Moments of Markov switching modelsq - Directory UMMdirectory.umm.ac.id/Data Elmu/jurnal/J-a/Journal...

Journal of Econometrics 96 (2000) 75}111

Moments of Markov switching modelsq

Allan Timmermann

Department of Economics UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0508, USA

Received 1 July 1997; received in revised form 1 April 1999

Abstract

This paper derives the moments for a range of Markov switching models. Wecharacterize in detail the patterns of volatility, skewness and kurtosis that these modelscan produce as a function of the transition probabilities and parameters of the underlyingstate densities entering the switching process. The autocovariance of the level and squaresof time series generated by Markov switching processes is also derived and we use theseresults to shed light on the relationship between volatility clustering, regime switches andstructural breaks in time-series models. ( 2000 Elsevier Science S.A. All rights reserved.

JEL classixcation: C1

Keywords: Markov switching; Higher-order moments; Mixtures of normals; Volatilityclustering

1. Introduction

Markov switching models have become increasingly popular in economicstudies of industrial production, interest rates, stock prices and unemploymentrates. However, so far no study has characterized in any detail the moments that

qTwo anonymous referees and an associate editor provided many useful suggestions for improve-ments of the paper. Discussions with Jim Hamilton and Martin Sola also were very helpful.

E-mail address: [email protected] (A. Timmermann)

0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 5 1 - 2

these models can generate. This is an important omission since Markov switch-ing models are often adopted by reseachers wishing to account for speci"cfeatures of economic time series such as the asymmetry of economic activity overthe business cycle (Hamilton, 1989; Neftci, 1984) or the fat tails, volatilityclustering and mean reversion in stock prices (Cecchetti et al., 1990; Pagan andSchwert, 1990; Turner et al., 1989) and interest rates (Gray, 1996; Hamilton,1988). These features translate into the higher-order moments and serial correla-tion of the data-generating process, so a characterization of the moments andautocorrelation function generated by Markov switching will allow researchersto better understand when to make use of this class of models. The contributionof this paper is to characterize the moments and serial correlation of the leveland the squared values of Markov switching processes.

Markov switching models belong to a general class of mixture distributions.Econometricians' initial interest in this class of distributions was based on theirability to #exibly approximate general classes of density functions and generatea wider range of values for the skewness and kurtosis than is obtainable throughuse of a single distribution. Along these lines, Granger and Orr (1972) and Clark(1973) considered time-independent mixtures of normal distributions as a meansof modeling non-normally distributed data. These initial models, however, didnot capture the time dependence in the conditional variance found in manyeconomic time series, as evidenced by the vast literature on ARCH models thatstarted with Engle (1982).

By allowing the mixing probabilities to display time dependence, Markovswitching models can be seen as a natural generalisation of the original time-independent mixture of normals model and we show that this feature enablesthem to generate a wide range of coe$cients of skewness, kurtosis and serialcorrelation even when based on a very small number of underlying states.

Regime switches in economic time series can be parsimomiously representedby Markov switching models by letting the mean, variance and possibly thedynamics of the series depend on the realization of a "nite number of discretestates. In increasing order of generality we consider in this paper three types ofmodels, each of which has been adopted in applied econometric studies. Thebasic Markov switching model is

yt"k

st#p

stet, (1)

where St"1, 2,2, k denotes the unobserved state indicator which follows an

ergodic k-state Markov process and etis a zero-mean random variable which is

identically and independently distributed over time. The number of states, k, isassumed to be "nite. This model has been used in empirical work by Engel andHamilton (1990). The second Markov switching model allows for state-indepen-dent autoregressive dynamics:

yt"k

st#

q+j/1

/j(y

t~j!k

st~j)#p

stet. (2)

76 A. Timmermann / Journal of Econometrics 96 (2000) 75}111

Hamilton (1989) used this type of autoregressive model to analyze growth in USGDP. Finally, we also consider a model with state-dependent dynamics in theautoregressive part:

yt"k

st#

q+j/1

/jst~j

(yt~j

!kst~j

)#pstet. (3)

For all models the errors are assumed to be independently distributed withrespect to all past and future realizations of the state variable, i.e.F(e

tDS

t`i)"F(e

t) for all values of i, where F (.) denotes the cumulative density

function of et.1 The stochastic transition probability matrix P that determines

the evolution in Stis given by

Prob(St`j

"j DSt"i, S

t~1"k,2,)"Prob(S

t`1"j D S

t"i)"p

ij,

0)pij)1,

k+j/1

pij"1 for all i, (4)

so that the states follow a homogenous Markov chain. In practice, if the processis not irreducible and not all states are visited with non-zero probability in thesteady state, then the moment analysis can simply be conducted on the subset ofstates occurring with non-zero stationary probability.2 The higher is p

ii, the

longer the process is expected to remain in state i. For this reason we shall referto p

iias measuring the &persistence' of the mixing of the underlying state

densities.We "nd in this paper that these persistence parameters are very important in

determining the higher order moments of the Markov switching process. Fur-thermore, once autoregressive parameters are introduced into the process as inthe second and third switching models, this gives rise to cross-product termsthat enhance the set of third- and fourth-order moments and the patterns inserial correlation and volatility dynamics that these models can generate. Evenlow-order autoregressive Markov switching processes with a small number ofstates provide the basis for very #exible econometric models. Our results proveuseful in understanding the literature on structural breaks and volatility cluster-ing since the models that have been adopted in this literature are closely relatedto Markov switching models with infrequent switches between states.

1Herein lies a key di!erence to ARCH models which is another type of time-dependent mixtureprocess. While Markov switching models mix a "nite number of states with di!erent mean andvolatility parameters based on an exogenous state process, ARCH models mix distributions withvolatility parameters drawn from an in"nite set of states driven by lagged innovations to the series.

2An analysis of the unconditional moments starting from the steady state need not assume thatthe Markov process is irreducible since, if either a single state or a block of states is absorbing, allother states will have zero steady-state probabilities.

A. Timmermann / Journal of Econometrics 96 (2000) 75}111 77

The paper is organized as follows. Section 2 provides results on the momentsof the basic Markov switching process without autoregressive terms. Section3 extends the results to the second model with state-independent, autoregressiveterms, while Section 4 analyses the moments of the most general model withstate dependence both in the mean, variance and autoregressive coe$cients.Section 5 reports the autocovariance structure generated by the three Markovswitching models and Section 6 discusses their relation to structural break andARCH models.

2. The basic Markov switching model

Let p"(n1,2,n

k)@ be the k-vector of steady-state (ergodic) probabilities that

solve the system of equations P@p"p. These probabilities can be computed asthe eigenvector (scaled so that its elements sum to one) associated with the uniteigenvalue of P@. Assuming that the state process started an in"nite number ofperiods back in time or that it is initialized by a random variable drawn from thestationary distribution, p is the vector of unconditional probabilities applying tothe k states. The following proposition provides the moments of the basicMarkov switching model.

Proposition 1. Suppose the stationary Markov switching process (1), (4) startedfrom its steady state characterized by the set of unconditional probabilities (p). Thenthe centered moments of the process are given by

E[(yt!k)n]"

k+i/1

ni

n+j/0

nC

jpjiE[ej

t](k

i!k)n~j,

wherenC

j"n!/(n!j)!j!. When e

tis t-distributed with q degrees of freedom, the

centered moments are

E[(yt!k)n]"

k+i/1

ni

n+j/0

nC

jpjiaj(k

i!k)n~j,

where

aj"G

qj@2b(j`12

, q~j2

)

b(12, q2)

if q'j and j is even,

0 otherwise

and b(.) is the Beta function. When et

follows a normal distribution we have

E[(yt!k)n]"

k+i/1

ni

n+j/0

nC

jpjibj(k

i!k)n~j,


where

bj"G

j@2<h/1

(2h!1) provided j is even and

0 otherwise.

A proof of Proposition 1 is given in the appendix. Since researchers are oftenparticularly interested in the variance, skewness and kurtosis of their data, wecharacterize these moments more explicitly for the empirically popular mixtureof normals model with two states:

Corollary 1. Suppose that there are two states and that the increments are Gaus-

sian. Then the unconditional variance (p2), coezcient of skewness (Jb1) and

coezcient of excess kurtosis (b2) of the basic Markov switching process (1), (4) are

given by

p2"(1!n1)p2

2#n

1p21#(1!n

1)n

1(k

2!k

1)2,

Jb1,

E[(yt!k)3]

(E[(yt!k)2])3@2

"

n1(1!n

1)(k

1!k

2)M3(p2

1!p2

2)#(1!2n

1)(k

2!k2

1)2N

((1!n1)p2

2#n

1p21#(1!n

1)n

1(k

2!k

1)2)3@2

,

b2,

E[(yt!k)4]!(3E[(y

t!k)2])2

(E[(yt!k)2])2

"

a

b,

where

a"3n1(1!n

1)(p2

2!p2

1)2#6(k

2!k

1)2n

1(1!n

1)(2n

1!1)(p2

2!p2

1)

#n1(1!n

1)(k

2!k

1)4(1!6n

1(1!n

1)),

b"((1!n1)p2

2#n

1p21#(1!n

1)n

1(k

2!k

1)2)2.

