CathyW.S.Chen , MikeK.P.So and RichardH.Gerlachdownload.xuebalib.com/26hfZ5TXabGL.pdf · Aust. N.Z....

Aust. N. Z. J. Stat. 47(4), 2005, 473–488

ASSESSING AND TESTING FOR THRESHOLD NONLINEARITYIN STOCK RETURNS

Cathy W.S. Chen1∗, Mike K.P. So2 and Richard H. Gerlach3

Feng Chia University, The Hong Kong University of Science and Technologyand The University of Newcastle

Summary

This paper proposes a test for threshold nonlinearity in a time series with generalized autore-gressive conditional heteroscedasticity (GARCH) volatility dynamics. This test is used toexamine whether financial returns on market indices exhibit asymmetric mean and volatil-ity around a threshold value, using a double-threshold GARCH model. The test adoptsthe reversible-jump Markov chain Monte Carlo idea of Green, proposed in 1995, to cal-culate the posterior probabilities for a conventional GARCH model and a double-thresholdGARCH model. Posterior evidence favouring the threshold GARCH model indicates thresh-old nonlinearity with asymmetric behaviour of the mean and volatility. Simulation experi-ments demonstrate that the test works very well in distinguishing between the conventionalGARCH and the double-threshold GARCH models. In an application to eight interna-tional financial market indices, including the G-7 countries, clear evidence supporting thehypothesis of threshold nonlinearity is discovered, simultaneously indicating an unevenmean-reverting pattern and volatility asymmetry around a threshold return value.

Key words: asymmetric mean reversion; asymmetric volatility model; Bayesian; double-thresholdGARCH models; Markov chain Monte Carlo method; reversible-jump; stock markets.

1. Introduction

The family of autoregressive conditional heteroscedastic (ARCH) and generalized ARCH(GARCH) models, respectively by Engle (1982) and Bollerslev (1986), has allowed conditionalvariance in time-series to be modelled so that the magnitude of volatility can be predicted frompast news and lagged conditional variance. However, these conventional, symmetric ARCHand GARCH formulations are not well suited to capturing an asymmetric response of volatil-ity, the phenomenon discovered by Black (1976) and subsequently confirmed by French,Schwert & Stambaugh (1987), Schwert (1989), Nelson (1991) and Glosten, Jagannathan &Runkle (1993), among others. Numerous refinements have been made to the GARCH model,

Received April 2004; revised September 2004; accepted November 2004.∗Author to whom correspondence should be addressed.1 Graduate Institute of Statistics and Actuarial Science, Feng Chia University, Taiwan.

e-mail: [email protected] Dept of Information and Systems Management, The Hong Kong University of Science and Technology.3 School of Mathematical and Physical Sciences, The University of Newcastle, Australia.Acknowledgments. The authors thank the Editor (Professor Rob J. Hyndman), an Associate Editor and tworeferees for their constructive comments and suggestions. C.W.S. Chen was supported by National ScienceCouncil (NSC) of Taiwan grants NSC93-2118-M-035-003. M. So was supported by Hong Kong RGC DirectAllocation grant 03/04.BM39. M. So and R. Gerlach were supported by NSC of Taiwan in a research visitto Feng Chia University. Gerlach was also supported by an ECA networking grant from The University ofNewcastle and the School of Mathematical and Physical Sciences. The authors thank Miss Ya-Jane Hsu for theinitial simulation study.

c© 2005 Australian Statistical Publishing Association Inc. Published by Blackwell Publishing Asia Pty Ltd.

474 CATHY W.S. CHEN, MIKE K.P. SO AND RICHARD H. GERLACH

incorporating an asymmetric volatility response into the conditional variance. These models,including the quadratic, the exponential, the threshold, and the modified GARCH, capturedifferential impacts on volatility from unexpected positive or negative shocks in the returnseries. Nam, Pyun & Avard (2001) also discovered an uneven or asymmetric mean revertingpattern in monthly return indices. Subsequently, Chen, Chiang & So (2003) investigated sixmajor market return indices using a double-threshold GARCH model, allowing asymmetryaround a threshold return level, in the mean and volatility equations. They found significantthreshold nonlinearity in both mean and volatility for these markets.

While various authors have successfully modelled and described the asymmetric volatil-ity and/or uneven mean reverting pattern, this does not automatically prove that these modelsdo better statistically than the conventional GARCH model. When tests on volatility asym-metry or threshold nonlinearity in volatility are performed, it is common that a zero thresholdvalue is implicitly assumed. For example, Engle & Ng (1993) proposed sign bias tests toexamine whether volatility is dependent on the sign and magnitude of returns. We can alsotest for nonlinearity by testing the null hypothesis H0: α−

1 = 0 under the GJR-GARCH modelof Glosten et al. (1993): ht = α0 + α1a

2t−1 + α−

1 I (at−1 ≤ 0)a2t−1 + βht−1 , where ht is the

conditional variance and at is the return innovation. Again, the threshold is taken to be zero inthis framework. It is theoretically unclear whether the above tests are still valid if the thresholdvalue is unknown and non-zero. Actually, our empirical study finds significant evidence thatthe threshold value is mostly negative. This paper provides a Bayesian test which generalizesexisting methods to allow for unknown threshold values; the Bayesian testing scheme allowsthe uncertainty in the threshold value to be incorporated, as well as a delay parameter in testingfor threshold nonlinearity. The popular existing tests in finance by Engle & Ng (1993) andGlosten et al. (1993) mainly focus on volatility asymmetry. As it is evident in the literaturethat mean asymmetry also exists (Li & Li, 1996; Liu, Li & Li, 1997; So, Li & Lam, 2002),we also looked into the threshold nonlinearity in the mean equation when developing our test.In short, we propose and demonstrate a test that is more general and flexible than the existingmethods by allowing for threshold nonlinearity in the conditional mean and variance.

