bayesian unit root test in double threshold heteroskedastic models

Comput Econ (2013) 42:471–490DOI 10.1007/s10614-012-9354-7

Bayesian Unit Root Test in Double ThresholdHeteroskedastic Models

Cathy W. S. Chen · Shu-Yu Chen · Sangyeol Lee

Accepted: 19 November 2012 / Published online: 15 December 2012© Springer Science+Business Media New York 2012

Abstract This paper aims to detect the presence of local non-stationarity of nonlinearautoregressive processes with heteroskedastic errors. A Bayesian test is developed totest for the unit root in multi-regime threshold autoregression with heteroskedasticity.To implement a test, a posterior odds analysis is proposed. Particularly, a mixture priorfor the autoregressive coefficient is used to alleviate the identifiability problem thatoccurs when time series has unit roots. The proposed method achieves a reliable infer-ence despite of the non-integrability problem in the likelihood function. A simulationstudy and two real data analysis are conducted for illustration. This paper successfullyproves the proposed model can accommodate different threshold values to cope withlocal non-stationarity and in addition, captures discrete time-varying properties.

Keywords Bayesian hypothesis testing · SETAR · GARCH · Unit-root test ·Markov chain Monte Carlo method · Posterior odds ratio

1 Introduction

Nonlinear time series modeling received much attention from researchers in the 1970sand various classes of nonlinear models were proposed during that period. As com-pared to linear models, nonlinear time series models provide a much wider spectrum ofpossible dynamics for economic and financial time series data. Among such models,the threshold autoregressive (TAR) model, proposed by Tong (1978, 1983) and Tongand Lim (1978), is well known to properly describe the characteristics of periodic time

C. W. S. Chen (B) · S.-Y. ChenDepartment of Statistics, Feng Chia University, Taichung, Taiwane-mail: [email protected]

S. LeeDepartment of Statistics, Seoul National University, Seoul, Korea

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472 C. W. S. Chen et al.

series. This model successfully captures dynamic behavior of time series by employ-ing a regime switching scheme characterized by thresholds. The threshold unit roottest problem is originally considered by Pham et al. (1991). Since then, the unit roottest designed to detect nonlinear characteristics of underlying models has become verypopular among researchers. Classical unit root tests such as those of Dickey and Fuller(1979), Phillips and Perron (1988), and Kwiatkowski et al. (1992) are all developedfor the linear time series model. However, since the stability of parameters in linearmodels is often questioned, more attention has been turned towards modeling nonlin-ear dynamics to effectively improve the accuracy of estimation and other statisticalinferences.

The non-stationarity in threshold autoregressive models has long been an importantissue: see Chen et al. (2003, 2011) for details. Little is known about the conditionsunder which self-exciting threshold autoregressive (SETAR) models are stationary(Franses and van Dijk 2000, p. 79); the SETAR model takes a modeling approachwhere the lagged-dependent variables play the role of the transition variables. Balkeand Fomby (1997) introduce the concept of threshold cointegration to capture themeaning of nonlinear adjustment processes; Taylor (2001) demonstrates that the lin-ear unit root hypothesis can be rejected in favor of stationary three-regime SETARmodels; Kapetanios et al. (2003) perform a test for the presence of non-stationarityagainst nonlinear but globally stationary exponential smooth transition autoregressiveprocesses. Further, Bec et al. (2004) consider modeling exchange rates by using astationary three-regime SETAR model that allows unit roots in middle regimes. Later,Kapetanios and Shin (2006) propose a unit root test based on the Wald statistic thatjointly uses the significance of the autoregressive parameters in both the lower andupper regimes.

The unit root test in nonlinear models assumes that the proposed Wald test has anasymptotic null distribution free from nuisance parameters as long as the thresholdparameters are known. Since the threshold parameters are unknown in real practice,though, Kapetanios and Shin (2006) make a further assumption that the grid set forthresholds can be selected. In addition to the problem of identifying the threshold para-meters, the Wald test also suffers from the trouble of generating a pivotal null limitdistribution needed to simplify its implementation. This task is nonstandard and com-plicated due to both the parameter identifiability problem and the non-stationary phe-nomenon under the null hypothesis. Thus, to overcome this difficulty and to improvethe Wald test, we take into consideration a Bayesian unit root test that can identify theexistence of a unit root and propose a testing procedure to detect the presence of localnon-stationarity.

Practitioners appreciate the usefulness of Bayesian methods to ease difficultiesin handling complicated statistical computing. This move is seemingly due to theadvent of inexpensive high speed computing facilities and the development of accu-rate stochastic integration methods, especially Markov chain Monte Carlo (MCMC)approaches. In fact, the unit root test can be performed based on the posterior oddsratio, which is approximately obtained by MCMC methods. This idea is similar tothat of the model selection method in Chen and So (2006). Li and Yu (2010) proposea unit root test in volatility dynamics in the context of stochastic volatility models.However, very little work has been done on formally detecting the presence of local

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Threshold Heteroskedastic Models 473

non-stationarity of nonlinear autoregressive processes with heteroskedastic errors andthe question of “which regime is local non-stationarity to a given data?” has beenraised frequently by many practitioners. To propose a formal Bayesian hypothesistesting procedure to answer this question is the major objective of this study.

The proposed unit root test is extended in many ways to better describe a financialtime series. Below we address the main motives of this study:

1. The models considered in existing literature have a limitation in that only onethreshold is taken into consideration. For example, as seen in Bec et al. (2004),the limiting null distribution is obtained under rather stringent conditions on themodel parameters. To cope with various situations, their model can be extendedto the one that accommodates multiple thresholds.

2. The threshold model in Bec et al. (2004, 2008) can be modified in a direction ofreflecting idiosyncratic heteroscedasticity (e.g., generalized autoregressive condi-tional heteroskedasticity (GARCH) models of Bollerslev 1986) and asymmetry inthe threshold nonlinear specifications. This approach additionally contributes tosimultaneously capturing discretely time-varying properties.

3. Bayesian tests can be adopted to test for the unit root against a threshold alternative.In this method, posterior credible intervals for model parameters are used to detectand specify threshold nonlinearity in the mean and/or volatility equations.

In this study, we particularly consider a multi-regime threshold GARCH model thatextends the TAR model of Chan et al. (1985) into a GARCH model to grasp thewell-known empirical evidence such as the clustering volatility, excess kurtosis, andvolatility and mean asymmetry. Li and Li (1996) model mean and volatility asymmetrysimultaneously by using a double threshold autoregressive conditional heteroscedas-ticity (ARCH) model for financial market returns. Chen and So (2006) investigate adouble-threshold GARCH model and propose a Bayesian method that allows simul-taneous inference for all unknown parameters. The multi-regime threshold GARCHmodel has been investigated by Chen et al. (2010) in which an explosive regime involatility is allowed. But the unit root test is not handled in the above-mentioned papers.In this study, we develop a Bayesian test for a unit root against a threshold alternative.A major contribution of our paper is to detect the presence of local non-stationarityand to estimate parameters simultaneously.