For a proof, see the appendix. It follows from Corollary 1 that, when theinnovations are drawn from a Gaussian density, a necessary condition for theMarkov switching process to generate skewness is that the means in the statesdi!er (k

1Ok

2) and that di!erences in the variances of the states alone are

insu$cient to generate skewness.3 This also suggests that Markov switching

3This is similar to the result in Bollerslev (1986) that the skewness of standard GARCH modelswithout leverage e!ects is zero.


models "tted to high-frequency "nancial data whose means are often very smallin all states may have trouble replicating successfully the skewness found inthese data.

It is useful to evaluate these expressions through some numerical examples.To establish a benchmark for some relevant parameter values, we report inTable 1 the maximum likelihood estimates from a simple two-state Markovswitching model "tted to monthly excess returns on the stock price index in fourcountries. In three out of four markets the switching model identi"es an &outlier'state in which excess returns are very volatile with a large negative mean anda &normal' state with low volatility and small positive mean returns. Thedi!erence between the volatility parameter in the two states can be very largeand is around three times higher in state 2 than in state 1 for the UK and USmarkets. Likewise the mean return di!erentials across the two states are verylarge in three of the four markets although the mean parameter is impreciselyestimated in the state with high volatility. In Germany the mean return para-meters in the two states are roughly of the same magnitude but of opposite signs.Based on this evidence we consider in our numerical examples a variety ofcombinations of di!erences in the relative size of the mean and volatilityparameters in two or more states.

Fig. 1 presents a plot of the coe$cients of skewness and excess kurtosis ofa two-state Markov switching process with parameters l

s"(1!1), r

s"(1 1).

Thus the skewness and kurtosis in Fig. 1 is entirely driven by di!erences in themean parameters of the model, while the variances are identical in the two states.To construct the "gure, these parameters were kept "xed and the probabilities ofstaying in the two states were varied over the grids [0.01, 0.99]. High, positivevalues of both skewness and excess kurtosis are obtained when the probabilityof staying in the second state (p

22) is around 0.9 and the probability of staying in

the "rst state (p11

) is not too high. Likewise, a large negative skewness anda large positive excess kurtosis is obtained when p

11is around 0.9 and p

22is

small. A large, negative excess kurtosis is obtained along the line where the twotransition probabilities are identical so that the process spends the same time inthe two states. This has the e!ect of moving some of the tail probability masstowards the centre and hence lowers the kurtosis.

To investigate the impact on the skewness and kurtosis of switching betweentwo states with di!erent means and volatilities, we used a second parametercon"guration, setting l

s"(1 !3), r

s"(2 4). This parameter con"guration is

close to what was found for three of the four models in Table 1. The plot, whichis presented in Fig. 2, is very di!erent from Fig. 1. Now a very large excesskurtosis is produced when the probability of staying in the low volatility state(p

11) is high and the probability of staying in the high volatility state (p

22)

is low. This case mixes large, but rare, outliers with a low dispersion distribution,thus generating the extreme kurtosis. Since the high volatility state alsohas a low mean, a high value of p

11and a low value of p

22generate large


Fig

.1.

Skew

ness

and

kur

tosis

ofth

etw

o-s

tate

Mar

kov

switch

ing

mode

l:e!

ect

of

di!er

ent

mea

npa

ram

eter

s.B

ased

onth

epa

ram

eter

valu

esk s"

(1,!

1),p s"

(1,1

),th

e"gu

replo

tsth

eco

e$ci

ents

ofs

kew

nes

san

dex

cess

kur

tosisof

atw

o-s

tate

Mar

kov

switch

ing

model

with

Gau

ssia

nin

nova

tions

asa

funct

ion

ofth

epr

obab

ility

ofst

ayin

gin

stat

e1

(P11

)an

dth

epro

bab

ility

ofst

ayin

gin

stat

e2

(P22

).


Tab

le1

Reg

ime

switch

ing

model"tted

toex

cess

stock

retu

rns

(mon

thly

retu

rns,

1970}19

97).

This

tabl

ere

port

sth

ere

sult

from

estim

atin

gth

efo

llow

ing

retu

rnm

odel

r t"k s t#

p s t)u

t,wher

ek

isth

em

ean,p

isth

est

andar

ddev

iation

,stis

thest

ateat

tim

etw

hich

follo

wsa

two-

stat

eM

arko

vsw

itch

ing

pro

cess

,and

u(t)

isin

depe

nde

ntly

and

iden

tica

llydi

stribu

ted

norm

alva

riab

le.The

mea

nan

dvo

latilit

ypar

amet

ers

are

repo

rted

inan

nua

lper

cent

age

term

s

Fra

nce

Ger

man

yU

KU

S

Est

imat

eSta

nda

rder

ror

Est

imat

eSta

nda

rder

ror

Est

imat

eSta

nda

rder

ror

Est

imat

eSta

nda

rder

ror

Stat

e1

Mea

n9.

44.

25.

23.

57.

83.

27.

82.

9V

olat

ility

17.6

1.1

13.7

0.9

15.7

0.8

13.2

0.7

Stay

erpro

babi

lity

%96

.62.

497

.21.

998

.50.

998

.31.

7St

eady-

stat

epr

obab

ility

%87

.275

.790

.894

.4

Stat

e2

Mea

n!

19.7

31.3

!5.

313

.1!

24.9

31.5

!39

.835

.1V

olat

ility

34.6

6.3

28.2

3.3

48.6

7.1

31.7

8.2

Stay

erpro

babi

lity

%76

.814

.091

.35.

785

.37.

571

.219

.6Ste

ady-

stat

epro

babili

ty%

12.8

24.3

9.2

5.6

Not

e:Thes

ere

sultsar

eba

sed

on

Morg

anSta

nle

yC

apital

Inte

rnat

iona

lind

exre

turn

sm

inus

therisk

-fre

eT

-bill

rate

.In

each

countr

yst

ate1

isth

est

atew

ith

the

hig

hes

tm

ean

retu

rn.


Fig

.2.

Skew

nes

san

dku

rtosis

ofth

etw

o-s

tate

Mar

kov

switch

ing

mode

l:e!

ect

ofdi!er

ent

mea

nan

dvo

latility

par

amet

ers.

Bas

edon

the

para

met

erva

lues

k s"(1

,!3)

,p s"

(2,4

),th

e"gu

replo

tsth

eco

e$ci

ents

ofsk

ewne

ssan

dex

cess

kurt

osis

ofa

two-

stat

eM

arkov

switch

ing

mode

lw

ith

Gau

ssia

nin

nov

atio

nsas

afu

nct

ion

ofth

epro

babi

lity

ofst

ayin

gin

stat

e1

(P11

)an

dth

epro

babili

tyofst

ayin

gin

stat

e2

(P22

).


negative skewness because of the presence in the distribution of large negativeoutliers.4

Researchers using Markov switching models sometimes identify states withsimilar means but very di!erent volatilities. To shed light on the sort of momentsthis situation gives rise to, Fig. 3 plots the excess kurtosis against combinationsof variance parameters in two states that vary between 0.1 and 10. The meanparameters are identical in the two states and P

11" 0.97, P

22" 0.75, so state

1 is highly persistent while state 2 is not, matching the estimates in Table 1. Sincethe mean parameters in the two states are the same, the skewness equals zerothroughout. However, a very substantial excess kurtosis is generated when p2

1is

very small and p22

is very high. In this case the process spends most of the time inthe low volatility regime but occasionally shifts to a high volatility state, thusincreasing the kurtosis.

