states. 0gain, if this is true, then it is

misleading to consider that period as a down period.

ere, I provide a more inclusive definition of /% and '#56 market states by fitting a

arkov switching model to excess returns (over Treasury bills on the market index.

!rom the literature in statistics, economics and finance, regime-switching models have

their advantages in identifying the market states.

#ften, we define model instability as a switch in a regression e8uation from one regime

to another. owever, in many cases, we may have little information on the dates at which

parameters change, and thus need to make inferences about the turning points as well as

the significance of parameter shifts. In statistics and econometrics, a lot research has been

done on switching models. In the sixties and seventies, many models have been

developed to deal with the issue, for example, Juandt (1)+*, 1)>3, 1)2, !arley and

inich (1)3, . ?im and "iegmund (1)*) and I.-. ?im and addala (1))1. 0n

interesting and important point of the above models is that the time at which a structural

change occurs is endogenous to the model.

amilton (1)*) proposes a tractable state-dependent arkov-switching model. 5e can

apply it to the important case of structural changes in the parameters of an autoregressive

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process. &ompared to the traditional models, there are at least three advantages of this

model. !irst, the model sorts data endogenously into regimes. "econd, unlike the

traditional models, in this latent variable model, we do not need to assume the

information set of the researcher, i.e., the econometrician, coincides with that of the agent

in the market. Thirdly, empirically, among other studies, =yden et al (1))* show that the

arkov-switching model can explain the temporal and distributional properties of stock

returns.

The model has drawn a lot attention in modeling structural changes in dependent data. In

economics, it has been used to identify business fluctuations, see amiltion (1)*)K to

study the changes in real interest rates, 4arcia and %erron (1))>. =ecently, the model has

been used extensively in finance area, esp. to model the nonlinear structure in time series

data. Turner et al (1)*) use the model to explain a time-varying risk premium in stock

returns. ethodologically, they consider a arkov-switching model which allows either

the mean or the variance or both to differ between the two regimes. amilton and Ain

(1))> adapt the model to capture the dynamics between the stock market and business

cycle. 4ordon and "t-0mour (2333 use a two-state arkov process to model risk

aversion, called as preference regimes, and link this model with the cyclical pattern of

asset prices.

"pecifically, some research has been conducted on identifying market states. Borrowing

the method from Turner et al (1)*), "challer and van 6orden (1)) find strong

evidence for switching behavior in stock returns in the /" market. By allowing for

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switch in both means and variances, they discover two distinct states7 in one state, the

excess returns are low and the variance is lowK in the other state, the excess return goes to

negative and the variance is high. The results agree with Brock et al (1))2 and aheu

and c&urdy (2333. !or the point of illustration, a specific description of the market

state from 6ielsen and #lesen (2333 will be cited in "ection III.

)))* Research Design and Data

.*, Data

The data are all 6L"@ and 0@M stocks listed on the &="% monthly file. To be

consistent with &ooper et al (233, the sample period is from anuary 1)2> to 'ecember

1))+. To identify the market states, I will collect monthly excess returns on the value-

weighted market index. @xcess return is the difference between the market returns on

&="% :alue-5eighted Index (ad$usted for dividends and the one-month Treasury-bill

rate.

.*& Methodology and Research Design

.*&*, 0n identiying market states

In this study, I will mainly follow amiltons (1)*) arkov-switching model to identify

two market states.

0 normal regression model without any switching is7

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,3(...N, 2 Ndiieexy tttt += (1

0 model with structural breaks in the parameters is7

....,2,,1, Ttexy tttt S =+=

(2

,3(N2

t

tS

Ne (

ttS SSt 10 += 1( (

where under regime 1, parameters are given by 1 and2

1 , and under regime 3,

parameters are given by 0 and2

3 . If the dates of switching or structural breaks are

know a priori, the above will be reduced to a dummy variable model. The problem here

is we do not know when tS is 3 or 1. The evolution of the discrete variable tS may be

dependent upon 1tS , 2tS , O, rtS , in which case the process of tS is named as an

r-th order arkov switching process.

ere, I adapt the standard amiltons (1)*) model into the following model7

tSttStt ttRFRRFR ++=

(111 (

,3(N 2 Nt (*

ttS SS

t 131( += ()

qSSpSS tttt ====== 33%r,11%r 11

(13

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where tS , e8uals to 3 or 1, follows a first order arkov switching process. tR is

monthly return on the &="% :alue-5eighted Index (ad$usted for dividends. tRF is

risk-free interest rate, here I use one-month Treasury bill rate. %lease note that this

model allows for the switching in means, and that an autoregressive term is included in

each state.