The organization of this paper is as follows. In Sect. 2, we introduce the schemeof the proposed unit root test and multiple-regime threshold GARCH models. InSect. 3, we describe the Bayesian inference and MCMC methods to estimate themodel parameters and to detect the unit root. In Sect. 4, we implement a simulationstudy that illustrates our method. In Sect. 5, we illustrate two examples of empiricalanalysis. The first example is the Chicago Board Options Exchange Volatility Index(VIX), and the second example is paid to describing dynamics of the S&P500 stockand stock index futures markets. The concluding remarks are provided in Sect. 6.

2 The Model

In the literature, the following three-regime SETAR model has been studied by manyauthors:

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yt =

⎧⎪⎨

⎪⎩

φ(1)1 yt−1 + at , yt−d < r1

φ(2)1 yt−1 + at , r1 ≤ yt−d < r2, t = 1, . . . , T,

φ(3)1 yt−1 + at , yt−d ≥ r2,

(1)

where at is an i.i.d. sequence with zero mean, variance σ 2a > 0 and a finite 4 + η

moment for some η > 0, and r1 < r2 are threshold parameters. The scheme in (1)appeals to an intuition in that stationary adjustments are allowed to asymmetricallyvary with regimes. When

φ(2)1 ≥ 1, |φ(1)

1 |, |φ(3)1 | < 1, (2)

the above model is proven to be locally non-stationary and globally ergodic: see Balkeand Fomby (1997) and Kapetanios and Shin (2006). This means that the thresholdcointegrating process is defined as a globally stationary process which possibly hasa unit root in the middle regime and is dampened in outer regimes. Note that thethree-regime SETAR model in (1) can be condensed to the model:

� yt ≡ yt − yt−1

= β1 yt−1 I {yt−1 ≤ r1} + β2 yt−1 I {r1 < yt−1 < r2}+β3 yt−1 I {yt−1 > r2} + at , (3)

where � stands for the first differences and I {.} denotes the indicator function,β1 = φ1 − 1, β2 = φ2 − 1, β3 = φ3 − 1, and yt−1 I {yt−1 ≤ r1}, yt−1 I {r1 <

yt−1 < r2}, yt−1 I {yt−1 > r2} are orthogonal to each other. Therefore, Eq. (2) can bereexpressed for Model in (3) as follows:

β2 = 0, β1 < 0, and β3 < 0. (4)

It is well known that financial market volatility changes over time and often exhibitsa volatility clustering property. A common way to effectively capture this phenomenonis to adopt GARCH models for modelling the time-varying conditional variance offinancial time series. In the next section, we consider a unit root test in double-thresholdGARCH models.

2.1 Double-threshold GARCH Models

In this study, we consider the double-threshold heteroscedastic model, proposed byChen and So (2006), as follows:

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yt = φ( j)0 + φ

( j)1 yt−1 + a( j)

t , if r j−1 ≤ zt−d < r j , (5)

at = √htεt , εt

i.i.d.∼ D(0, 1),

h( j)t = α

( j)0 +

m j∑

i=1

α( j)i a2

t−i +k j∑

l=1

β( j)l ht−l , (6)

for t = 1, . . . , T and j = 1, . . . , g, where g is the number of regimes; zt are observedthreshold variables; r j are the threshold values with −∞ = r0 < r1 < · · · < rg = ∞;the integer d is the threshold lag. Further, ht is V ar(yt |y1, . . . , yt−1) and D(0, 1)

stands for a standardized error distribution with mean 0 and variance 1. An interestingfeature of multi-regime threshold GARCH models is that they can capture the volatilitylevel, mean reversion, volatility persistence, and asymmetries in the average return; infact, all mean and volatility parameters can change between regimes.

Chan et al. (1985) derive the necessary and sufficient conditions for the model in(5) with i.i.d. errors a( j)

t and d = 1. According to their analysis, if φ(1)0 , φ

(g)0 , φ

(1)1 and

φ(g)1 satisfy one of the following regularity conditions, yt is stationary:

1. φ(1)1 < 1, φ

(g)1 < 1 and φ

(1)1 φ

(g)1 < 1;

2. φ(1)1 = 1, φ

(g)1 < 1 and φ

(1)0 > 0;

3. φ(1)1 < 1, φ

(g)1 = 1 and φ

(g)0 < 0;

4. φ(1)1 = 1, φ

(g)1 = 1 and φ

(g)0 < 0 < φ

(1)0 ;

5. φ(1)1 φ

(g)1 = 1, φ

(1)1 < 0 and φ

(g)0 + φ

(g)1 φ

(1)0 > 0.

It is noteworthy that regardless of the behavior of yt in the interior regime, in which yt

can display a unit root or an explosive behavior, the nature of yt in the upper and lowerregimes determines whether yt is stationary or not. Due to this reasoning, even whenthe autoregressive coefficient φ1 is equal to 1 everywhere, yt can still be stationarysince the drift parameters act to push the series back towards the equilibrium band (i.e.,φ

(1)0 > 0 and φ

(g)0 < 0). However, one has to keep in mind that the above conditions

may not suffice to ensure the stationarity of model (5) since the GARCH errors areinvolved.

Chen et al. (2010) provide a parsimonious representation of well-known stylizedfeatures of financial time series and facilitates statistical inference in the presence ofhigh or explosive persistence and dynamic conditional volatility. They investigate thefeature of double threshold GARCH models such that overall stationarity does notrequire the model to be covariance stationary in each regime: on the contrary, thelimit cycle behavior that this class of models is able to demonstrate arises from thealternation of explosive, dormant, and rising regimes. It is required to enforce somesufficient conditions on the parameters to ensure positive variance and covariancestationarity. Also, Wong and Li (2001) and Medeiros and Veiga (2009) have shownthat multi-regime time series processes may have explosive regimes, yet still remainstationary and ergodic.

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In fact, to provide stationarity, Chen and So (2006) restrict the parameters by

maxj

|φ( j)1 | < 1 (7)

α( j)0 > 0, α

( j)i ≥ 0, and β

( j)l ≥ 0

m j∑

i=1

α( j)i +

k j∑

l=1

β( j)l < 1, (8)

where j = 1, . . . , g (see Ling 1999 for the proof). In our case, though, the firstcondition should be relaxed to cover the unit root in the regimes. Indeed, the stationaritycondition is worth investigating theoretical within the context of our model, and thus,we leave this as a task for our future research.