Section 2 of the paper describes the double-threshold GARCH (DTGARCH) model used inthis study. Section 3 presents the Bayesian methods for estimating the DTGARCH model. Sec-tion 4 gives the reversible-jump methods and the Bayesian test for comparing the DTGARCHand GARCH models. Section 5 presents a simulation study illustrating the performance ofthe methods in sections 3 and 4. Section 6 reports empirical findings from eight major stockmarkets, and Section 7 provides conclusions from this study.

2. Double-threshold model

We use a double-threshold heteroscedastic model, based on the nonlinearity argumentof Tong (1978, 1990) and Tong & Lim (1980), that allows asymmetry in both the conditionalmean and the volatility equations, around a threshold value. The double-threshold DTGARCHmodel, DTGARCH(p1, . . . , pg; q1, . . . , qg;m1, . . . , mg), is described as follows:

yt = φ(j )0 +

pj∑i=1

φ(j )i yt−i + at , rj−1 ≤ yt−d < rj ,

at = √htεt , εt

d= D(0, 1) , ht = α(j )0 +

qj∑i=1

α(j )i a2

t−i +mj∑i=1

β(j )i ht−i ,

c© 2005 Australian Statistical Publishing Association Inc.

TESTING FOR THRESHOLD NONLINEARITY IN STOCK RETURNS 475

where j = 1, . . . , g, d is a positive integer, and D(0, 1) denotes a distribution with mean 0and variance 1. The number of regimes is assumed to be g. The threshold values rj satisfy−∞ = r0 < r1 < · · · < rg = ∞ and so rj, j = 1, . . . , g, form a partition of the space ofyt−d . The positive integer d is often called the delay or threshold lag. It is straightforwardto add an exogenous variable into the mean equation, as in Chen et al. (2003), but we choosenot to as we wish to focus on threshold nonlinearity in individual markets separately, ratherthan market integration. To allow heteroscedasticity in yt , we have a GARCH formulation inthe conditional variance ht . Standard restrictions on the variance parameters apply:

α(j )0 > 0, α

(j )i , β

(j )i ≥ 0 and

qj∑i=1

α(j )i +

mj∑i=1

β(j )i < 1 . (1)

Liu et al. (1997) have also considered a double-threshold GARCH model; they used htrather than yt as the threshold variable. Our model generalizes the model of Li & Li (1996)by having a GARCH component. It also extends the model in Brooks (2001) to have a moreflexible error distribution D(0, 1). Typical empirical evidence in the literature tells us that εt isusually fat-tailed, thus assuming normality of errors, as in Li & Li (1996) and Brooks (2001),may be unrealistic in the return percentile and volatility forecasts, and may have an impacton the significance levels of any tests of model parameters. In the empirical applications ofthis paper, we consider D(0, 1) = t∗ν , a standardized t-distribution (i.e. t∗ν = √

(ν − 2)/ν tν)which captures the usual conditional leptokurtosis observed in financial return data.

This model allows general asymmetric reactions to past information, not simply the usualincrease in volatility following negative stock returns, as in Nelson (1991). Both the meanand volatility equations in this model can change in a positive or negative direction, follow-ing an observed yt−d on either side of some threshold values, rj, j = 1, . . . , g. Our modelcan capture asymmetries in the average return, average volatility, mean reversion and volatil-ity persistence. When the data suggest the type and direction of asymmetric response, theresponse can induce the usual finding of negative correlation between returns and volatility.

Two additional features are associated with this model. Firstly, the threshold value of r ina two-regime model is not necessarily 0 as per the GJR-GARCH model in Glosten et al. (1993).Second, the threshold lag d is not necessarily equal to 1 as implied by most conventionalmodels (LeBaron, 1992; Koutmos, 1999; Bekaert & Wu, 2000). In our model, the values ofrj , j = 1, . . . , g, and d, along with the other parameters, are estimated simultaneously bythe Markov chain Monte Carlo (MCMC) method, as we discuss next.

3. Bayesian inference

Parameter estimation in homoscedastic threshold models is usually performed in twosteps; see e.g. Tong & Lim (1980) or Li & Li (1996). For fixed d and r, the other parametersare estimated in the first step. Then, estimates of d and rj can be determined by minimizing aninformation criterion (Tong & Lim, 1980; Tong, 1990), by minimizing a nonlinearity statisticand observing relevant t-ratios (Tsay, 1989; Li & Li, 1996), or by minimizing a conditionalsum of squared errors (Tsay, 1998). In this paper, we propose a Bayesian approach to performsimultaneous inference for all parameters, including threshold values and lag. We generatesamples from the joint posterior distribution of the model parameters via MCMC methods.One important advantage of the Bayesian approach is that we can estimate rj , d and other un-known model parameters at the same time, allowing uncertainty about these threshold and lagparameters to be reflected in our analysis. Furthermore, any prior information regarding the



parameters can also be incorporated into the prior distribution, such as the standard stationarityor positivity constraints for GARCH model parameters as above in (1).

Let φj = (φ(j )0 , . . . , φ(j )pj

), αj = (α(j )0 , . . . , α(j )qj

, β(j )1 , . . . , β(j )mj

) and r = (r1, . . . , rg−1)

denote the mean, variance and threshold parameter vectors respectively. Denote the completevector of parameters by θ . Let d0 denote the maximum delay and s = max{p1, . . . , pg, d0}.The conditional likelihood function of the model is given by

p(ys+1,n | θ) =n∏

t=s+1

( g∑j=1

1√ht

pε

(yt − µt√ht

)Ijt

), (2)

µt = E(yt | y1,t−1) = φ(j )0 +

pj∑i=1

φ(j )i yt−i ,

where Ijt denotes the indicator variable I (rj−1 ≤ yt−d < rj ) and pε denotes the probability

density function of εt . The dependence of the conditional likelihood on y1,s is ignored forconvenience. If εt

d= N(0, 1), as is usually assumed in the literature, (2) reduces to

p(ys+1,n | θ) =n∏

t=s+1

( g∑j=1

1√2πht

exp(

− (yt − µt)2

2ht

)Ijt

).