These "ndings are also representative of mixture processes with more thantwo states. Suppose that the asssumptions of Proposition 1 hold and that theincrements are normally distributed. We refer to such a process as MSI. Then itfollows from the proposition that the higher-order moments have the conve-nient representation

E[(yt!k)n]"p@M@

nmn, (5)

where mnis the (n#1)-vector (1

nC

12nC

j2nC

n~11)@, and

Mn"C

(k1!k)n . . (k

k!k)n

0 . . 0

p21(k

1!k)n~2 . . p2

k(k

k!k)n~2

0 . . 0

3p41(k

1!k)n~4 . . 3p4

k(k

k!k)n~4

. . . .

bnpn1

. . bnpnk

Dis the (n#1)xk matrix of moments E[pj

stejt(k

st!k)n~j] for each of the normal

state densities entering the mixture distribution. It is clear from (5) and the formof M

nthat the moment expressions are symmetric with respect to the underlying

4Clark (1973) showed that time-independent mixtures of normals can generate skewness andkurtosis beyond that of a single Gaussian distribution. Corollary 1 readily allows us to measure thee!ect on these moments of Markovian dependence in the state probabilities. We can illustrate thispoint using the grid points and parameter values underlying Fig. 2. Under time-independentswitching p

22"1!p

11and the coe$cient of excess kurtosis ranges from !0.02 to 0.22 while with

state-dependent switching the range is much wider at !0.02 to 2.47.


Fig. 3. Kurtosis of the two-state Markov switching model: e!ect of di!erent volatility parameters.Based on the parameter values k

s"(1,1), and letting the state variances go from 0.1 to 10, the "gure

plots the coe$cient of excess kurtosis of a two-state Markov switching model with Gaussianinnovations. The probability of staying in state 1 (P

11) and the probability of staying in state 2 (P

22)

were set equal to 0.97 and 0.75, respectively.

state densities. To illustrate this point, Fig. 4 plots the coe$cients of skewnessand kurtosis as a function of p

11and p

22for the 3-state case with p

13"

p23"0.1, p

31"p

32"p

33"1/3, l

s"(1 !3 0), r

s"(2 4 3). Hence a third

&intermediate' state is mixed with the two states from Fig. 2. Naturally, the samehigh values of skewness and kurtosis are not reached in Figs. 2 and 4 sincep11

and p22

are now constrained to be less than 0.90, but the shapes of the two"gures in this range of p

11, p

22-values are very similar.


Fig

.4.

Skew

ness

and

kur

tosis

ofth

eth

ree-

stat

eM

arkov

switch

ing

mode

l:e!

ect

ofdi!

eren

tm

ean

and

vola

tilit

ypar

amet

ers.

Bas

edon

the

para

met

erva

lues

k s"(1

,!3,

0),p s"

(2,4

,3),

the"gu

replo

tsth

eco

e$ci

ents

ofs

kew

nes

san

dex

cess

kur

tosisofa

thre

e-st

ate

Mar

kov

switch

ing

mode

lwith

Gau

ssia

nin

nova

tion

sas

afu

nction

oft

he

prob

abili

tyofs

tayi

ngin

stat

e1

(P11

)an

dth

epro

bab

ility

ofs

tayi

ngin

stat

e2

(P22

).The

tran

sition

pro

bab

ilities

forst

ate

3al

leq

ual

1/3,

whi

leP

13"

P23"

0.10

.


3. The simple autoregressive Markov switching model

Before dealing with the general case of Eq. (2) that involves q lags of (yt!k

st),

we "rst consider the more tractable "rst-order autoregressive model with normal-ly distributed increments. This case demonstrates the e!ect of additional termsentering the expression for the moments that researchers pay most attention to.Notice from (2) that when q"1, the Markov switching process becomes

yt"k

st#/

1(y

t~1!k

st~1)#p

stet, (6)

which we shall refer to as MSII. Upon substituting backward we get

yt!k

st"

=+i/0

/i1pst~i

et~i

. (7)

From the assumption that E[et~i

DSt]"E[e

t~i]"0 it still holds that

E[yt~1

!kst~1

DSt~1

]"0. Thus the "rst moment of yt

is unchanged andE[y

t]"p@E[y

tDS

t]"p@l

s, where E[y

tDS

t] is the k-vector whose ith element

consists of E[ytDS

t"i].

To state concisely the moments for this process, let i be a k-vector of ones,while I

kis the k-dimensional identity matrix, ( is the element-by-element

multiplication operator and B is the (k]k) matrix of transition probabilities forthe &time-reversed' Markov chain that moves back in time:

Prob(St"jDS

t`1"i)"b

ij, 0)b

ij)1,

k+j/1

bij"1. (8)

Since Prob(St"jWS

t`1"i)"Prob(S

t`1"iDS

t"j)Prob(S

t"j)"

Prob(St"jDS

t`1"i)Prob(S

t`1"i), the &backward' transition probability

matrix B is related to the &forward' transition probabilities as follows:5

bij"p

jiAnj

niB, (9)

assuming ni'0, and b

ij"0 if j belongs to a group of absorbing states while

i does not. Again, if this condition is not satis"ed, the analysis can be performedon the sub-set of states for which the steady-state probabilities exceed zero.Using this notation, Proposition 2 presents expressions for the second to fourthcentered moments of the "rst-order autoregressive Markov switching model:6

5 If bij"p

ijfor all i and j, then the process is said to be time reversible. Since the diagonal elements

of B and P are identical, the probability of remaining in a given state is always the same regardless ofwhether time moves forward or backward. Furthermore, in the case with two states it is easy toverify from the de"nitions of n

1and n

2that the Markov chain will be time reversible, although this

does not hold generally for processes with several states.6The proposition requires that (I

k!/

1B) is invertible. Since D/

1D(1 this will automatically be

satis"ed for all transition probability matrices since B has a single eigenvalue equal to unity and itsremaining eigenvalues are less than one.


Proposition 2. Suppose yt

follows the xrst-order autoregressive Gaussian Markovswitching process

yt"k

st#/

1(y

t~1!k

st~1)#p

stet, D/

1D(1, e

t&IIN(0,1),

and assume that the process started from its steady-state distribution. Then thevariance, skewness and kurtosis of y

tare given by

E[(yt!k)2]"p@A(ls

!ki)((ls!ki)#

r2s

1!/21B,

E[(yt!k)3]"p@((l

s!ki)((l

s!ki)((l

s!ki))

#3/21p@((B ) (I

k!/2

1B)~1r2

s)((l

s!ki))

#3p@((ls!ki)(r2

s),

E[(yt!k)4]"p@((l

s!ki)((l

s!ki)((l

s!ki)((l

s!ki))

#6p@((ls!ki)((l

s!ki)( r2

s)

#p@(Ik!/4

1B)~1(3r4

s#6/2

1(B ) (I

k!/2

1B)~1r2

s)(r2

s)

#6/21p@((B(I

k!/2

1B)~1r2

s)((l

s!ki)((l

s!ki)).

Comparing Proposition 2 to the results for the case with normal incrementsstated in Proposition 1, we see that the unconditional variance is increased bythe persistence in the y

tprocess. This is no di!erent from the usual autoregres-

sive model without a Markov switching e!ect. Turning next to the skewness,a new term re#ecting the expectation of the cross-product term (k

st!k)

(yt~1

!kst~1

)2 enters into this expression. This term will be negative when lowmean states tend to follow high variance states and will otherwise be positive. Todemonstrate this e!ect, Fig. 5 plots skewness for the same parameter values as inFig. 2 and with /

1" 0.9. A comparison of Figs. 2 and 5 shows that introducing

an autoregressive term into the model produces a very di!erent pro"le of theskewness. Large negative skew is still obtained when p

11and p

22are both high

so that the process does not switch very often and the negative termE[(k

st!k)p2

st] dominates. For lower values of p

11and p

22the process switches

more often and the positive term E[(kst!k)(y

t~1!k

st~1)2] #attens the skewness

pro"le compared to Fig. 2.Three components change in the expression for the kurtosis once an autore-

gressive lag is introduced in the Markov switching process. First, the contribu-tion to kurtosis from p4

ste4tis scaled by a factor (I

k!/4

1B)~1. Furthermore, terms

re#ecting the expectation of (yt~1

!kst~1

)2 times (kst!k)2 or p2

stalso contribute

to kurtosis. Hence, if high volatility states are followed by states with either highvolatility or a mean parameter far away from the unconditional mean, this willtend to create fat tails and increase kurtosis. The highest kurtosis now occurs


Fig

.5.

Ske

wnes

san

dkurt

osis

ofth

etw

o-s

tate

auto

regr

essive

Mar

kov

switch

ing

mod

el:e!

ect

of

auto

regr

essive

par

amet

er.Bas

edon

the

par

amet

erva

lues

k s"(1

,!3)

,p s"

(2,4

),/

1"0.