3 Bayesian Inference

In order to overcome identifiability problem, we propose a Bayesian unit roottest that can specify the existence of a unit root in any regimes. Bayesian esti-mation needs to indicate a likelihood and a prior distribution on the modelparameters. We mostly use non-informative prior distributions so that the datainference is implemented via the likelihood function. We denote the full parame-ter vector by θ = (φ

(1)0 , . . . , φ

(g)0 , φ

(1)1 , . . . , φ

(g)1 ,α1, . . . ,αg, r, ν)

′, where α j =

(α( j)0 , . . . , α

( j)mj , β

( j)1 , . . . , β

( j)k j )

′and r = (r1, . . . , rg−1)

′. Our prior settings are asfollows: α j follows a uniform prior, i.e., p(α j ) ∝ I (S j ) for j = 1, . . . , g; S j is theset of α j that satisfies the restrictions in (8); the degree of freedom (DOF) ν is repa-rameterized to ν∗ = ν−1 with uniform prior I (v∗ ∈ [0, 0.25]), which restricts ν > 4so that the first four moments of the error distribution are finite; a Gaussian prior isassumed for the drift parameter φ

( j)0 ∼ N (φ j0, σ

2j ), constrained for mean stationarity,

where φ j0 = 0 ; the autoregressive coefficient φ( j)1 follows a mixture normal prior

distribution:

φ( j)1 |δ j ∼ δ j C0 + (1 − δ j )N(φ j0, σ

2j ), (9)

δ j ={

1 with probability p j

0 with probability (1 − p j ),

where C0 denotes a degenerate distribution with all its mass at 1. The δ j is a latentvariable with δ j = 0 or δ j = 1 based on whether regime j has a unit root or not,and follows a Bernoulli distribution with probability p j . In the two regime case, thestandard flat prior on the single threshold limit r1 is Unif(u, l), where (u, l) are chosenas suitable quantiles of the sample vector z. Such a prior needs an extension whenmore than two regimes are considered. A natural extension to general g regimes isa uniform prior for r = (r1, . . . , rg−1)

′, constrained to the region A, which satisfiesboth (i) I (B) = I (r1 < · · · < rg−1) and (ii) each regime must contain at least h % ofthe sample vector z (to ensure a sufficient sample size for valid inference), and h > 0

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is chosen by the user. The constant prior density is then the inverse of the integral:

C =∫ ∫

A

I (B)dr1 · · · drg−1. (10)

Define s as the maximum delay and y = (ys+1, . . . , yT )′. The conditional likeli-

hood function of the model is

L( y|θ) =T∏

t=s+1

⎡

⎣g∑

j=1

1√ht

pε

(yt − μt√

ht

)

I j t

⎤

⎦, (11)

where μt = φ( j)0 + φ

( j)1 yt−1 and I j t is the indicator variable I (r j−1 ≤ zt−d < r j ).

If εti.i.d.∼ t∗ν , where t∗ is the standardized t-distribution which captures the usual

conditional leptokurtosis in financial data. Eq. (11) becomes

L( y | θ) =T∏

t=s+1

⎧⎨

⎩

g∑

j=1

(ν+12 )

( ν2 )

√(ν − 2)π

1√ht

[

1 + (yt − μt )2

(ν − 2)ht

]− ν+12

I j t

⎫⎬

⎭.

Bayesian inference in mixture setup is mostly done via a data augmentation (Tannerand Wong 1987). Let δ = (δ1, δ2, . . . , δg)

′ denote latent mixture indicators. The

augmented model for ( y, δ) has a joint density. The conditional posterior for φ( j)1 is

p(φ( j)1 |δ j , y, θ−φ

( j)1

) ∝ L( y|δ j , θ)π(φ( j)1 |δ j ), j = 1, . . . , g, (12)

where θ−φ( j)1

be the parameter vector θ excluding the element φ( j)1 .

The conditional posterior probability function of δ j is a Bernoulli distribution withprobability

P(δ j =1| y, θ)= a

a+b,

where a = P(φ( j)1 | y, θ−φ

( j)1

, δ j =1)p j and b= P(φ( j)1 | y, θ−φ

( j)1

, δ j = 0)(1− p j ).

(13)

We note here that the idea of predictive updating proposed by Chen (1999) is applicableto the selection of subset autoregression. We can iteratively update δ j by drawing fromits predictive distribution P(δ j | y, θ). The computation is equivalent to integrating outthe AR coefficients under suitable prior distribution. We calculate how many δ j ’s areequal to 1 in MCMC iterations and its percentage. We use the percentage of δ j = 1 inthe MCMC iterations to estimate the posterior probability p j . Notice that the higher

the value of p j is, the closer φ( j)1 is to 1: in other words, the regime j has a unit root

with high possibility.

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The conditional posteriors for each remaining parameter group are non-standard.We thus incorporate the Metropolis and Metropolis–Hastings (MH) methods to drawthe MCMC iterates for the other parameter groups. See Chib (1995) for the discussionon the MH method. To speed up convergence and allow optimal mixing, we employan adaptive MH-MCMC algorithm that combines a random walk Metropolis and anindependent kernel MH algorithm.

We implement the following MCMC sampling algorithm to sample from the jointposterior density. MCMC algorithm as follows:

Step1: Set l = 0 and specify an initial value for δ = 0.Step2:

Step 2a: Sample φ( j) = (φ( j)0 , φ

( j)1 ) from p(φ( j)| y, δ,φ− j ,α, r, ν),where j =

1, . . . , gStep 2b: Sample α j from p(α j | y, δ,φ,α− j , r, ν),where j = 1, . . . , gStep 2c: Sample r from p(r| y, δ,φ,α, ν)

Step 2d: Sample ν from p(ν| y, δ,φ,α, r)Step 2e: Sample δ j from a Bernoulli distribution with probability p∗

j , where j =1, . . . , g and p∗

j = P(δ j = 1| y, θ) follows from Eq. (13)Step 3: Count δ j = 1, j = 1, . . . , gStep 4: Set l = l + 1 and if l ≤ N go to step 2 else go to step 5Step 5: Calculate the value of P O R = B̂ F× prior odds ratio

The first M iterations are discarded as a burn-in period, with the remaining N − Miterations used for posterior inference.

3.1 Bayesian Unit-Root Test

Bayesians consider hypotheses about multiple parameters usually by using Bayesfactors. They are the Bayesian analogues of likelihood ratio tests. The basic intuitionis that prior and posterior information are combined in a ratio that provides evidencein favor of one model specification verses another. Bayes Factors are very flexible,allowing multiple hypotheses to be compared simultaneously and nested models arenot required in order to make comparisons.