Moreover, if εtd= t∗ν , then (2) becomes

p(ys+1,n | θ) =n∏

t=s+1

( g∑j=1

�( 12 (ν + 1))

�( 12ν)

√(ν − 2)π

1√ht

(1 + (yt − µt)

2

(ν − 2)ht

)−(ν+1)/2Ijt

).

In setting prior distributions, we have sought to be as uninformative as possible, mainlysatisfying the necessary constraints on parameters, so that the sample data and model like-lihood dominate our inferences. For the delay d, we assume a uniform discrete prior overd = 1, . . . , d0 , reflecting prior ignorance. To apply the constraint on α

(j )0 in (1), we adopt

the prior α(j )0

d= U(0, b), where b is a pre-specified positive constant. Similarly, the pa-

rameters in αj other than α(j )0 follow a constrained uniform prior defined by the indicator

I (Sj ), j = 1, 2, where Sj is the set αj that satisfies (1). We assume prior independence

between α(j )0 and the other parameters in αj . In threshold modelling, it is important to set

a minimum sample size in each regime to generate meaningful inferences. We choose theconditional prior (r | d) ∝ I (A), where A specifies that each regime contains at least h%of yt−d . For example, in a two-regime model (g = 2), having h = 10 means that the prioron r = r1 | d d= U(c10(yt−d), c90(yt−d)), where c10(yt−d) and c90(yt−d) are the 10th and90th percentiles of yt−d . For ν, the degrees of freedom (df), we reparameterize by definingτ = ν−1. The prior for τ is U(0, 1

4 ), so that ν > 4. This ensures that the first four momentsof the distribution of εt are finite. Finally, we assume the normal prior φj

d= Npj(φj0,Vj ),

where usually we set φj0 = 0 and V −1j to be a matrix with ‘large’ numbers on the diagonal,

to ensure a reasonably flat distribution.The MCMC sampling scheme requires posterior distributions for groups of parameters,

conditional on the sample data and the other parameters. Multiplying the likelihood (2) andthe priors, using Bayes’ rule, leads to these conditional posteriors, which are, in general, notof a standard form and require us to use techniques such as the Metropolis–Hastings (MH)method (Metropolis et al., 1953; Hastings, 1970) to achieve the desired sample. We use thefollowing parameter groupings: the mean parameters in each regime φj for j = 1, . . . , g;



the variance parameters in each regime αj ; the threshold parameters r. The delay d and df νare simulated separately to these groups. All random generation for ν is done in the τ-space,inverting back to ν at the end.

We rely mostly on the random walk Metropolis method, using a normal proposal densitywith variance a�, to simulate from the relevant posterior distributions. This is a standardMCMC method, details of which can be found in Chen & So (2006). To yield good conver-gence properties, the choices of � and a for each parameter vector are important. Thesechoices can be made to ensure good coverage for each conditional posterior distribution, with� usually chosen with sufficiently large diagonal terms, while the coverage and samplingefficiency can also be tuned using the acceptance rate in the burn-in period of the MCMCsample; see Gelman, Roberts & Gilks (1996). A suitable value of a can usually be selectedby having an acceptance rate of between 25% and 50% in the burn-in period.

We give specific details below for the methods for the delay parameter d, the meanparameters φ and the GARCH parameters α. We refer the reader to the general Metropolisrandom walk method for the parameters ν and r.

The delay parameter d is obtained by sampling from a multinomial posterior distribution,

p(d = j | ys+1,n, θ−d) = p(ys+1,n | d = j, θ−d)∑d0i=1 p(y

s+1,n | d = i, θ−d)(j = 1, . . . , d0) .

For the mean parameters φ, we make a specific choice for the proposal variance matrix� in the random walk algorithm. Let πj denote the ordered time index for the set of timepoints in regime j , i.e. πj = t : t = s + 1, . . . , n, Ijt = 1 and aj =

∑nt=s+1 Ijt denotes the

number of observations in regime j . In the simplest case that εt is normal and ht is constant,the posterior reduces to

p(φj | ys+1,n, θ−φj) ∝

∏πj

1√2πht

exp(

− (yt − µt)2

2ht

)p(φj )

∝ |V ∗j |−1/2 exp

( − 12 (φj − φ∗

j )TV ∗

j−1

(φj − φ∗j )

), (3)

where V ∗j = (XT%−1X + V −1

j )−1, φ∗j = V ∗

j (V−1j φj0 + XT%−1Y ),

X =

1 yπ1−1 . . . yπ1−pj

1 yπ2−1 . . . yπ2−pj

......

. . ....

1 yπaj−1 . . . yπaj−pj

and � =

hπ10 . . . 0

0 hπ2. . . 0

......

. . ....

0 0 . . . hπaj

.

The posterior (3) gives a multivariate normal distribution in φj , if and only if each ht isconstant in φj ; this is not the case for GARCH models, and (3) is a non-standard distributionas both V ∗ and � depend on φj via ht . However, we anticipate that V ∗ can still containuseful information about the conditional correlations among the components of φj , even whenpε is non-normal. So, we choose � to be V ∗, but with all ht in � replaced by the samplevariance of yt .

The GARCH parameter αj also requires special mention here. Denote the target density

p(αj | ys+1,n, θ−αj) by f(αj ). After carrying out the random walk Metropolis algorithm many



times for GARCH models, we have discovered that the correlations of the MCMC iterates ofαj can decay very slowly, indicating slow convergence of the Markov chain. This seems

especially true when∑

α(j )i + ∑

β(j )i is close to 1, as often occurs for financial data. To

enhance the convergence and efficiency properties we revise the proposal density once, afterM iterations. We perform the random walk Metropolis for M iterations, choosing � withsufficiently large diagonal terms to cover the region defined in (1), and then form the samplemean µα and sample covariance matrix �α , using the first M iterates of αj . Then we applythe following independent kernel MH algorithm from the (M + 1)th iteration onwards:

Step 1: Form µα and �α using the first M iterates of αj .