9th

e"gu

repl

ots

the

coe$

cien

tsof

skew

nes

san

dex

cess

kurt

osis

ofa

two-s

tate

auto

regr

essive

Mar

kov

switch

ing

mode

lw

ith

Gau

ssia

nin

nova

tions

asa

func

tion

ofth

epr

obab

ility

ofst

ayin

gin

stat

e1

(P11

)an

dth

epro

bab

ility

ofst

ayin

gin

stat

e2

(P22

).


when both p11

and p22

are high but not too close to one so that rare jumpsresulting from mean shifts also add to the tail mass.

Proposition 2 states results for a "rst-order autoregressive process whosemoments are analytically tractable to derive. Similar analytical expressions forhigher-order autoregressive Markov switching processes contain the same typesof components and are not very insightful to derive as they quickly becomeintractable. However, we still need to have a method for obtaining the momentsof speci"c higher order processes that researchers have in mind. We demonstratehow to do this in the case of the second moment of such processes and note thatthe skewness and kurtosis can also be derived using similar techniques. In thegeneral case with an arbitrary, but "nite, number of autoregressive lags, thesecond moment of the Markov process can be derived in one of two ways. First,from (2) we have

E[(yt!k)2]"E[(k

st!k)2]#

q+j/1

/2jE[(y

t~j!k

st~j)2]#E[p2

ste2t]

#2q+i;j

q+j/1

/i/jE[(y

t~i!k

st~i)(y

t~j!k

st~j)]. (10)

Applying the steady-state probabilities to the "rst and third terms we get

E[(yt!k)2]"p@((l

s!ki)((l

s!ki))#

q+j/1

/2jE[(y

t~j!k

st~j)2]#p@r2

s

#2q+i;j

q+j/1

/i/jE[(y

t~i!k

st~i)(y

t~j!k

st~j)]. (11)

This equation involves variances and covariances of the process relative to thestate-speci"c means. To derive an expression for these terms, we exploit theassociated Yule}Walker equations. Multiplying (2) through by (y

t~1!k

st~1),

2,(yt~q

!kst~q

), we get a system of equations

E[(yt!k

st)(y

t~m!k

st~m)]"

q+i/1

/iE[(y

t~i!k

st~i)(y

t~m!k

st~m)],

m"1,2,q. (12)

An extra term p@r2sappears on the right-hand side of (12) when m"0. This gives

(q#1) equations that can be used to jointly determine E[(yt!k

st)2],2,

E[(yt!k

st)(y

t!k

st~q)]. These terms can then be substituted into (11) to get the

unconditional variance of the process around k.An alternative, and as far as we know, new method for obtaining second

moments of Markov switching processes combines the companion form of theautoregressive model with the technique of expanding the state space of aMarkov process proposed by Cox and Miller (1965). This is very convenient


from a computational point of view and does not require solving a set of q#1equations in the autocovariances. We can always write (2) as

Cyt

yt~1

yt~2)

)

yt~q`1

D"Ckst

kst~1

kst~2

)

)

kst~q`1

D#C/

1/2

) ) /q~1

/q

1 0

1 0

1

0 1

1 0D C

yt~1

!kst~1

yt~2

!kst~2

yt~3

!kst~3

)

)

yt~q

!kst~q

D#C

pst

0 ) ) 0 0

0 0

0 0

0

0 0

0D C

et

et~1

et~2)

)

et~q`1

D, (13)

or, in matrix form,

zt"l

sHt#W(z

t~1!l

sHt~1

)#+sHtet, (14)

where lsHt

now lies in the kH"k ) q dimensional state space formed as theCartesian product S

t]S

t~1]S

t~q`1of the original q state spaces.

Taking expectations of ztaround the unconditional mean vector we get the

q]q covariance matrix

E[(zt!ki)(z

t!ki)@]"E[(l

sHt!ki)(l

sHt!ki)@]

#WE[(zt~1

!lsHt~1

)(zt~1

!lsHt~1

)@]W@

#EC+sHtete@t+ @

sHt D , (15)

where we used that zt~1

!lsHt~1"+=

i/0Wi+

sHt~i~1

et~i~1

is uncorrelated with(k

sHt!ki) and +

sHtet. Notice also from (14) that

E[(zt!l

sHt)(z

t!l

sHt)@]"WE[(z

t~1!l

sHt~1

)(zt~1

!lsHt~1

)@]W@

#EC+sHtete@t+ @

sHt D, (16)

where E[+sHtete@t+@

sHt] is a q]q matrix whose "rst element is E[p2

st] with all other

elements being zero. We refer to this matrix as <. It then follows that, provided


zt

is a stationary process so E[(zt!l

sHt)(z

t!l

sHt)@]"E[(z

t~1!l

sHt~1

)(zt~1

!lsHt~1

)@], the q2-vector of second moments centered around the statemeans is

vec(E[(zt!l

sHt)(z

t!l

sHt)@])"(W?W) ) vec(E[(z

t!l

sHt)(z

t!l

sHt)@])

#vecAEC+ sHtete@t+ @

sHt DB, (17)

so that

vec(E[(zt!l

sHt)(z

t!l

sHt)@])"(I

q2!(W?W))~1vec(<). (18)

Substituting this back into (15) we get the q2-vector of unconditional secondmoments:

vec(E[(zt!ki)(z

t!ki)@])"vec(E[(l

sHt!ki)(l

sHt!ki)@])

#(Iq2#(W?W)(I

q2!(W?W))~1)vec(<).

(19)

This expression is very convenient for computation of the variance of higher-order autoregressive Markov switching models. It also allows calculation ofautocovariances from the cross-product terms. For example, E[(y

t!k)

(yt~1

!k)] will be the second element of the vector in (19).

4. State-dependent autoregressive dynamics

Again we initially consider the "rst-order autoregressive model and thendemonstrate how to generalize the results to the less tractable, but similar,general case. The underlying model (which we shall refer to as MSIII) is

yt"k

st#/

1st~1(y

t~1!k

st~1)#p

stet, D/

st~1D(1, e

t&IIN(0,1). (20)

Backwards substitution gives

yt!k

st"

=+l/1A

l<j/1

/1st~j

pst~jBet~l

#pstet, (21)

so that E[yt!k

st]"0 and E[y

t]"p@ls"k. To state parsimoniously the mo-

ments of this process, it is convenient to de"ne the (k]k) diagonal matrix ofstate-speci"c autoregressive coe$cients

U"C/11

0

0 /12

)

) )

/1kD,


where /1r

is the "rst-order autoregressive coe$cient in state r. Proposition3 provides the second to fourth moments of this process:

Proposition 3. Consider the xrst-order Gaussian Markov switching process witha state-dependent autoregressive term

yt"k

st#/

1st~1(y

t~1!k

st~1)#p

stet, e

t&IIN(0,1), D/

st~1D(1,

and suppose that the process started from its steady state. The variance, skewnessand kurtosis of y

tare given by

E[(yt!k)2]"p@((l

s!ki)((l

s!ki)#U2(I

k!BU2)~1r2

s#r2

s),

E[(yt!k)3]"p@((l

s!ki)((l

s!ki)((l

s!ki))

#3p@((BU2(Ik!BU2)~1r2

s)((l

s!ki))

#3p@((ls!ki)(r2

s),

E[(yt!k)4]"p@((l

s!ki)((l

s!ki)((l

s!ki)((l

s!ki))

#6p@((BU2(Ik!BU2)~1r2

s)((l

s!ki)((l

s!ki))

#6p@((ls!ki)((l

s!ki)(r2

s)

#p@U4(Ik!BU4)~1(3r4

s#6(BU2(I

k!BU2)~1r2

s)(r2

s)

#6p@((BU2(Ik!BU2)~1r2

s)(r2

s)#3p@r4

s.

To demonstrate the e!ect on the coe$cients of skewness and kurtosis of havingdi!erent autoregressive parameters in the di!erent states, Fig. 6 keeps theparameters from Fig. 5 but sets /

11"0.99 and /

12"0.81, so that, compared

with the choice of /1

in Fig. 5, the serial correlation is now 0.09 higher in state1 and 0.09 lower in state 2. The skewness is now positive for high values ofp11

and p22

and is otherwise negative.7 The kurtosis still peaks when p11

andp22

are large but is now #at on a larger part of the grid. Letting the auto-regressive coe$cients di!er between the two states can signi"cantly changethe skewness and kurtosis pro"les and a comparison of Figs. 2, 5 and 6 illustratethe complexity of the interaction between the Markov switching parametersand the autoregressive parameters in determining the higher-ordermoments.

7 The positive skewness for high values of the transition probabilities is caused by the termE[(k

st!k)/2

1st~1(y

t~1!k

st~1)2] which is positive when a switch occurs from state 2 to state 1. This

term is higher in Fig. 6 (/1st~1

"0.99 for st~1

"2) than in Fig. 5 (/1"0.90).