Suppose that we observe data y to test two competing models. Let M1 be themodel formulated in the null hypothesis and let M2 be the model formulated underthe alternative hypothesis, relating these data to two different sets of parameters, θ1and θ2. The posterior odds ratio (P O R) in favor of M1 over M2 is:

p(M1| y)p(M2| y)

= p( y|M1)

p( y|M2)× π(M1)

π(M2). (14)

where p( y|Mi ) is the marginal likelihood under model Mi and π(Mi ) is prior prob-ability for Mi . That is, “POR = Bayes Factor (BF) × Prior Odds Ratio”. Here, themarginal likelihood p( y|Mi ) can be expressed as

p( y|Mi ) =∫

p( y|θ i , Mi )p(θ i |Mi )dθ i , i = 1, 2. (15)

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More precisely, Bayesian unit root testing is via threshold specification, we considerthe hypothesis testing problem as follows:

H0 : φ( j)1 = 1 for all j ∈ J vs.

H1 : φ( j)1 ∈ (−1, 1) for some j ∈ J , (16)

where J is a nonempty subset of {1, 2, . . . , g}. We set the latent variable vectorδ = (δ1, . . . , δg) to be indexed with the parameter of interest φ

( j)1 that is in the

hypothesis. When under null hypothesis H0, the φ( j)1 is equal to 1, the corresponding

δ j in δ is equal to 1. Also, we denote by p( y|δ, θ) the likelihood function of theobserved data.

Unfortunately, while Bayes Factors are rather intuitive, as a practical matter theyare often quite difficult to calculate. Following the idea of Li and Yu (2012), the Bayesfactor can be derived as follows:

B F =∫

Ω2∪Ωδ

p( y, δ|θ1, M1)

p( y, δ|θ2, M2)p(δ, θ2| y, M2)dφ1dθ2dδ = E

{p( y, δ|θ1, M1)

p( y, δ|θ2, M2)

}

,

(17)

where Ωk and Ωδ are the support of θk and δ, respectively, and the expectation is takenwith respect to the posterior distribution p(δ, θ2| y, M2). Then, the Bayes factor canbe approximated by

B̂ F = 1

N

N∑

n=1

{p( y, δ(n)|θ (n)

1 , M1)

p( y, δ(n)|θ (n)2 , M2)

}

, (18)

where {δ(n), θ(n)1 }, n = 1, 2, . . . , N , are generated by the MCMC techniques, from

the posterior distribution p(δ, θ1| y, M1). When the prior odds ratio is known, one caneasily obtain the posterior odds ratio as in (14). Hence, the posterior odds can be givenby

P O R = p(M1| y)p(M2| y)

≈ B̂ F × proior odds ratio. (19)

Without any prior information on which model gives a better fit, we set the prior oddsratio equal to 1, so that P O R is reduced to the Bayes factor. The larger the value ofP O R is, the more evidence we have in favor of M1 against M2; conventionally, whenP O R > 1, we conclude that M1 is preferred to M2. Alternatively, we can use a mixedprior distribution with a random weight for the unit root test. This practice is followedby Li and Yu (2010). It can be expected that a large weight is assigned to the unit rootprocess when the data are generated from a unit root process.

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4 Simulation Study

To examine the effectiveness of the proposed sampling scheme in inference and a unitroot test, we conduct a simulation study to evaluate the proposed methodology. Tothis end, we consider a three-regime SETAR model with AR(1) in mean equation andGARCH(1,1) in volatility which is given as follows:

yt =

⎧⎪⎨

⎪⎩

φ(1)0 + φ

(1)1 yt−1 + at , yt−d < r1

φ(2)0 + φ

(2)1 yt−1 + at , r1 ≤ yt−d < r2, t = 1, . . . , T,

φ(3)0 + φ

(3)1 yt−1 + at , yt−d ≥ r2

at = √htεt , εt

i.i.d.∼ D(0, 1),

ht =

⎧⎪⎨

⎪⎩

α(1)0 + α

(1)1 a2

t−1 + β(1)1 ht−1, yt−d < r1

α(2)0 + α

(2)1 a2

t−1 + β(2)1 ht−1, r1 ≤ yt−d < r2

α(3)0 + α

(3)1 a2

t−1 + β(3)1 ht−1, yt−d ≥ r2.

(20)

We consider modelling a simulated time series using a double threshold model withpossibly a unit root in the middle regime. There are two scenarios considered in thissimulation study.

1. Case 1 is considered in (20) with standardized Student-t errors. Data sample sizes,T, of 500, 1,000, 1500, and 2,000 are chosen.

2. Case 2 is considered in (20) with standardized Student-t error distributions.It is an important task in any econometric study to test for model mis-specification.Considering the middle regime without a unit root, 0.20, 0.50, 0.70 and 0.80 arechosen as the parameter for φ

(2)1 , and when there is a possibility of a unit root, φ(2)

1is closer to 1 ( 0.90, 0.95, 0.98 and 1.0 ). Note that we need to ensure a sufficientsample size to make inference, and we set different threshold values for the modelof the middle regime which may or may not have a unit root.

200 data sets are simulated for each model specified above in this simulation study.The sample of size T = 2000 is simulated for Case 2. The MCMC sample sizeis N = 20, 000 with burn-in iterations, M = 8, 000. We set the initial values asfollows: δ = (0, 0, 0),φ j = (0, 0), ν = 200, and α j = (0.1, 0.1, 0.5). The initialvalues for the thresholds r1 and r2 are chosen to be the 10th and 90th percentiles,respectively, to ensure sufficient observations in each regime for valid inference. Weset the hyper-parameters φ0 j = 0 and σ 2

j = 10. For α j , we employ the random walkMetropolis method for the first M MCMC iterations (here, M denotes the size of theburn-in sample) and then switch to an independent kernel MH method with a Gaussianproposal distribution. The overall method is adaptive since the independent kernel MHproposal’s mean and variance-covariance are chosen to be the sample mean and samplevariance-covariance of the iterates for the burn-in period. To enhance and confirm theMCMC convergence, we employ trace and ACF plots for the MCMC samples.