The parameter estimates can be obtained by numerically maximi9ing the log likelihood

function. The likelihood function and the maximi9ing procedure are standard for

regime-switching models and described in both amilton (1))


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In their figure 2 is the probability that observation t is in state 3 given the information on

current and past stock returns available at time t.

It is natural to use a +3; probability cutoff point to delineate between /% and '#56

market. That is, at time t, if the smoothed probability of the /% state is say 3., I label

the market state then as an /% state. If the smoothed probability is 3.


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NDN

i

it P1

=

(11

!or example, suppose the lagged period has 13 months, and this comprises months

(not necessary contiguous where the market is in /% state (probability of /% state Q

3.+ and months in which the market is in down state. Then, the probability of the

market being in the up state over the last 13 months is 3.. Based on this, I classify the

prior market as being in an /% state. Aikewise, if the same probability is say 3.


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factors and a constant. The three factors are a the excess return of the value-weighted

market index over the one-month Treasure bill rate (?T=!, b the small-minus-big

return premium ("B and the high-book-to-market-minus-low-book-to-market return

premium (A. Thus, the risk-ad$usted profits are

tktktkkt

adj

kt HMLSMBMKTRFRR .21SSS = (1

where ktR is the raw profit for the strategy in the holding-period month k, for k1, 2,O

>3, in calendar month t. The S s are the estimated loading of the time-series of raw

profits in holding-period-month k on their respective factors. The monthly raw or !ama-

!rench-ad$usted profits are cumulated to form the holding-period profits (&0=.

=

++ =2

1

2

U

,

K

Kk

ktkKt RCAR (1, (1, 12, and (1, >3.

.*&*. 0n analy1ing the proits across market states

&ooper et als (233 results may be contaminated by the contemporaneous market state.

In particular, it is not clear whether the strong momentum that they find following /%

states is due to delayed overreaction (their explanation or simply due to the fact that

momentum happens to be strong during a holding period in which the market is also in an

/% state.

a. &ontingency table

To disentangle the prior and contemporaneous market state influences, I first use a 2 x 2

contingency table to document the distribution of momentum profits by prior and

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contemporaneous market states. Table 1 presents the contingency table where I document

in each cell7 1 ean raw &0=, 2 ean risk-ad$usted &0= (using !! model,

%roportion of holding periods where &0= is greater than 3, and


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=egression 2

tttKt DconDlagCAR +++=

+ 2132

=egression

tttttKt DconDlagDconDlagCAR ++++=+ ((.2132

5here7

2KtCAR

+ is cumulative excess returns of the momentum portfolio which is formed at the

beginning of month t and held for ?2month. ?2 >, 12 or >3.Dlag 1 if the market is /%prior toholding period and 3 otherwise

Dcon 1 if the market is /% dringthe holding period and 3 otherwise.

tt DconDlag U is an interactive variable to measure the impact on &0= of the combined

effects of an /% market in both prior and contemporaneous periods.

ere, regression 1 is similar to the tests in &ooper et al (233 to see whether the mean

momentum profits following /% and '#56 markets are e8ual. The overreaction

theories predict that 1 is significantly positive, when ?2is > or 12. 5hen ?2 is >3, 1

is significantly negative. In regression 2 and , we can ask many interesting 8uestions. Is

2 significant or notG If yes, what happens to the significance of 1 , when 2 is

addedG 5hat is the sign of 2 G ow about . G

c. =egressions using a &ontinuous easure of arket "tate

!ollowing &oopers et. al.(233, I perform &0= regressions on the lagged market returns

as well as contemporaneous holding-period market returns. 0gain, we can ask similar

8uestions about the signs and magnitudes of the s.

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ielsen, S., and Olesen. $. O. (000), "Re'i!esCitchin' stoc returns and !ean reversion",orin' paper, @epart!ent o% 8cono!ics, 3openha'en &usiness School.

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