Fig

.6.

Skew

ness

and

kurt

osis

of

the

two-s

tate

auto

regr

essive

Mar

kov

switch

ing

mod

el:e!

ect

ofdi!

eren

tau

tore

gres

sive

para

met

ers.

Bas

edon

the

para

met

erva

lues

k s"(1

,!3)

,p s"

(2,4

),/

11"

0.99

,/12"

0.81

,th

e"gu

replo

tsth

eco

e$ci

ents

ofsk

ewnes

san

dex

cess

kurt

osis

of

atw

o-s

tate

auto

regr

essive

Mar

kov

switch

ing

mode

lwith

Gau

ssia

nin

nov

atio

nsas

afu

nct

ion

ofth

epro

babili

tyof

stay

ing

inst

ate1

(P11

)and

thepro

bab

ility

ofs

tayi

ng

inst

ate

2(P

22).


Second moment results for higher-order autoregressive processes can againbe obtained through the companion form of y

tin the expanded state space:

Cyt

yt~1

yt~2)

)

yt~q`1

D"Ckst

kst~1

kst~2

)

)

kst~q`1

D#C/

1st~1/

2st~2) ) ) /

qst~q

1 0

1 0

1

0 1

1 0D C

yt~1

!kst~1

yt~2

!kst~2

yt~3

!kst~3

)

)

yt~q

!kst~q

D#C

pst

0 ) ) 0 0

0 0

0 0

0

0 0

0D C

et

et~1

et~2)

)

et~q`1

D, (22)

or, in matrix form,

zt"l

sHt#W

sHt~1

(zt~1

!lsHt~1

)#+sHtet. (23)

To analyze this case we "rst derive an expression for E[(zt!l

sHt)(z

t!l

sHt)@].

Let vec(E[(zt!ik

sHt)2 D SH

t"i]) be short-hand notation for the q2-vector of ex-

pectations of (zt!ik

sHt)(z

t!ik

sHt)@ conditional on SH

t"i, let <

i,

vec(E[+sHtete@t+@

sHtDSH

t"i]) be a q2 vector of conditional expectations of the

residual term and let Wibe the q]q matrix W

sHt~1

conditional on SHt~1

"i. As inSection 3, the dimension of the expanded state space is kH"kq. Stacking themoment conditions resulting from (23), we get the kHq2-vector

Cvec(E[(z

t!ik

sHt)2 DSH

t"1])

vec(E[(zt!ik

sHt)2 DSH

t"2])

2

vec(E[(zt!ik

sHt)2 D SH

t"kH])D

"(BH?Iq2)C

(W1?W

1)vec(E[(z

t~1!ik

sHt~1

)2 D SHt~1

"1])

(W2?W

2)vec(E[(z

t~1!ik

sHt~1

)2 D SHt~1

"2])

2

(WkH?W

kH)vec(E[(z

t~1!ik

sHt~1

)2 D SHt~1

"kH])D


#C<

1<

2

2

<kHD (24)

"(BH?Iq2)WK C

vec(E[(zt~1

!lsHt~1

)2 D SHt~1

"1])

vec(E[(zt~1

!lsHt~1

)2 D SHt~1

"2])

2

vec(E[(zt~1

!lsHt~1

)2 DSHt~1

"kH])D#C<

1<

2

2

<kHD,

where BH is the (kH]kH) matrix of backward transition probabilities while WK isthe (kHq2)](kHq2) block-diagonal matrix formed as

WK "CW

1?W

10

W2?W

20 2

WkH?W

kHD.

Under stationarity of the ytprocess we thus have

Cvec(E[(z

t!ik

sHt)2 DSH

t"1])

vec(E[(zt!ik

sHt)2 DSH

t"2])

2

vec(E[(zt!ik

sHt)2 D SH

t"kH])D"(I

m!(BH?I

q2)WK )~1C

<1<

2

2

<kHD, (25)

where m"kHq2. The q2-vector of unconditional second moments centeredaround the state means can now be extracted from (25) by pre-multiplying bya (q2]q2kH) matrix K given by

K"((pH)@?Iq2)

"Cn1

0 2 n2

0 0 2 nkH 2 0

0 n1

0 2 n2

0 2 0 2 0

2 2 2 2 2 2 2 2 2 2

0 2 n1

0 2 n2 2 0 2 n

kHD ,

so that

E[(zt!l

sHt)(z

t!l

sHt)@]"K(I

m!(BH?I

q2)WK )~1(<

1<

22<

kH)@. (26)


Finally, the unconditional variance of ztfollows from (23) as

vec(E[(zt!ki)(z

t!ki)@])"vec(E[(l

sHt!ki)(l

sHt!ki)@])

#K )WK (Im!(BH?I

q2)WK )~1C

<1<

2.

<kHD#vec(<),

(27)

where<"E[+sHtete@t+@

sHt]. Again these expressions are easy to implement, do not

require solving a set of Yule}Walker equations, and should prove useful toresearchers wanting to characterize the moments of this quite complex class ofMarkov switching processes with state-dependent autoregressive parameters.

5. Autocorrelations of the Markov switching processes

To fully understand the dynamic patterns that the Markov switching modelsgenerate, we need to characterize their autocovariance functions. Just howstrong the serial correlation is for a given set of parameters can be computedfrom Proposition 4:8

Proposition 4. The autocovariance functions of the stationary Gaussian Markovswitching processes MSI, MSII and MSIII starting from their steady state are givenby

E[(yt`n

!k)(yt!k)]

"p@((Bn(ls!ki))((l

s!ki)) (MSI)

p@((Bn(ls!ki))((l

s!ki))#/n

1p@(I

k!/2

1B)~1r2

s(MSII)

p@((Bn(ls!ki))((l

s!ki))#p@C

n(I

k!BU2)~1r2

s(MSIII)

8We remind the reader that the three Markov switching models are

MSI: yt"k

st#p

stet, e

t&IIN(0,1),

MSII: yt"k

st#/

1(y

t~1!k

st~1)#p

stet, e

t&IIN(0,1), D/

1D(1,

MSIII: yt"k

st#/

1st~1(y

t~1!k

st~1)#p

stet, e

t&IIN(0,1), D/

1st~1D(1.


where Cnis the (k]k) diagonal matrix

Cn"C

c1,n

0

0 c2,n

.

. .

ck,nD

with diagonal elements cr,n"E[(<n~1

i/0/

1st`i) DS

t"r], so that C

1"U.

Notice that we can compute the elements of Cnas

cr,n"/

1r

k+

j1/1

k+

j2/1

2

k+

jn~1/1

Prj1

Pj1j22

Pjn~2jn~1

/j1/js2

/jn~1

which is the product of the probabilities and the associated autoregressivecoe$cients on paths emanating from state r. For higher order autocovariances,the expression for C

nwill be quite complex since it re#ects the entire set of

possible paths followed by the process over the n-period horizon.Again the basic Markov switching model with two states provides some

intuition for the result since it allows us to considerably simplify theautocovariance function

E[(yt!k)(y

t~n!k)]"n

1(1!n

1)(k

1!k

2)2vec(Pn)@v

1, (28)

where v1"((1!n

1),!(1!n

1),!n

1, n

1)@. Of particular interest is the "rst-

order autocovariance for the levels of the process which is given by(k

1!k

2)2n

1(1!n

1)(p

11#p

22!1). This expression has an intuitive interpreta-

tion since the autocovariance will be positive provided the presence of theprocess in the two states is persistent (p

11#p

22'1), otherwise a negative

autocovariance will result. Without a Markov switching e!ect (p11"1!p

22)

there will be no "rst order autocorrelation in the process.Many studies "nd signi"cant serial correlation in the squared values of

economic time series. The success of ARCH models in empirical work can beexplained by the fact that these models can generate such autocorrelationpatterns. We show in the following Proposition that Markov switching modelscan also give rise to autocorrelation in the squares of a time series:

Proposition 5. The autocovariance function of the squared values of the stationaryMarkov switching process starting from its steady state is given by

E[(y2t!E[y2

t])(y2

t~n!E[y2

t~n])]

"p@((Bn(r2s#l2

s))((l2

s#r2

s))!(E[y2

t])2 (MSI)


"p@((Bn(Ik!/2

1B)~1r2

s)(l2

s)#p@((Bnl2

s)(l2

s)

#/2n1

p@(Ik!/4

1B)~1(3r4

s#6/2

1((B(I

k!/2

1B)~1r2

s)(r2

s))

#/2n1

p@(((Ik!/2

1B)~1r2

s)(l2

s)

#p@An+i/1

/2(n~i)1

(Bi(Ik!/2

1B)~1r2

s#Bil2

s)(r2

sB# 4/n

1p@(Bn(((I

k!/2

1B)~1r2

s)(l

s))(l

s)!(E[y2

t])2 (MSII)

"p@((Bn(Ik!BU2)~1r2

s)(l2

s)#p@((Bnl2

s)(l2

s)

#p@B2,n

(Ik!BU4)~1(3r4

s#6(BU2(I

k!BU2)~1r2

s)(r2

s)

#p@((B2,n

(Ik!BU2)~1r2

s)(l2

s)

#p@An+i/1

(Bi(I

k!BU2)~1r2

s#B

il2s)B

# 4p@(Cn(((I

k!BU2)~1r2

s)(l

s)!(E[y2

t])2, (MSIII)

where B2,n

is a diagonal k]k matrix whose rth diagonal element is given byB

2,n[r, r]"E[(<n~1

i/0/2

1stì) DS

t"r], B

iis a diagonal matrix with elements

Bi[r, r]"E[(<n~1

j/i/21st`j

)p2stì

D St"r] and C

nis a diagonal matrix with C

n[r, r]"

E[(<n~1i/0

/1stì

)kst`n

D St"r].