The p̃ j listed in Tables 1, 2, 3 and 4 are the percentage of δ j = 1 in the MCMC

iterations, and the higher the value of p̃ j , the closer φ̂1( j)

is to 1; in other words,the regime j has a unit root with high possibility. The second-to-last row shows the

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averages of the P O Rs in favor of the first listed model (the model formulated inthe null hypothesis) over the 200 replications, with average standard errors. A mixedprior specification with random weights for π(Mi ) is employed. For the purpose ofsensitivity, we set the prior probability for Model 1 based on p̃ j . To avoid givingover-weights, the random weight π(Mi ) is restricted to (0.25, 0.75). The parameterestimates from 200 replications are summarized in Tables 1 and 2, which illustrate eachtrue value together with the averages of posterior means, medians, standard deviations,2.5th and 97.5th percentiles for each parameter. “Freq.( %)” in the last row of Tables 1and 2 denotes the number of cases that P O R > 1 (in brackets) out of 200 replicationsand “( %)” denotes its percentage (in parentheses).

Table 1 Simulation results for Case 1 with various sample sizes and obtained from 200 replications

Par. True T = 500 T = 1,000

Med Std Lower Upper Med Std Lower Upper

φ(1)0 −0.90 −0.6865 0.3218 −1.3714 −0.1217 −0.9842 0.2259 −1.4448 −0.5593

φ(1)1 0.50 0.5812 0.1543 0.2525 0.8491 0.4739 0.1042 0.2631 0.6715

φ(2)0 0.10 0.0990 0.0261 0.0479 0.1507 0.1077 0.0186 0.0714 0.1441

φ(2)1 1.00 0.9950 0.0104 0.9716 0.9993 1.0000 0.0003 1.0000 1.0000

φ(3)0 0.20 0.2577 0.2225 −0.1764 0.6959 0.2262 0.1523 −0.0723 0.5248

φ(3)1 0.30 0.2474 0.1970 −0.1436 0.6289 0.2909 0.1348 0.0261 0.5549

α(1)0 0.04 0.0260 0.0177 0.0037 0.0707 0.0465 0.0254 0.0101 0.1068

α(1)1 0.02 0.1114 0.0922 0.0145 0.3653 0.0708 0.0621 0.0083 0.2445

β(1)1 0.82 0.7483 0.1251 0.4464 0.9248 0.7152 0.1288 0.4188 0.9098

α(2)0 0.03 0.0468 0.0195 0.0182 0.0935 0.0634 0.0245 0.0268 0.1214

α(2)1 0.04 0.0840 0.0568 0.0135 0.2293 0.0758 0.0433 0.0168 0.1837

β(2)1 0.86 0.7577 0.0933 0.5417 0.9017 0.7100 0.1052 0.4663 0.8711

α(3)0 0.01 0.0141 0.0131 0.0011 0.0511 0.0183 0.0166 0.0014 0.0643

α(3)1 0.05 0.1026 0.0721 0.0169 0.2910 0.0917 0.0548 0.0202 0.2304

β(3)1 0.80 0.6893 0.1248 0.4153 0.8900 0.6777 0.1144 0.4250 0.8684

r1 −1.50 −1.3141 0.1767 −1.5454 −0.9064 −1.4931 0.1011 −1.6820 −1.2946

r2 0.80 0.7819 0.0430 0.6733 0.8287 0.8001 0.0124 0.7746 0.8201

ν 7.00 7.9418 85.6641 4.7154 34.3750 6.9092 3.6937 4.8239 11.7609

p̃1 0.0774 0.0034 0.0712 0.0823 0.0000 0.0000 0.0000 0.0000

p̃2 0.8488 0.0196 0.8091 0.8727 0.9942 0.0026 0.9882 0.9963

p̃3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

H0 : φ(2)1 = 1

POR 1.4004 0.4058 0.4655 1.7178 1.7173 0.0813 1.3312 1.7186

Freq. ( %) [147] (73.5 %) [200] (100 %)

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Table 1 continued

Par. True T = 1500 T = 2,000

Med Std Lower Upper Med Std Lower Upper

φ(1)0 −0.90 −0.8192 0.1947 −1.2115 −0.4493 −0.8465 0.1699 −1.1861 −0.5199

φ(1)1 0.50 0.5323 0.0932 0.3452 0.7104 0.5220 0.0809 0.3609 0.6782

φ(2)0 0.10 0.0975 0.0144 0.0692 0.1259 0.0994 0.0123 0.0753 0.1236

φ(2)1 1.00 0.9989 0.0013 0.9963 0.9998 0.9997 0.0006 0.9988 1.0000

φ(3)0 0.20 0.1844 0.1265 −0.0644 0.4332 0.1828 0.1060 −0.0253 0.3906

φ(3)1 0.30 0.3117 0.1120 0.0914 0.5318 0.3173 0.0941 0.1329 0.5025

α(1)0 0.04 0.0361 0.0205 0.0073 0.0854 0.0360 0.0207 0.0069 0.0855

α(1)1 0.02 0.0599 0.0464 0.0080 0.1849 0.0486 0.0367 0.0068 0.1472

β(1)1 0.82 0.7817 0.1010 0.5461 0.9323 0.7977 0.0957 0.5718 0.9386

α(2)0 0.03 0.0578 0.0208 0.0258 0.1060 0.0578 0.0199 0.0273 0.1038

α(2)1 0.04 0.0629 0.0328 0.0153 0.1420 0.0585 0.0276 0.0171 0.1240

β(2)1 0.86 0.7301 0.0917 0.5211 0.8750 0.7339 0.0863 0.5368 0.8678

α(3)0 0.01 0.0153 0.0143 0.0011 0.0555 0.0153 0.0139 0.0011 0.0538

α(3)1 0.05 0.0756 0.0409 0.0190 0.1776 0.0721 0.0346 0.0213 0.1555

β(3)1 0.80 0.7095 0.1004 0.4839 0.8736 0.7184 0.0919 0.5106 0.8696

r1 −1.50 −1.4309 0.0843 −1.5628 −1.2482 −1.4614 0.0609 −1.5554 −1.3201

r2 0.80 0.8009 0.0080 0.7849 0.8138 0.7999 0.0049 0.7907 0.8089

ν 7.00 7.2369 1.4838 5.2737 10.9977 7.1907 1.6032 5.4451 10.3420

p̃1 0.0100 0.0000 0.0099 0.0100 0.0051 0.0001 0.0050 0.0053

p̃2 0.9825 0.0030 0.9758 0.9853 0.9948 0.0014 0.9915 0.9959

p̃3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

H0 : φ(2)1 = 1

POR 1.7180 0.1633 1.0164 1.7187 1.7182 0.0744 1.5286 1.7188

Freq.(%) [198] (99%) [200] (100%)

As anticipated, Table 1 shows that as the sample size increases, the standard errorsdecrease and the posterior mean estimates get closer to their true values. When thesample size is greater than 1, 000, as seen in Table 1, the percentage of correct decisionis higher than 0.99. This means that the sample size is at least 1,000 and the Bayesianunit root test is precise. As seen in Table 2, the percentage of the P O R > 1 is observedto be less than 5 % for φ

(2)1 =0.2, 0.5, 0.7, and 0.8. Thus, there is no doubt that the null

hypothesis should be rejected, and the percentage of correct decisions is higher than95 %. On the other hand, when the φ

(2)1 is greater than or equal to 0.98, the percentage

of the middle regime with a unit root is higher than 0.95. This is quite natural in thata high probability of favoring a unit root in the middle regime is anticipated as φ

(2)1

gets close to 1.