Step 2: At iteration i > M, generate a point α∗j from the independent kernel,

α∗j

d= N(µα,�α) .

Step 3: Accept α∗j as α

[i]j with probability

p = min(

1,f(α∗

j ) g(α[i−1]j )

f(α[i−1]j ) g(α∗

j )

),

where α[i]j is the i th iterate of αj and g(α) is the evaluated normal proposal density

with mean µα and covariance matrix �α . Otherwise, set α[i]j = α

[i−1]j . The advantage

of using the matrix �α is that the posterior correlations among the components of αjcan be accounted for, thus allowing potentially improved coverage and convergenceproperties.

4. Model selection using reversible-jump MCMC

The major goal of this study is to test for threshold nonlinearity in a financial time series.In the current setting, we do this by selecting between a conventional GARCH model, withno threshold component, and a double-threshold GARCH model. This test can be done viaBayesian model selection criteria, based on calculating the posterior probabilities for twocompeting models. These quantities allow the comparison of non-nested models of potentiallydifferent dimensions, and can be written as p(Mj | y1,n), j = 1, 2, where M1 and M2 hererepresent the GARCH and DTGARCH models respectively.

Much work in the Bayesian literature has been done to estimate these posterior quantities,see Kass & Raftery (1995) for example. We need to integrate over the parameter space:

p(Mj | y1,n) ∝ p(y1,n | Mj )p(Mj ) = p(Mj )

∫p(y1,n | θj ,Mj )p(θj | Mj ) dθj .

This integral has proved difficult to estimate in general. Kass & Raftery (1995) used a Lapla-cian approximation to estimate the integral directly, while Gerlach, Carter & Kohn (1999) tookan MCMC approach to numerical integration, for time series models. We adopt the reversible-jump (RJ) MCMC method of Green (1995) — a very useful method allowing jumps betweenmodels of different dimensions inside an MCMC sample, maintaining the necessary conditionsfor convergence. This allows a sample to be obtained from the model space p(M | y1,n), andthe posterior probabilities for competing models to be calculated by numerically performing



the integral above, as part of a single run of the MCMC method. Applications of this RJMCMCmethod can be found in Richardson & Green (1997) and Dellaportas & Forster (1999).

The RJMCMC method is similar to the MH algorithm in that it requires a proposal distri-bution describing how to jump from one model to another, and jumps are either accepted orrejected based on a probability rule. To set up the jump distribution in a two-model situation,as in this paper, requires a one-to-one bijective transformation to be defined between the twomodels M1 and M2 . In general this bijection can be quite complex, but here we define thebijection u1 = θ2 and u2 = θ1 , thus implying a simple transformation Jacobian of 1 betweenmodels and ensuring the necessary condition that the dimension in each space is the same.Here, θ1 = (φ,α, τ) and θ2 = (φj,αj, r, d, τ), j = 1, . . . , g. To propose a jump betweenmodels, we need a proposal distribution to generate a set of parameter values for each model.We choose this to be the independent normal proposal distributions, built up during the MCMCsampling scheme, already specified in Section 2. In other words, for each parameter subset(except the lag d), as specified in the MCMC sampling scheme in Section 2, we use the first Miterations to build a proposal distribution for that parameter vector, while the discrete uniformdistribution is used for the delay parameter d. We multiply these proposals together to obtainthe required proposal kernel density, denoted q1(u1). See Green (1995) for further details ofthis method.

To calculate the probability of accepting a jump we also need each model likelihood, theprior distributions p(θk |Mk), the prior model probabilities p(Mk), the prior probability ofconsidering the jump for model Mk , j(Mk), k = 1, 2, and the Jacobian of the transformation,in this case 1. We set p(M1) = p(M2) = 0.5 to reflect prior ignorance about model choiceand j(M1) = j(M2) = 1 to allow jumps between models in each MCMC iteration.

Thus, in our case, the probability of accepting a jump from model 1 to model 2 can bewritten as

p = p(y1,n | M2, θ2)p(θ2 | M2)q2(u2)

p(y1,n | M1, θ1)p(θ1 | M1)q1(u1),

with the parameter proposal distributions q1 and q2 chosen as above, to be independent be-tween models respectively. The jump from model 2 to model 1 is accepted with probabilityp−1. For the simulation of the parameter vector (φ,α, τ) under a symmetric AR-GARCHmodel, we use the same method as above to construct independence proposals N(µφ,�φ),

N(µα,�α), N(µτ, σ2τ ) from the first M iterates of (φ,α, τ), and use the product of these

proposals as the kernel density q2(u2). In summary, the jumping scheme from (Mi) to (Mk) is:

(i) draw θk from kernel density qi(ui), and accept the jump with probability min{1, pk−i};(ii) if accepted in (i), update θk ; otherwise, update θi .

At the completion of the RJMCMC sampling scheme, we have a Monte Carlo sample ofmodels from the model space thus giving an estimated posterior probability for each competingmodel. This probability provides a direct test of the threshold nonlinear model in this case.

5. Simulation study

In a review of various market volatility models, Bollerslev, Chou & Kroner (1992) reportthat the GARCH(1, 1) model appears to be sufficient to describe the volatility evolution ofmost stock-return series. Motivated by this, the following two competing models are con-sidered in this simulation study, the AR-GARCH(1, 1) model (p = 1, q = m = 1) andDTGARCH(1, 1; 1, 1; 1, 1) model (g = 2, pj = 1, qj = mj = 1).