Proofs of Propositions 4 and 5 are provided in the appendix. To compute theautocorrelation, the autocovariances need to be scaled by the variance of y2

t,

E[(y2t!E[y2

t])2]. For the three models, this can be shown to be given by

E[(y2t!E[y2

t])2]"E[y4

t]!(E[y2

t])2

"p@l4s#6p@(l

s(l

s(r2

s)

#3p@r4s!(var(y

t)#E[y

t]2)2 (MSI)

"p@l4s#6/2

1p@((B(I

k!/2

1B)~1r2

s)(l

s(l

s)

#6p@(l2s(r2

s)

#p@(Ik!/4

1B)~1(3r4

s#6/2

1(B(I

k!/2

1B)~1r2

s)(r2

s)

!(var(yt)#E[y

t]2)2 (MSII)

"p@l4s#6p@((BU2(I

k!BU2)~1r2

s)((l2

s#r2

s))

# 6p@(l2s(r2

s)#p@U4(I

k!BU4)

](3r4s#6(BU2(I

k!BU2)~1r2

s)(r2

s)

# 3p@r4s!(var(y

t)#E[y

t]2)2. (MSIII)


Intuition is provided by the autocovariance of y2t

in the basic two-state modelwhich is given by

E[(y2t!E[y2

t])(y2

t~n!E[y2

t~n])]

"n1(1!n

1)(k2

2!k2

1#p2

2!p2

1)2vec(Pn)@v

1. (30)

Hence the "rst-order autocovariance is E[(y2t!E[y2

t])

(y2t~1

!E[y2t~1

])]"n1(1!n

1)(k2

2!k2

1#p2

2!p2

1)2(p

11#p

22!1). Again this

expression has the intuitive interpretation that there will be positive autocorre-lation in y2

tif the state mixing process is persistent (p

11#p

22'1), otherwise the

squared values of the process will be negatively autocorrelated. If p11"1!p

22,

there will be no "rst-order serial correlation in the squared values of the series. Ifthere is no switching between the two regimes, i.e. n

1(1!n

1)"0, there will be

no autocorrelation in the squared values of the series.Fig. 7 plots the "rst-order autocorrelation in the squared levels of a Markov

switching process for the case where ls"(1, 1) and r

s"(2, 4). Since the two

mean parameters are identical across states, there is no serial correlation in thelevel of the series. Re#ecting the di!erent volatility parameters, quite high serialcorrelation in the squares of the series is obtained, however, when the persistenceof the two states is high.

6. Discussion of results

Our results on the moments of the regime switching models show that theMarkovian dependency in the mixture probabilities signi"cantly expands thescope for asymmetry and fat tails that can be generated by time-independentmixture models. This may help to explain the relative success that these modelshave had in applications to economic time series that clearly display suchfeatures.

Our theoretical results are also closely related to the literature on structuralbreaks and persistence. Structural breaks are usually thought of as one-o!events that introduce non-stationarities in the data. In the context of a Markovswitching model a structural break can be modeled as follows. Consider thefollowing partitioned matrix of transition probabilities:

P"CP

1P2

P3

P4D,

where, say, P1is a k

1]k

1matrix, P

4is a k

2]k

2matrix and k

1#k

2"k. If some

element of P2is nonzero while the elements of P

3all equal zero, then once a state

ordered k1#1 or higher is reached then the process can never return to a state

ordered k1or lower, so this event represents a break from a switching model that

mixes one block of states to another model mixing a di!erent block of states. If


Fig. 7. Autocorrelation in the squared levels of a two-state Markov switching model. Based on theparameter values k

s"(1, 1), p

s"(2, 4), the "gure plots the "rst-order autocorrelation of the squared

levels of a two-state Markov switching model with Gaussian innovations as a function of theprobability of staying in state 1 (P

11) and the probability of staying in state 2 (P

22).

the values of P2

and P3

are close to, but not all equal to, zero then there existsa stationary &hyper model' that includes both blocks of states, but it will be verydi$cult in any "nite sample to distinguish between a structural break andinfrequently occurring regime switches.

The parameter values for which the Markov switching models seem bestcapable of generating volatility clustering and autocorrelation, i.e. those valuesrepresenting infrequent mixing of regimes with quite di!erent mean and vari-ance parameters, are also those for which the switching models most closelyresemble structural break models. Thus, our "ndings are clearly related to theliterature on structural breaks and persistence in the "rst and second momentsof time series. Perron (1989) considers a single exogenous shift in the level orslope of the trend function of a time series while Perron (1990) investigates thecase of an exogenous break in the mean level. This latter case closely corres-ponds to a Markov switching model with k"2, p

1"p

2, and k

1Ok

2. In both


cases such a break in the time series is found to lead to higher rates ofnon-rejection of the null of a unit root since it introduces persistence in the seriescentered around the sample mean. As the size of the mean shift gets larger, theestimated autoregressive parameter in a model without serial correlation butwith a single shift in the mean gets closer to one.9

We carried out an experiment to show that these studies are closely related toProposition 4. Setting p2

1"p2

2"1, p

11"p

22"0.99, k

1"1, and varying k

2we

obtained the following population moments for the "rst-order autocorrelationof the process:

First-order autocorrelationValue of k

2MSI MSII (/

1"0.9)

2 0.196 0.9043 0.495 0.9164 0.688 0.930

Even when there is no autocorrelation in any of the states (as for MSI), a shift inthe mean can still produce very substantial persistence around the uncondi-tional mean, and the e!ect is an increasing function of the size of the shift. As theshift gets large, and consistent with Perron (1990), the autocorrelation coe$cientfor this case with rare shift gets closer to one.

The proposition that persistence in the squared level of a Markov switchingprocess can be the result of rare, large shifts in the unconditional variance is alsorelated to earlier empirical "ndings. Lamoureux and Lastrapes (1990) studysecond moments and "nd that the existence of deterministic structural shifts inthe unconditional variance, when not accounted for, will increase the persistenceof squared residuals.10 They show that the resulting upward bias in GARCHestimates of persistence of variance can be quite substantial. For a number ofindividual stock return series once a structural break in the intercept term in theconditional variance equation is accounted for, the estimated persistence de-clines from an average of 0.98 to an average of 0.82. Indeed, Lamoureux andLastrapes propose to use Markov switching models as a way of handlingmisspeci"ation problems due to occasional shifts in the conditional variance.Likewise, Diebold (1986) suggests that the very high persistence in the condi-tional variance observed in GARCH models may re#ect a failure to include

9 In related work, Hendry and Neale (1991) use Monte Carlo methods to quantify the e!ect thatthe introduction of shifts in the intercept of an autoregressive process has on the loss in the power ofstandard unit root tests.

10Yet another possibility is that a misspeci"ed mean leads to overrejection of the null of noconditional heteroskedasticiy as recently reported by Lumsdaine and Ng (1997).


dummies for shifts in the intercept in the variance equation caused by exogenousshocks such as monetary policy regime changes.11

Appendix

This appendix contains the proofs of the propositions and the corollary.

Proof of Proposition 1. From the law of iterated expectations we have

E[(yt!k)n]"E[E[(y

t!k)nDS

t]]"

k+i/1

niE[(k

i#p

iet!k)n]

"

k+i/1

ni

n+j/0

nC

jpji(k

i!k)n~jE[ej

t], (A.1)

where we used Newton's binomial formula and the assumption that thesteady-state probabilities apply. The expressions for the cases where e

tfollows a t-distribution or a normal distribution are based on the moment-generating functions for these distributions. For example, from the moment-generating function of the normal distribution we have that E[ej

t]"0, if j is odd

and

E[ejt]"

j@2<h/1

(2h!1),bj,

otherwise. Substituting this expression into (A.1) we get the result.