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Table 2 Simulation results for Case 2 and the frequency of selecting the local non-stationarity of nonlinearmodels based on 200 replications with a sample size of 2,000

Par. φ(2)1 =0.20 φ

(2)1 =0.50 φ

(2)1 =0.70 φ

(2)1 =0.80

True Med Std Med Std Med Std Med Std

φ(1)0 −0.90 −0.7135 0.1150 −0.8330 0.1144 −0.8990 0.1057 −0.9074 0.0948

φ(1)1 0.50 0.5791 0.0553 0.5292 0.0576 0.4995 0.0541 0.4949 0.0486

φ(2)0 0.10 0.1005 0.0127 0.0982 0.0131 0.1006 0.0131 0.0993 0.0134

φ(2)1 0.2026 0.0343 0.5018 0.0346 0.6991 0.0319 0.8016 0.0312

φ(3)0 0.20 0.1839 0.1199 0.1995 0.1092 0.2112 0.1084 0.2046 0.1053

φ(3)1 0.30 0.3069 0.1172 0.3026 0.1074 0.2895 0.1048 0.2949 0.1024

α(1)0 0.04 0.0281 0.0147 0.0372 0.0179 0.0446 0.0211 0.0590 0.0255

α(1)1 0.02 0.0518 0.0308 0.0460 0.0313 0.0443 0.0305 0.0461 0.0326

β(1)1 0.82 0.8295 0.0722 0.7982 0.0847 0.7730 0.0949 0.7103 0.1144

α(2)0 0.03 0.0483 0.0180 0.0512 0.0188 0.0575 0.0220 0.0612 0.0226

α(2)1 0.04 0.0556 0.0308 0.0610 0.0315 0.0622 0.0314 0.0616 0.0318

β(2)1 0.86 0.7774 0.0774 0.7620 0.0816 0.7331 0.0963 0.7167 0.0994

α(3)0 0.01 0.0105 0.0099 0.0123 0.0116 0.0150 0.0135 0.0176 0.0154

α(3)1 0.05 0.0666 0.0355 0.0719 0.0365 0.0716 0.0361 0.0746 0.0382

β(3)1 0.80 0.7506 0.0955 0.7362 0.0961 0.7232 0.0947 0.7074 0.1011

r1 −1.10 −1.0549 0.0104 −1.0814 0.0127 −1.0977 0.0103 −1.0992 0.0116

r2 0.70 0.6715 0.0490 0.6615 0.0780 0.6898 0.0416 0.6939 0.0220

ν 7.00 7.2781 1.2348 7.2026 1.2516 7.1097 1.1917 7.1687 1.1936

p̃1 0.1449 0.0001 0.0400 0.0000 0.0000 0.0000 0.0050 0.0000

p̃2 0.0100 0.0000 0.0050 0.0000 0.0000 0.0000 0.0385 0.0103

p̃3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

H0 : φ(2)1 = 1

P O R 0.3333 0.1381 0.3334 0.098 0.3334 0.0000 0.3334 0.1831Freq.(%) [2] (1%) [1] (0.5%) [0] (0%) [4] (2%)

Furthermore, to explore sensitivity to prior information, we re-estimate the systemby setting the prior odds ratio equal to 1; the hypothesis testing decision result remainsthe same. To save space, we only display the results for random weights, and the resultsfor the equal weight are available from the authors upon request.

5 Empirical Study

We illustrate the proposed method using two examples. The first example, based onthe VIX data from the period covering the subprime crisis, and the second exampleis the price differential between stock and stock index futures markets (the basis).

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Table 2 continued

Par. φ(2)1 =0.90 φ

(2)1 =0.95 φ

(2)1 =0.98 φ

(2)1 =1.0

True Med Std Med Std Med Std Med Std

φ(1)0 −0.90 −0.8674 0.1755 −0.9651 0.1739 −0.7634 0.1677 −0.8465 0.1699

φ(1)1 0.50 0.5233 0.0829 0.4745 0.0820 0.5583 0.0806 0.5220 0.0809

φ(2)0 0.10 0.0936 0.0123 0.0946 0.0124 0.1005 0.0125 0.0994 0.0123

φ(2)1 0.9218 0.0250 0.9908 0.0093 0.9983 0.0035 0.9997 0.0006

φ(3)0 0.20 0.2096 0.1127 0.2000 0.1101 0.2032 0.1062 0.1828 0.1060

φ(3)1 0.30 0.2885 0.1001 0.2975 0.0978 0.2964 0.0943 0.3173 0.0941

α(1)0 0.04 0.0318 0.0185 0.0359 0.0202 0.0351 0.0199 0.0360 0.0207

α(1)1 0.02 0.0468 0.0353 0.0501 0.0368 0.0512 0.0371 0.0486 0.0367

β(1)1 0.82 0.8114 0.0890 0.7908 0.0956 0.7999 0.0922 0.7977 0.0957

α(2)0 0.03 0.0560 0.0198 0.0558 0.0195 0.0556 0.0196 0.0578 0.0199

α(2)1 0.04 0.0597 0.0282 0.0555 0.0267 0.0579 0.0283 0.0585 0.0276

β(2)1 0.86 0.7494 0.0839 0.7524 0.0823 0.7448 0.0853 0.7339 0.0863

α(3)0 0.01 0.0134 0.0127 0.0143 0.0132 0.0147 0.0135 0.0153 0.0139

α(3)1 0.05 0.0700 0.0333 0.0730 0.0342 0.0690 0.0337 0.0721 0.0346

β(3)1 0.80 0.7279 0.0885 0.7172 0.0882 0.7288 0.0895 0.7184 0.0919

r1 −1.50 −1.4549 0.0323 −1.5031 0.0405 −1.4006 0.0703 −1.4614 0.0609

r2 0.80 0.7983 0.0098 0.7994 0.0065 0.7994 0.0055 0.7999 0.0049

ν 7.00 6.9402 1.1257 6.9543 1.1525 7.2408 1.2334 7.1907 1.6032

p̃1 0.0000 0.0000 0.0000 0.0000 0.0166 0.0011 0.0051 0.0001

p̃2 0.3646 0.0187 0.9007 0.0107 0.9658 0.0050 0.9948 0.0014

p̃3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

H0 : φ(2)1 = 1

P O R 0.5869 0.3463 1.6788 0.3634 1.7163 0.2273 1.7182 0.0744

Freq.(%) [71] (35.5%) [189] (94.5%) [197] (98.5%) [200] (100%)

We would like to investigate the role of double threshold heteroskedastic models inexplaining the daily dynamics of the S&P500 index-index futures basis.