To start the MCMC, the initial GARCH parameters αj are selected randomly with thecondition in (1) being satisfied. In all the experiments in this section, we implement 40 000iterations; that is, we apply the random walk MH to draw αj and form the normal kernelsN(µαj

,�αj) from the first 8000 iterations. We then carry out 12 000 iterations for posterior

inference and use the last 20 000 iterations for model selection.We consider the following combinations of sample size and prior settings:

1. sample size of n: 1000 and 2000.2. four choices of prior of α[j]

0 : U(0, 0.5s2), U(0, s2), U(0, 5s2) and U(0, 10s2), wheres2 is the sample variance.

For each model, prior and sample size combinations, 100 replications are generated and thenanalysed. This helps determine the sensitivity of our model selection procedure to variousprior settings and to sample size.

The prior used for φ is Np(0, 0.1I ), where I denotes the identity matrix. Kim, Shep-hard & Chib (1998) used the transformed Beta prior, φ1

d= 2Be(a, b) − 1, which is useful ifφ1 is potentially close to −1 or 1. Much literature evidence has shown this is not the casefor financial return data. In addition, choosing the parameters as a = b = 4.5 is practicallyequivalent to the normal prior used here. We choose the prior for r to follow U(Q1,Q3),

where Q1 and Q3 are the first and third sample quartiles, ensuring at least 25% of the sampleis in each regime. The maximum lag for d is set to 3. Initial values for these parameters werechosen randomly from these prior distributions. The true models considered are listed below.

Model 1: The true model is GARCH(1, 1) with

yt = 0.003+0.30yt−1 +at , ht = 0.10 +0.05a2t−1 +0.85ht−1 , at = √

htεt , εtd= t∗7 .

Models 2–4 and 6 below are all examples of the double-threshold GARCH model. We set thevalues of αj and βj in these models according to two criteria:

1. αj + βj is larger in negative regime than in positive regime;2. the unconditional variance is larger in negative regime than in positive regime.

These criteria reflect those found in studies using threshold nonlinear models; see e.g. Koutmos(1999) and Brooks (2001).

Model 2: The true model is DTGARCH with

yt ={ −0.002 − 0.30yt−1 + at yt−1 ≤ −0.2 ,

0.003 + 0.30yt−1 + at yt−1 > −0.2 ;

ht ={

0.50 + 0.10a2t−1 + 0.85ht−1 yt−1 ≤ −0.2 ,

0.05 + 0.05a2t−1 + 0.85ht−1 yt−1 > −0.2.

Model 3: The true model is DTGARCH with the same mean equation as model 2 and

ht ={

0.50 + 0.10a2t−1 + 0.85ht−1 yt−1 ≤ −0.2 ,

0.10 + 0.05a2t−1 + 0.85ht−1 yt−1 > −0.2 .

Model 4: The true model is DTGARCH with the same mean equation as model 2 and

ht ={

0.50 + 0.10a2t−1 + 0.85ht−1 yt−1 ≤ −0.2 ,

0.10 + 0.05a2t−1 + 0.88ht−1 yt−1 > −0.2 .



Table 1

Summary statistics based on n = 2000 and the prior on U(0, s2), obtained from 100replications. The entries for d are the mean of 100 posterior modes while those for the

other parameters are the mean (and standard deviation) of the 100 posterior means

Model 1 Model 2 Model 3 Model 4

φ(1)0 0.03 0.0288 –0.02 –0.0300 –0.02 0.0250 –0.02 0.0040

(0.0122) (0.1054) (0.1134) (0.1324)

φ(1)1 0.30 0.2968 –0.30 –0.3184 –0.30 –0.2865 –0.30 –0.2929

(0.0217) (0.0734) (0.0723) (0.0724)

α(1)0 0.10 0.2409 0.50 0.6298 0.50 0.7138 0.50 0.6623

(0.1134) (0.1771) (0.2254) (0.2576)

α(1)1 0.05 0.0670 0.10 0.1237 0.10 0.1218 0.10 0.1219

(0.0213) (0.0335) (0.0339) (0.0316)

β(1)1 0.85 0.6933 0.85 0.7864 0.85 0.7684 0.85 0.7951

(0.1229) (0.0717) (0.0798) (0.0684)

φ(2)0 0.03 0.0268 0.03 –0.0052 0.03 0.0531

(0.0573) (0.0710) (0.0863)

φ(2)1 0.30 0.3004 0.30 0.3044 0.30 0.2928

(0.0374) (0.0405) (0.0413)

α(2)0 0.05 0.1004 0.10 0.1658 0.10 0.1928

(0.0632) (0.1034) (0.1210)

α(2)1 0.05 0.0628 0.05 0.0657 0.05 0.0645

(0.0215) (0.0229) (0.0221)

β(2)1 0.85 0.8204 0.85 0.8120 0.88 0.8463

(0.0381) (0.0466) (0.0406)

d 1 1 1

Model 5: The true model’s mean equation is like that of model 2 (except for the thresholdvariable) and its volatility equation is from GJR-GARCH(1, 1),

yt ={ −0.002 − 0.30yt−1 + at at−1 ≤ 0 ,

0.003 + 0.30yt−1 + at at−1 > 0 ;ht = 0.1 + 0.1a2

t−1 + 0.05 I (at−1 ≤ 0) a2t−1 + 0.84ht−1 .

Model 6: The true model is the same as model 2, except that εtd= t∗4 .

Models 2, 3 and 4 have been chosen to have decreasing differences in the level of un-conditional variance in each regime. The unconditional variance in the negative regime formodel 2 is 10, compared to 0.5 in the positive regime. For model 3, the positive regime vari-ance is 1, while for model 4 the positive regime variance is 1.4. This decreasing discrepancyin variance is intentionally introduced to see the effect on the posterior probability of identi-fying the threshold nonlinearity. Model 5 is a GJR-GARCH model; it has a different thresholdvariable (at−1) and is chosen to illustrate the performance of the threshold test when the truemodel has threshold nonlinearity, but not exactly as specified by the DTGARCH model.