Proof of Corollary 1. First consider the variance. From the law of iteratedexpectations we have

E[(yt!k)2]"E[E[(y

t!k)2DS

t]]

"n1E[(k

1#p

1et!k)2]#(1!n

1)E[(k

2#p

2et!k)2]

"(1!n1)p2

2#n

1p21#(1!n

1)(k

2!k)2#n

1(k

1!k)2

"(1!n1)p2

2#n

1p21#n

1(1!n

1)(k

2!k

1)2. (A.2)

11 In principle, ARCH and Markov switching e!ects could be combined to produce a highly#exible, nonlinear mixture model as proposed by Hamilton and Susmel (1994). Our results canexplain why such an extension may be necessary because the Markov switching model only appearsto be able to generate limited persistence in the squared values of a time series.


We next compute the skewness and kurtosis of this model:

E[(yt!k)3]"E[E[(y

t!k)3DS

t]]

"(1!n1)E[(k

2#p

2et!k)3]#n

1E[(k

1#p

1et!k)3]

"n1(1!n

1)(k

1!k

2)M3(p2

1!p2

2)#(1!2n

1)(k

2!k

1)2N,

(A.3)

and hence the coe$cient of skewness is given by

E[(yt!k)3]

(E[(yt!k)2])3@2

"

n1(1!n

1)(k

1!k

2)M3(p2

1!p2

2)#(1!2n

1)(k

2!k

1)2N

((1!n1)p2

2#n

1p21#n

1(1!n

1)(k

2!k

1)2)3@2

.

(A.4)

To compute the coe$cient of kurtosis we proceed as follows:

E[(yt!k)4]"E[E[(y

t!k)4DS

t]]

"(1!n1)E[(k

2#p

2et!k)4]#n

1E[(k

1#p

1et!k)4]

"(1!n1)M3p4

2#(k

2!k)4#6p2

2(k

2!k)2N

#n1M3p4

1#(k

1!k)4#6p2

1(k

1!k)2N. (A.5)

Simplifying the expression by using that k"n1k1#(1!n

1)k

2, we get the

coe$cient of excess kurtosis

E[(yt!k)4]!3(E[(y

t!k)2])2

(E[(yt!k)2])2

,

a

b,

a"3n1(1!n

1)(p2

2!p2

1)2#6(k

2!k

1)2n

1(1!n

1)(2n

1!1)(p2

2!p2

1)

#n1(1!n

1)(k

2!k

1)4(1!6n

1(1!n

1)),

b"((1!n1)p2

2#n

1p21#n

1(1!n

1)(k

2!k

1)2)2. (A.6)

Proof of Proposition 2. Squaring ytaround its unconditional mean and taking

expectations we have

E[(yt!k)2]"E[(k

st!k)2#/2

1(y

t~1!k

st~1)2#p2

ste2t]

#2/1Cov(k

st!k, y

t~1!k

st~1)#2Cov(k

st!k, p

stet)

#2/1Cov(y

t~1!k

st~1, p

stet). (A.7)


From (7), the assumption that etis iid and the independence at all leads and lags

between etand S

t, it follows that all three covariance terms are equal to zero so

that

E[(yt!k)2]"p@E[(y

t!ki)((y

t!ki)DS

t]

"p@(ls!ki)((l

s!ki)#p@/2

1A=+i/0

/2i1Pir2

sB#p@r2s

"p@((ls!ki)((l

s!ki)#/2

1(1!/2

1)~1r2

s#r2

s)

"p@A(ls!ki)((l

s!ki)#

r2s

1!/21B, (A.8)

where i is a k-vector of ones, ( is the element-by-element multiplicationoperator and E[(y

t!ki)((y

t!ki)DS

t] stacks the vector E[(y

t!k)(

(yt!k)DS

t"i] for i"1,2,k. The third equality uses the property of the

steady-state probabilities that p@Pi"[email protected] third centered moment can be derived using the result that

E[(yt!k)3]"E[(k

st!k)3]#3/2

1E[(k

st!k)(y

t~1!k

st~1)2]

#3E[(kst!k)p2

ste2t], (A.9)

which can be veri"ed by expanding Eq. (6) around k. To derive an expression forthe second term notice that

E[(yt!k

st)2DS

t"i]"/2

1

k+j/1

E[(yt~1

!kst~1

)2DSt~1

"jWSt"i] '

Prob(St~1

"jDSt"i)#p2

i

"/21

k+j/1

E[(yt~1

!kst~1

)2DSt~1

"j] ) bij#p2

i, (A.10)

since the expectation of (yt~1

!kst~1

)2 conditional on St~1

is the same as itsexpectation conditional on S

t~1and S

t. Stacking the equations resulting from

setting i"1,2,k, we have

E[(yt!l

st)2D S

t]"/2

1)B )E[(y

t~1!l

st~1)2D S

t~1]#r2

s, (A.11)

so that, under stationarity of the process,

E[(yt~1

!lst~1

)2D St~1

]"(Ik!/2

1B)~1 )r2

s, (A.12)

where Ik

is the k-dimensional identity matrix. Using this in (A.11) and notingthat E[(y

t~1!l

st~1)2D S

t]"B(I

k!/2

1B)~1 )r2

s, we get

E[(yt!k)3]"p@((l

s!ki)((l

s!ki)((l

s!ki))

#3/21p@((B(I

k!/2

1B)~1r2

s)((l

s!ki))

#3p@((ls!ki)(r2

s). (A.13)


The fourth centered moment is given by

E[(yt!k)4]"E[(k

st!k)4]#6/2

1E[(k

st!k)2(y

t~1!k

st~1)2]

#6E[(kst!k)2p2

ste2t]#/4

1E[(y

t~1!k

st~1)4]

#6/21E[(y

t~1!k

st~1)2p2

ste2t]#E[p4

ste4t]. (A.14)

Most of these terms are simple to evaluate, but notice that

E[(yt!l

st)4D S

t]"/4

1B )E[(y

t~1!l

st~1)4D S

t~1]

#3r4s#6/2

1E[(y

t~1!l

st~1)2r2

stD S

t].

From (6) we also have

E[(yt~1

!lst~1

)2r2stD S

t]"(B )E[(y

t~1!l

st~1)2D S

t~1])(r2

s

"(B ) (Ik!/2

1B)~1r2

s)(r2

s.

Thus, provided the process is stationary, we get

E[(yt!l

st)4D S

t]"(I

k!/4

1B)~1(3r4

s#6/2

1(B ) (I

k!/2

1B)~1r2

s)(r2

s).

(A.15)

Using this together with the equation E[(yt!k)4]"p@E[(y

t!ki)4DS

t], (A.14)

becomes

E[(yt!k)4]"p@((l

s!ki)((l

s!ki)((l

s!ki )((l

s!ki))

#6/21p@((B(I

k!/2

1B)~1r2

s)((l

s!ki)((l

s!ki))

#6p@((ls!ki)((l

s!ki)(r2

s)

#/41p@(I

k!/4

1B)~1(3r4

s#6/2

1(B ) (I

k!/2

1B)~1r2

s)(r2

s)

#6/21p@((B ) (I

k!/2

1B)~1r2

s)(r2

s)#3p@r4

s. (A.16)

Collecting terms we get the expression in Proposition 2.

Proof of Proposition 3. The unconditional variance of the process underlyingMSIII can be evaluated from the expression

E[(yt!k)2]"E[(k

st!k)2#/2

1st~1(y

t~1!k

st~1)2#p2

ste2t]

#2Cov(kst!k, p

stet)#2Cov(k

st!k, /

1st~1(y

t~1!k

st~1))

#2Cov(/1st~1

(yt~1

!kst~1

), pstet). (A.17)

Again the covariance terms are zero so only the second term has changed from(A.7). To evaluate this term we condition on S

tto get the set of equations

E[(yt!l

st)2D S

t]"B )U2E[(y

t~1!l

st~1)2D S

t~1]#r2

s, (A.18)


where U is the k]k diagonal matrix de"ned just before Proposition 3. Understationarity of y

twe have

E[(yt~1

!lst~1

)2D St~1

]"(Ik!BU2)~1r2

s, (A.19)

from which the variance of yteasily follows from (A.17):

E[(yt!k)2]"p@((l

s!ki)((l

s!ki))#p@(U2(I

k!BU2)~1r2

s#r2

s).