5.1 Chicago Board Options Exchange Volatility Index (VIX)

VIX is the symbol for the measure of implied volatility of the Standard & Poor’s 500index options. Since its introduction in 1993, VIX has been considered by many to bethe world’s premier barometer of investor sentiment and market volatility. VIX is oftenreferred to as the “investor fear gauge”. In this section, we consider the VIX impliedvolatility index. Previous studies suggest that the derivative market in the US maylead the spot market, and the VIX index carries better information helpful to improve

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1030

5070

Orig

inal

Ser

ies

−15

−5

05

10

2002 2004 2006 2008 2010 2012

1st D

iffer

ence

Ser

ies

Fig. 1 Time series plot of daily VIX from March 12, 2002 to March 12, 2012. Upper panel is for originalseries, and lower panel is for the change series of VIX

volatility forecasting performance. To fit models to the VIX index, Becker et al. (2006),Hung et al. (2009) and Aizenman and Pasricha (2010) use the original series withouttaking difference, while Shu and Zhang (2012) and Konstantinidi and Skiadopoulos(2011) use a modified series (with the first difference). Since this particularly bearsinside a controversial argument concerning the stationarity of the VIX series, weanalyze both the VIX series with and without taking difference to test for its stationarityand the existence of unit roots in the regimes. The stock price indices are obtainedfrom Data stream International from March 12, 2002 to March 12, 2012. Figure 1presents the time series plot of the VIX series with and without taking difference.The time period includes a financial turmoil period of late 2008–2009 triggered bythe US subprime mortgage crisis. We observe that the sharp increase in volatility inNovember 2008 with an intra-day high of 80.86 in Fig. 1.

Table 3 summarizes the results for VIX, and the parameter estimates are underH0 : φ

(2)1 = 1. The left panel reports a summary of the VIX series without taking the

first difference. In this study, we consider four scenarios and use Bayesian unit roottesting. In Table 3, the results shown on the left demonstrate that the VIX series hasunit roots based on P O R. The P O R values for all the scenarios appear to be greaterthan 1, which suggests that we need to take the first difference for the VIX series. Theresults on the right, however, show that taking the difference for the VIX series resultsin a model without unit roots.

The estimates of volatility parameters in the right panel are significant. This demon-strates that financial market volatility changes over time and often exhibits a volatilityclustering property. The estimates of threshold limits, r1 and r2, are significantly wellseparated: r1 and r2 are estimated to be approximately 15 and 27, respectively. Seem-ingly, the VIX hints of irrational exuberance when the VIX falls below the lowerthreshold of around 15. The right panel reports Bayesian estimates for the changeseries of VIX. The other interesting phenomenon is that the AR coefficients of upper

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Table 3 Parameter estimates for daily VIX and change series of VIX from the model in (20)

yt = V I Xt yt = V I Xt − V I Xt−1

Mean Med Std Lower Upper Mean Med Std Lower Upper

φ(1)0 −0.0002 −0.0042 0.0201 −0.0329 0.0395 −0.1496 −0.1486 0.1863 −0.5508 0.2209

φ(1)1 1.0000 1.0000 0.0000 1.0000 1.0000 −0.0252 −0.0242 0.0832 −0.1954 0.1373

φ(2)0 −0.0537 −0.0514 0.0259 −0.1116 −0.0063 0.0314 0.0323 0.0266 −0.0199 0.0823

φ(2)1 1.0000 1.0000 0.0000 1.0000 1.0000 0.0308 0.0372 0.0751 −0.1367 0.1682

φ(3)0 −0.3235 −0.3237 0.0777 −0.4643 −0.1748 0.0243 0.0234 0.0577 −0.0877 0.1405

φ(3)1 1.0000 1.0000 0.0000 1.0000 1.0000 −0.1980 −0.1974 0.0458 −0.2878 −0.1097

α(1)0 0.2468 0.2461 0.0304 0.1875 0.3074 0.1245 0.0930 0.1132 0.0023 0.4083

α(1)1 0.1829 0.1781 0.0794 0.0603 0.3638 0.0366 0.0315 0.0277 0.0017 0.1145

β(1)1 0.1454 0.1375 0.0516 0.0699 0.2396 0.8852 0.8869 0.0407 0.8086 0.9630

α(2)0 0.2562 0.2507 0.0599 0.1507 0.3802 0.0113 0.0110 0.0066 0.0006 0.0243

α(2)1 0.2819 0.2812 0.0463 0.1943 0.3771 0.0499 0.0426 0.0356 0.0013 0.1125

β(2)1 0.5997 0.6004 0.0524 0.4917 0.6963 0.8472 0.8496 0.0183 0.8065 0.8803

α(3)0 0.4674 0.4444 0.1706 0.2007 0.8724 0.1898 0.1852 0.0413 0.1170 0.2714

α(3)1 0.2093 0.2057 0.0448 0.1342 0.3069 0.1624 0.1617 0.0270 0.1042 0.2177

β(3)1 0.7565 0.7598 0.0466 0.6528 0.8367 0.8311 0.8326 0.0283 0.7764 0.8936

r1 13.8449 13.9616 0.3019 13.2439 14.2225 −1.2741 −1.3628 0.2178 −1.5153 −0.7461

r2 25.8133 25.4659 0.8023 25.0662 28.2662 0.3614 0.3312 0.1102 0.2340 0.6531

ν 4.2623 4.2086 0.2186 4.0089 4.8205 4.4652 4.4106 0.3190 4.0290 5.1861

p̃1 1.0000 1.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

p̃2 1.0000 1.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

p̃3 1.0000 1.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

H0 : φ(2)1 = 1 P O R = 1.7185 P O R∗ = 0.3333

H0 : φ(2)1 = φ

(3)1 = 1 P O R = 1.7118 P O R∗ = 0.3331

H0 : φ(1)1 = φ

(2)1 = 1 P O R = 1.7151 P O R∗ = 0.3333

H0 : φ(1)1 = φ

(2)1 = φ

(3)1 = 1 P O R = 1.7183 P O R∗ = 0.3330

and lower regimes turn out to have negative values. This effect shows that the VIXcan be used as a hedge. Indeed, the change series of VIX is an appropriate hedgingtool since it has a strong negative correlation in the upper regime.