Table 1 displays the true values, posterior means and standard deviations of the modelparameter estimates, plus average posterior modes of d, from 100 replications for models 1–4,



Table 2

The frequencies and probabilities of selecting the DTGARCH modelout of 100 replications for the prior U(0, s2)

n = 1000 n = 2000Model Prior r df Prob. Freq. r df Prob. Freq.

1 U(0, 1s2) — 7.81 0.0822 5 — 7.22 0.0710 6— (6.75) (0.1977) — (1.17) (0.2045)

U(0, 5s2) — 7.47 0.0545 5 — 7.33 0.0072 0— (2.81) (0.1866) — (1.19) (0.0404)

2 U(0, 1s2) –0.06 7.47 0.9998 100 –0.13 7.43 1.0000 100(0.32) (2.30) (0.0006) (0.18) (1.32) (0.00002)

U(0, 5s2) –0.06 7.34 0.9905 99 –0.12 7.07 1.0000 100(0.32) (3.15) (0.0829) (0.20) (1.14) (0.00003)

3 U(0, 1s2) –0.14 7.39 0.9897 99 –0.18 7.04 1.0000 100(0.23) (2.54) (0.1000) (0.13) (1.12) (0.00002)

U(0, 5s2) –0.12 7.23 0.9994 100 –0.21 7.22 1.0000 100(0.22) (3.22) (0.0027) (0.12) (1.19) (0.0001)

4 U(0, 1s2) 0.01 7.87 0.9988 100 –0.14 6.81 1.0000 100(0.41) (11.12) (0.0075) (0.32) (1.06) (0.00003)

U(0, 5s2) –0.002 7.17 0.9851 99 –0.12 7.23 0.9900 99(0.40) (4.22) (0.0674) (0.29) (1.17) (0.1000)

5 U(0, 1s2) 0.01 7.85 0.9851 99 0.07 7.10 0.9998 100(0.32) (2.63) (0.0948) (0.26) (1.13) (0.0018)

U(0, 5s2) –0.01 7.85 0.9727 98 –0.02 7.05 1.0000 100(0.29) (4.90) (0.1434) (0.25) (1.75) (0.00003)

6 U(0, 1s2) –0.19 4.20 0.9255 92 –0.20 4.37 0.9426 94(0.10) (0.18) (0.2523) (0.09) (0.28) (0.2233)

U(0, 5s2) –0.15 4.50 0.9543 96 –0.17 4.40 0.9569 96(0.16) (0.41) (0.1904) (0.10) (0.29) (0.1956)

for sample size n = 2000 only and prior U(0, s2) to save space. The other prior formulationsdid not significantly change the results (available from the authors). The sampling schemeperforms reasonably well, with all parameter averages not significantly different from theirtrue values. All 100 replications correctly identify the posterior mode of d = 1.

Table 2 reports on the performance of the threshold nonlinearity test. We list results forpriors U(0, s2), U(0, 5s2) only and for n = 1000 and 2000. The entries for r (r = −0.2 formodels 2–4 and 6, r = 0 for model 5), df (ν = 7, except model 6 where ν = 4), and Prob. arethe average of the posterior mean estimates of r, ν and the posterior probability of choosingthe DTGARCH model, respectively, over the 100 replications. The numbers in parenthesesin the r and df columns are the means of the 100 posterior standard deviations, while in theProb. column they are the standard deviations of the 100 estimated posterior probabilities. Theestimates for r and ν are significantly more precise for the larger sample size, and performwell over all models. In particular, no problem is observed for model 6, with the estimates ofν all close to the true value 4 and with relatively small standard error, despite this true valuebeing outside the range of the prior.

Model selection has worked consistently well across all six models (and prior specifica-tions). For data simulated from model 1, the DTGARCH model is preferred only about 5% ofthe time on average, regardless of sample size or prior specification, as measured by the Freq.



column indicating the number of times out of 100 that the DTGARCH model had the highestposterior probability. The standard errors on this probability estimate are quite high over thereplications, indicating that there is some degree of model uncertainty, as would be expected.This is a good result, as the nonlinearity test has not specifically corrected for the complexityof the competing models, yet the simpler model still correctly wins out in this case about95% of the time. For models 2–6, the DTGARCH model is preferred about 95–99% of thetime for n = 1000 and even more strongly preferred for n = 2000, with very low standarderrors across replications. The results are more consistent and more strongly in favour of theDTGARCH model for the larger sample size and for model 2 compared to models 3, 4 and 6.Model 2 had the largest discrepancy between variances for each regime, while model 4 hadthe smallest discrepancy. Finally, when the true model is GJR-GARCH, as in model 5, theposterior probabilities strongly favour the DTGARCH model over the GARCH(1, 1) model,detecting the threshold nonlinearity despite the DTGARCH not being the true model.

In summary, this study illustrates a favourable estimation and model selection perfor-mance for the methods discussed in this paper. Specifically, the model selection test showsgood performance at distinguishing between a conventional and a double-threshold GARCHmodel, for reasonable sample size and in both directions. This performance is not sensitive toa changing prior specification on the model parameters, is robust to a slightly different GJR-GARCH model specification, and improves with increasing sample size and for increasingdiscrepancy between levels of variance in each regime.

6. Empirical study

The data consist of daily closing values for eight stock indices from 1 January 1993 to 14November 2002. The data include the G-7 financial markets, namely the Standard & Poor 500Index (United States), the CAC 40 (France), the Dax 30 (Germany), the FTSE 100 (UK), theNikkei 225 Index (Japan), the Milan MIBTel Index (Italy), and the Toronto SE 300 (Canada),plus the Taiwan Weighted Index (Taiwan). All the data were taken from Datastream Interna-tional. Daily stock-return series are generated by taking the logarithmic difference of the dailystock index times 100; that is, yt = (logPt−logPt−1)×100%, where Pt is daily closing price.