(A.20)

To derive the third moment, notice that the only new term relative to (A.9) is3E[(k

st!k)/2

1st~1(y

t~1!k

st~1)2]. Conditioning on S

tand applying the steady-

state probabilities, we get

E[(kst!k)/2

1st~1(y

t~1!k

st~1)2]

"p@((B )U2(Ik!BU2)~1r2

s)((l

s!ki)). (A.21)

Inserting this into (A.9) and using the similar results for MSII, the skewnessresult follows immediately. The fourth moment of MSIII can be evaluated fromthe expression

E[(yt!k)4]"E[(k

st!k)4]#6E[(k

st!k)2/2

1st~1(y

t~1!k

st~1)2]

#6E[(kst!k)2p2

ste2t]#E[/4

1st~1(y

t~1!k

st~1)4]

#6E[/21st~1

(yt~1

!kst~1

)2p2ste2t]#E[p4

ste4t]. (A.22)

The second, fourth and "fth terms in this expression have changed comparedwith (A.14). We derive these expressions as follows:

E[(kst!k)2/2

1st~1(y

t~1!k

st~1)2]

"p@((BU2(Ik!BU2)~1r2

s)((l

s!ki)((l

s!ki)), (A.23)

E[/21st~1

(yt~1

!kst~1

)2p2ste2t]"p@((BU2(I

k!BU2)~1r2

s)(r2

s). (A.24)

Finally, conditioning on Stwe get the system of equations

E[(yt!l

st)4D S

t]"BU4 )E[(y

t~1!l

st~1)4D S

t~1]#3r4

s

#6(BU2(Ik!BU2)~1r2

s)(r2

s. (A.25)

Hence

E[(yt!l

st)4D S

t]"(I

k!BU4)~1(3r4

s#6(BU2(I

k!BU2)~1r2

s)(r2

s),

(A.26)

and the fourth term entering (A.22) is p@U4 times this expression. Inserting theseterms into (A.22), we get the kurtosis expression stated in Proposition 3.


Proof of Proposition 4. We "rst compute E[(yt!k)(y

t~1!k)] for MSI and then

demonstrate how to generalize the result. Notice that

E[(yt!k)(y

t~1!k)]

"E[((kst!k)#p

stet)((k

st~1!k)#p

st~1et~1

)]

"E[(kst!k)(k

st~1!k)]"p@((B(l

s!ki))((l

s!ki)). (A.27)

When we consider the nth-order autocovariance of yt, the only part of the

calculation that changes is the transition probabilities, i.e. instead of using B weuse Bn.

To compute the autocovariances of MSII, notice that

E[(yt!k)(y

t~1!k)]"E[((k

st!k)#/

1(y

t~1!k

st~1)#p

stet)

((yt~1

!kst~1

)#(kst~1

!k))]. (A.28)

Obviously, E[pstet(y

t~1!k

st~1)]"E[p

stet(k

st~1!k)]"0, and, from the indepen-

dence of et~i

and St,

E[((kst!k)(y

t~1!k

st~1))]"E[(k

st~1!k)/

1(y

t~1!k

st~1)]"0. (A.29)

This leaves us with the terms E[(kst!k)(k

st~1!k)#/

1(y

t~1!k

st~1)2], and

hence

E[(yt!k)(y

t~1!k)]"p@((B(l

s!ki))((l

s!ki))

#/1p@(I

k!/2

1B)~1r2

s. (A.30)

This can easily be generalized to obtain the nth-order autocovariance by notingthat

yt`n

!k"kst`n

!k#/n1(y

t!k

st)#

n+i/1

/n~i1

pst`i

et`i

, (A.31)

so that the autocovariance becomes

E[(yt`n

!k)(yt!k)]"E[(k

st`n!k)(k

st!k)#/n

1(y

t!k

st)2]

"p@((Bn(ls!ki))((l

s!ki))#/n

1p@(I

k!/2

1B)~1r2

s.

(A.32)

To obtain the "rst-order autocovariance of the MSIII process, we need toevaluate E[(k

st!k)(k

st~1!k)#/

1st~1(y

t~1!k

st~1)2]:

E[(yt!k)(y

t~1!k)]"p@((B(l

s!ki))((l

s!ki))#p@U(I

k!BU2)~1r2

s.

(A.33)

Second- and higher-order autocorrelations do not follow as easily from thisexpression as (A.32) follows from (A.31), however, due to the state dependence in


/. To evaluate the nth-order autocovariance, we need to compute

E[(yt`n

!k)(yt!k)]"E[(k

st`n!k)(k

st!k)]

#ECAn~1<i/0

/1st`iB(yt

!kst)2D. (A.34)

This expression can be written as stated in Proposition 4.

Proof of Proposition 5. Again we "rst derive the result for the simple MS model(MSI). Note that for this process

y2t"(k2

st#2k

stpstet#p2

ste2t), (A.35)

y2t~1

"(yt~1

!kst~1

)2#2kst~1

(yt~1

!kst~1

)#k2st~1

, (A.36)

so their expected cross-product is

E[y2ty2t~1

]"E[k2st(y

t~1!k

st~1)2#k2

stk2st~1

#p2ste2t(y

t~1!k

st~1)2#p2

ste2tk2st~1

]

"p@((Br2s)(l2

s)#p@((Bl2

s)(l2

s)#p@((Br2

s)(r2

s)

#p@((Bl2s)(r2

s), (A.37)

where we used that, for the simple Markov switching model,E[(y

t!l

s)2D S

t]"r2

s. The expression in Proposition 5 follows by collecting

terms. Once again, the general expression for E[(y2t!E[y2

t])(y2

t~n!E[y2

t~n])]

can be derived by substituting Bn for B.For MSII, the "rst-order autocovariance of y

tcan be based on

y2t!E[y2

t]"k2

st#/2

1(y

t~1!k

st~1)2#p2

ste2t#2k

st/

1(y

t~1!k

st~1)

#2kstpstet#2/

1(y

t~1!k

st~1)p

stet!E[y2

t], (A.38)

and

y2t~1

!E[y2t~1

]"(yt~1

!kst~1

)2#2kst~1

(yt~1

!kst~1

)#k2st~1

!E[y2t~1

].

(A.39)

Using that the expected value of cross-product terms in which etenters linearly is

zero we get

E[(y2t!E[y2

t])(y2

t~1!E[y2

t~1])]

"E[k2st(y

t~1!k

st~1)2#k2

stk2st~1

#/21(y

t~1!k

st~1)4#/2

1(y

t~1!k

st~1)2k2

st~1

#p2ste2t(y

t~1!k

st~1)2#p2

ste2tk2st~1

#4/1kstkst~1

(yt~1

!kst~1

)2]

!(E[y2t])2. (A.40)


A similar expression holds for MSIII, substituting /1st~1

in place of /1. To get

the nth-order autocovariance we use that, for MSII,

y2t`n

"k2st`n

#/2n1

(yt!k

st)2#A

n+i/1

/(n~i)1

pstì

etìB

2

#2kst`n

/n1(y

t!k

st)#f (e

t`n,2,e

t`1), (A.41)

where f ( ) ) is a linear function of its arguments and hence will be uncorrelatedwith terms dated period t or earlier. For MSIII, the similar expression is morecomplex:

y2t`n

"k2st`n

#An~1<i/0

/21stìB(yt!k

st)2#A

n+i/1An~1<j/i

/1st`jBpstì

etìB

2

#2kst`nA

n~1<i/0

/1stìB(yt!k

st)#g(e

t`n,2,e

t`1), (A.42)

where again g( ) ) is a linear function of its arguments and we de"ne<n~1

i/n/1stì~1

,1. Using these expressions, Eq. (A.40) can be extended to obtainthe n-period autocovariances:

E[(y2t`n

!E[y2t`n

])(y2t!E[y2

t])]

"E[k2st`n

(yt!k

st)2#k2

st`nk2st#/2n

1(y

t!k

st)4

#/2n1

(yt!k

st)2k2

st(A.43)

#((yt!k

st)2#k2

st)A

n+i/1

/2(n~i)1

p2stìB

#4/n1kst`n

kst(y

t!k

st)2]!(E[y2

t])2 (MSII)

"E[k2st`n

(yt!k

st)2#k2

st`nk2st#(y

t!k

st)4

n~1<i/0

/21stì

#(yt!k

st)2k2

st

n~1<i/0

/21stì

#((yt!k

st)2#k2

st)

n+i/1An~1<j/i

/21st`jBp2

stì

#4kst`n

kst(y

t!k

st)2

n~1<i/0

/1stì

]!(E[y2t])2 (MSIII)

which can be written as stated in Proposition 5.


References

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