5.2 S&P 500 Basis

The second example examines the relationship between spot and stock index futuresmarkets. Over the last two decades, a large body of both theoretical and empiricalresearch focusing on the dynamic relationship between spot and futures prices in

123


stock index futures markets has developed in literature (e.g., Modest and Sundaresan1983; Figlewski 1984). In particular, a number of empirical studies have focused on thepersistence of deviations from the cost of carry and have investigated the relationshipbetween the spot and futures prices in the context of vector autoregressions. Recently,an interesting strand of literature has developed that allows for nonlinear mean rever-sion in the basis. It suggests that the dynamic relationship between spot and futuresprices could be characterized by a nonlinear equilibrium-correction model. See, forexample, Sarno and Valente (2000) and Brooks and Garrett (2002). Therein, the basisis defined as bt = ft − st , where the ft is the stock index futures price at time t and st

is the stock index price. The test for mispricing is typically concerned with the task ofidentifying whether or not there exist profitable arbitrage opportunities that arises asa result of stock and stock index futures processes moving out of line with each other.

In this example, the data used here is daily closing prices for the S&P 500 stockindex and S&P 500 stock index futures contract over the period of March 12, 2002 toMarch 12, 2012. The futures series is continuous, and is constructed being rolled overinto the next contract in the month prior to expiration. The S&P 500 basis is plottedin Fig. 2.

Table 4 summarizes the results for the S&P 500 basis. The parameter estimatesare under H0 : φ

(2)1 = 1. Here, we analyze whether such dynamics can be explained

as the result of different regimes within which no arbitrage is triggered and outsideof which an arbitrage occurs. The rationale for the existence of different regimes inthis context is that the basis (adjusted for carrying costs if necessary), which is veryimportant in the arbitrage process, can fluctuate within the middle regime determinedby transaction costs without actually triggering arbitrage. Therefore, as long as thebasis is in the middle regime, the dynamics of the basis are of little concern. Weinterpret threshold unit-root problems in the last four rows of Table 4 as the nullmodels. The AR coefficient, φ(2)

1 = 0.9148, is higher than 0.9, but not closer to 1. The

threshold unit-root in the second regime model (H0 : φ(2)1 = 1) is not supported since

P O R < 1. The basis fluctuates in the middle regime (within boundaries) withoutactually triggering arbitrage. This result is similar to the conclusion by Brooks andGarrett (2002).

pric

e

2002 2004 2006 2008 2010 2012

−15

−10

−5

05

1015

20

Fig. 2 The time plot of daily S&P 500 basis from March 12, 2002 to March 12, 2012

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Table 4 Parameter estimatesfor the S&P 500 basis from themodel in (20)

Mean Med Std Lower Upper

φ(1)0 −3.0167 −3.0029 0.5635 −4.1838 −1.9622

φ(1)1 0.2774 0.2792 0.0993 0.0686 0.4663

φ(2)0 −0.0995 −0.0984 0.0408 −0.1814 −0.0199

φ(2)1 0.9148 0.9146 0.0149 0.8851 0.9443

φ(3)0 1.4248 1.4246 0.5540 0.3228 2.5231

φ(3)1 0.8279 0.8277 0.0502 0.7250 0.9263

α(1)0 0.2261 0.1533 0.2294 0.0051 1.0556

α(1)1 0.2036 0.1970 0.0593 0.1035 0.3356

β(1)1 0.7199 0.7256 0.0713 0.5684 0.8435

α(2)0 0.4281 0.4324 0.1736 0.0877 0.7627

α(2)1 0.2216 0.2276 0.0621 0.0869 0.3269

β(2)1 0.7609 0.7539 0.0663 0.6486 0.9079

α(3)0 1.0289 1.0076 0.7017 0.0621 2.5246

α(3)1 0.0781 0.0585 0.0701 0.0012 0.2479

β(3)1 0.6596 0.6554 0.2071 0.2806 0.9635

r1 −4.3745 −4.3616 0.2232 −4.8714 −3.9605

r2 7.5544 7.5679 0.1922 7.1874 7.8761

ν 5.1345 5.0970 0.4879 4.2737 6.2159

p̃1 0.0000 0.0000 0.0000 0.0000 0.0000

p̃2 0.0525 0.0495 0.0198 0.0282 0.0807

p̃3 0.0000 0.0000 0.0000 0.0000 0.0000

H0 : φ(2)1 = 1 P O R = 0.3582

H0 : φ(2)1 = φ

(3)1 = 1 P O R = 0.3572

H0 : φ(1)1 = φ

(2)1 = 1 P O R = 0.3579

H0 : φ(1)1 = φ

(2)1 = φ

(3)1 = 1 P O R = 0.3570

When the basis crosses the upper threshold (≈7.55), futures are regarded as beingovervalued, and thus, a proper strategy would be to take a short position in futures anda long position in stocks. On the contrary, when the basis crosses the lower threshold(≈ −4.37), a proper strategy is to take a long position in futures and a short positionin stocks.

6 Concluding Remarks

The main objective of this paper is to detect the presence of local non-stationarityof nonlinear autoregressive processes with heteroskedastic errors. The testing proce-dure is based on the mixture prior and posterior odds ratio. The mixture prior for the

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autoregressive coefficient has been used to alleviate the usual identifiability problemthat arises when the AR(1) parameter is close to 1. The method for computing the BFis numerically stable and easy to implement in this paper. We employ the MCMC sam-pling method to perform the approximation process more conveniently and efficientlyand to simultaneously estimate the threshold values and all other parameters.

We use both the simulated and empirical data to illustrate our proposed method.This method appears to successfully allow reliable inference, and further, to overcomethe non-integrability problem of the likelihood function. We demonstrate effectiveperformance of the Bayesian hypothesis test in our simulation study. Two examplesof empirical analysis are demonstrated. Our results are confirmed by detecting a localrandom-walk in the middle regime in the VIX series, which suggests that profitablearbitrage opportunities will not be present. In the future, the model can be extended toinclude a smooth transition function as in the work of Gerlach and Chen (2008), ratherthan sharp transition, as a threshold to capture smooth mean and variance asymmetriesin financial markets.

Acknowledgments We thank the Editor-in-Chief, Hans M. Amman, for his thoughtful consideration.Cathy W.S. Chen and Sangyeol Lee are respectively supported by the grants from the National ScienceCouncil of Taiwan (NSC) (NSC 101-2118-M-035-006-MY2) and the National Research Foundation ofKorea (NRF) funded by the Korea government (MEST) (No. 2012R1A2A2A01046092).

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