To understand the characteristics of each market we briefly discuss standard summarystatistics. Each market except Japan has a positive average daily return, between 0.01% and0.03%, over the sample period, Japan having an average −0.025% return. All markets failthe standard Jarque–Bera normality test, with each distribution of returns being negativelyskewed except Japan and Taiwan, and exhibiting leptokurtosis in each case.

For each market, two competing models considered here are given as follows:

AR-GARCH model yt = φ0 + φ1yt−1 + at , where at = √htεt , with

ht = α0 + α1a2t−1 + β1ht−1 ;

DTGARCH model yt ={φ(1)0 + φ

(1)1 yt−1 + at yt−d ≤ r ,

φ(2)0 + φ

(2)1 yt−1 + at yt−d > r , where at = √

htεt with

ht ={α(1)0 + α

(1)1 a2

t−1 + β(1)1 ht−1 yt−d ≤ r ,

α(2)0 + α

(2)1 a2

t−1 + β(2)1 ht−1 yt−d > r .

For both models εtd= t∗ν . The parameter estimates for the double-threshold model, applied to



Table 3

Estimates of parameters for daily stock index returns of S&P 500, FTSE 100,CAC 40, DAX, TSE 300, MIBTel, Nikkei 225 and TAIEX

US UK France Germany Canada Italy Japan TaiwanS&P 500 FTSE 100 CAC 40 DAX TSE 300 MIBTel Nikkei 225 TAIEX

φ(1)0 –0.0501 –0.0248 –0.3189 –0.1226 –0.0444 0.0018 –0.1821 –0.2661

(0.0341) (0.0597) (0.1112) (0.1078) (0.0292) (0.0567) (0.1170) (0.1704)

φ(1)1 0.0345 –0.0102 –0.1714 –0.1190 0.0389 0.0206 –0.1398 –0.1128

(0.0441) (0.0551) (0.0700) (0.0691) (0.0423) (0.0478) (0.0688) (0.0825)

φ(2)0 0.1134 0.0453 0.0466 0.0573 0.1201 0.0437 –0.0394 0.0108

(0.0285) (0.0581) (0.0323) (0.0286) (0.0250) (0.0866) (0.0302) (0.0348)

φ(2)1 0.0891 0.0320 0.0267 0.0197 0.0816 0.0422 –0.0453 0.0272

(0.0435) (0.0518) (0.0322) (0.0332) (0.0394) (0.0573) (0.0305) (0.0290)

α(1)0 0.0251 0.0323 0.0797 0.1349 0.0247 0.1094 0.2124 0.5269

(0.0083) (0.0140) (0.0327) (0.0400) (0.0076) (0.0328) (0.0507) (0.1832)

α(1)1 0.0984 0.0947 0.0937 0.0973 0.1018 0.1488 0.0879 0.0957

(0.0198) (0.0140) (0.0154) (0.0189) (0.0200) (0.0261) (0.0183) (0.0408)

β(1)1 0.8954 0.8998 0.8981 0.8952 0.8922 0.8374 0.8958 0.8659

(0.0213) (0.0155) (0.0176) (0.0150) (0.0212) (0.0324) (0.0244) (0.0616)

α(2)0 0.0036 0.0059 0.0254 0.0171 0.0036 0.0599 0.0097 0.1182

(0.0027) (0.0041) (0.0099) (0.0087) (0.0031) (0.0385) (0.0080) (0.0556)

α(2)1 0.0674 0.0263 0.0153 0.0583 0.0694 0.0964 0.0260 0.0757

(0.0179) (0.0127) (0.0088) (0.0141) (0.0195) (0.0245) (0.0095) (0.0225)

β(2)1 0.9071 0.9372 0.9332 0.8957 0.9040 0.8149 0.9210 0.8340

(0.0177) (0.0155) (0.0130) (0.0150) (0.01854) (0.0385) (0.0124) (0.0449)

r –0.0536 –0.2032 –0.6581 –0.5755 –0.0215 0.2243 –0.6591 –0.8145(0.0832) (0.2953) (0.0432) (0.0858) (0.0300) (0.4323) (0.0804) (0.0792)

ν 6.2940 17.0253 21.2685 12.4463 6.3138 13.5357 6.5697 5.6178(0.7031) (5.0330) (12.4181) (2.5706) (0.7128) (3.5860) (0.8420) (0.6949)

d 1 1 1 1 1 1 1 1

each of the eight markets, are summarized in Table 3. Again, we list only the results based onthe prior specification U(0, s2). Results were not significantly different across the four priorsand are summarized as follows.

In each market, theAR parameter in the mean equation was lower in the ‘negative’regime,changing from positive to negative sign in four markets, representing a clear threshold asym-metric pattern. Thus, mean reversion was faster in the negative regime, i.e. following ‘bad’news. While there was a clear nonlinear pattern, most estimates were not significantly differ-ent from 0. Threshold nonlinearity was observed in the mean–intercept parameter in all eightmarkets, this parameter estimate changing from positive to negative sign in six of the eightmarkets, following a large enough negative return. This information is summarized in Table 4,giving posterior mean estimates for the average return in each regime, plus estimates of theposterior probability that µ2 < µ1 , formed using the MCMC sample of estimates for φ0 and

φ1 in each regime, to form a sample of regime mean estimates i.e. µ(j )

k = φ(j )

0(k)/(1 − φ(j )

1(k)),

j = 1, 2, where k refers to the k th MCMC iterate. We use the proportion of times that



Table 4

Estimates of means and unconditional variances for each regime

S&P 500 Prob. FTSE 100 Prob. CAC 40 Prob. DAX Prob.

µ(1) –0.0508 0.000 –0.0219 0.178 –0.268 0.002 –0.110 0.0597µ(2) 0.124 0.0444 0.0473 0.0585H(1) 26.691 0.0093 28.254 0.0007 50.926 0.0039 17.987 0.0022H(2) 0.176 0.162 0.484 3.717

TSE 300 Prob. MIBTel Prob. Nikkei 225 Prob. TAIEX Prob.

µ(1) –0.0453 0.000 0.0039 0.346 –0.155 0.119 –0.231 0.0270µ(2) 0.130 0.0413 –0.0381 0.0107H(1) 21.473 0.0118 41.552 0.0019 65.500 0.0005 81.984 0.000H(2) 1.034 0.670 0.175 1.265

µ(2)k − µ

(1)k < 0, as the probability estimate. These probabilities can be interpreted in a

similar fashion to a classical P-value in a one-sided test for a difference between means.Table 4 also reports µ(1) and µ(2), the sample mean obtained from the MCMC samples forµ(1)k and µ

(2)k respectively. As an apparent reaction to bad news or negative returns, the

markets each recorded a decrease in average returns in the negative regime, with six of theeight markets changing from a positive mean return estimate to a negative mean return. Fourof the eight markets, namely US, France, Canada and Taiwan, recorded significantly highermean returns in the positive regime, at the 5% level, based on the probability estimates inthe column Prob. This illustrates a definite difference in average return behaviour for mostmarkets following ‘good’ or ‘bad’ news.

Threshold nonlinear behaviour was also observed in the volatility equation in each mar-ket. This is mainly seen in the variance intercept (α0) and ARCH parameters (α1), and lessso in the GARCH parameters (β1). The estimates for α0 increased by a factor of at least 2and at most 22, following a return below the threshold value, compared to the positive regime,while the ARCH estimates also increased slightly in each of the eight markets following badnews. The strong threshold asymmetric pattern in volatility is summarized in Table 4, givingposterior mean estimates for the unconditional volatility (H(j ) = α

(j )0 /(1 − α

(j )1 − β

(j )1 ),

j = 1, 2) in each regime and estimated posterior probabilities that the average volatility levelin the negative regime is smaller than that in the positive regime. These probabilities are esti-mated like those above for the mean returns, using the MCMC sample to give the proportion ofiterates where the ratio of average volatility level in the negative regime to that in the positiveregime is smaller than 1. Here we can see that volatility level increases significantly followingbad news, with all probabilities close to or below 1%. Volatility following bad news increasesby factors of between 5 (Germany) and 380 (Japan) times the volatility following good news.This behaviour was observed consistently in all eight markets examined.

The threshold parameter (r) estimate was negative in seven of the eight markets; signifi-cantly so in four markets. This illustrates that the threshold is somewhat below the commonlyassumed value of 0 and that ‘small’ negative returns seem not to produce a significant nonlin-ear market reaction. The delay parameter d was estimated using a posterior mode, and wasfound to be one day in all markets. This confirms the use of this value by most asymmetricstudies in the literature. Finally, the estimates for the df parameter are reasonably consistentacross two groups of markets. The US, Canada, Japan and Taiwan markets all have df closeto 6, indicating significant departures from normality in the fitted errors for the DTGARCH



Table 5

The posterior probabilities for selecting the double threshold GARCH

US UK France GermanyPrior S&P 500 FTSE 100 CAC 40 DAX

U(0, 0.5s2) 1.00 1.00 1.00 1.00U(0, s2) 1.00 1.00 1.00 1.00U(0, 5s2) 1.00 1.00 0.99 1.00U(0, 10s2) 1.00 1.00 0.98 1.00

Canada Italy Japan TaiwanPrior TSE 300 MIBTel Nikkei 225 TAIEX

U(0, 0.5s2) 1.00 0.76 1.00 1.00U(0, s2) 1.00 0.59 1.00 1.00U(0, 5s2) 1.00 1.00 1.00 1.00U(0, 10s2) 1.00 1.00 1.00 0.99

model. Italy, France, Germany and the UK have estimated df between 12 and 22, indicatingerror distributions closer to normality.

Table 5 presents estimated posterior probabilities in favour of the DTGARCH model overthe conventional GARCH model, across the four prior specifications considered. Except forItaly, the results seem insensitive to the prior specification. In particular, the DTGARCH modelis strongly preferred over the GARCH model in all eight markets, having posterior probabilityvery close to 1 in most cases, and above 0.5 in every case. Only Italy has any posteriorprobability estimates below 0.98 and it is the only market to show any real difference acrossprior specifications.

The results above suggest that markets and/or investors react differently to good and badnews, with the nonlinear threshold model preferred to the conventional GARCH model. Thenonlinear reaction occurs only after a significant negative previous day’s return and resultsin a quicker mean reversion, lower mean and higher volatility in returns. Various theorieshave been put forward to try to explain this behaviour; e.g. higher costs to market makers fornon-reaction in a falling market, stronger aversion to downside risk for investors, etc.; seeKoutmos (1999). It is unclear whether this asymmetric behaviour is the result of rational orirrational investor behaviour, or whether it represents an exploitable market inefficiency.

7. Conclusion

This paper has tested the hypothesis that stock returns exhibit threshold nonlinearity, usinga Bayesian model selection procedure based on the reversible-jump method. This threshold be-haviour means that stock returns react asymmetrically to past information, around a thresholdreturn level. A simulation study suggests favourable estimation and model selection perfor-mance for this method. Specifically, the model selection test has shown good performanceat distinguishing between a conventional GARCH and a double-threshold GARCH model forreasonable sample size. This performance was not sensitive to a changing prior specificationon the model parameters and improved with sample size. Empirical evidence from eightfinancial market indices suggests that both the conditional mean and conditional variance ofreturns exhibit threshold nonlinearity, responding asymmetrically around a negative thresholdreturn level, at a lag of one time period. The double-threshold model was strongly preferredover the conventional GARCH model in all eight markets. The asymmetric pattern was one



of lower mean returns and significantly higher volatility immediately following significantnegative returns. This result was consistent across the G-7 financial markets and the Taiwanmarket.

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