Bayesian unit-root tests for Stochastic Volatility models

Statistical Methodology 6 (2009) 189–201 Contents lists available at ScienceDirect Statistical Methodology journal homepage: www.elsevier.com/locate/stamet Bayesian unit-root tests for Stochastic Volatility models Zeynep I. Kalaylıoğlu a , Sujit K. Ghosh b,∗ a Department of Statistics, Middle East Technical University, Ankara, Turkey b Department of Statistics, North Carolina State University, Raleigh, NC 27695-8203, USA article info Article history: Received 23 April 2007 Received in revised form 10 May 2008 Accepted 25 July 2008 Keywords: BUGS software Gibbs sampling Markov Chain Monte Carlo Stochastic Volatility models Unit-root test a b s t r a c t In this article, we consider Bayesian inference procedures to test for a unit root in Stochastic Volatility (SV) models. Unit-root tests for the persistence parameter of the SV models, based on the Bayes Factor (BF), have been recently introduced in the literature. In contrast, we propose a flexible class of priors that is noninformative over the entire support of the persistence parameter (including the non-stationarity region). In addition, we show that our model fitting procedure is computationally efficient (using the software WinBUGS). Finally, we show that our proposed test procedures have good frequentist properties in terms of achieving high statistical power, while maintaining low total error rates. We illustrate the above features of our method by extensive simulation studies, followed by an application to a real data set on exchange rates. © 2008 Elsevier B.V. All rights reserved. 1. Introduction In finance, it is very common to see that the mean corrected return on holding an asset has time dependent variation (i.e. volatility). Researchers have been interested in modeling the time dependent feature of unobserved volatility. A model that is commonly used to model such features, is known as the Autoregressive Conditional Heteroscedastic (ARCH) model (see [4]). An extension is to consider Generalized ARCH (GARCH) models, which have been found very popular to model the volatility over time. There is an extensive literature on estimation and testing procedures for GARCH models (e.g., see [2]). However, in ARCH and GARCH models, given the past observations (returns), volatility is a deterministic function of the past observations. This feature may not be appropriate for some real ∗ Corresponding author. Tel.: +1 9195151950; fax: +1 9195157591. E-mail addresses: [email protected] (Z.I. Kalaylıoğlu), [email protected] (S.K. Ghosh). 1572-3127/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.stamet.2008.07.002 190 Z.I. Kalaylıoğlu, S.K. Ghosh / Statistical Methodology 6 (2009) 189–201 data sets such as stock exchange rates, where we expect the volatilities to vary stochastically as a function of past observations. An alternative approach is to use Stochastic Volatility (SV) models. In the SV models, the logarithm of the unobserved volatility is modeled as a stochastic process. A popular choice is to model the log-volatility as an AR(1) process. Extensions to higher order AR process may also be considered. However, for illustrative purpose only, we restrict our attention to only AR(1) processes for log-volatilities within the SV models. Jacquier, Polson and Rossi [10] developed Bayesian and frequentist methods for estimating the parameters of SV models. The posterior distributions of the parameters (and hence the Bayes estimates) are obtained using a Markov Chain Monte Carlo (MCMC) algorithm that is based on treating the unobserved volatilities as missing observations. The maximum likelihood estimation of the parameters as well as the method of moment estimation are also considered by Jacquier et al. [10]. They compare these estimation procedures for SV models based on the bias and the root mean squared errors of the estimators. Their simulation results show that their Bayesian estimation technique performs better than the other two procedures for SV models. They also illustrate their methodology by fitting four series; equal-weighted (EW) and value-weighted (VW) indices of NYSE, and three decile portfolios corresponding to the 1, 5, and 10 deciles of stock. By using their MCMC algorithm they obtain the Bayes estimators under squared error loss, i.e. the posterior mean of the persistence parameter (henceforth denoted by φ ) of the volatility processes. It turns out that the posterior mean of φ for the four series are 0.95 (s.d. 0.013), 0.93 (s.d. 0.016), 0.91(s.d. 0.046), and 0.93 (s.d. 0.022), respectively, along with posterior standard deviations (s.d.) in parenthesis. The striking feature being that in all four series φ is expected to be close to unity. It is to be noted that the priors used for φ had support only on the stationarity region, i.e. uniform on the interval (0, 1). We suspect that such priors down-weight the posterior mass of φ at unity. In Section 2 we introduce priors that overcome such limitations of previous Bayesian procedures. There are several other real data examples in the literature where this φ parameter of the process for the volatility is estimated to be close to unity. For instance, see [13,14]. This raises the question of the presence of a unit-root in the SV models. Unfortunately this feature of the SV models has not received much attention in the literature. There has been only few attempts to test for the presence of a unit root in SV models. Harvey, Ruiz and Shephard [9] considered a SV model to fit four exchange rates. They obtained the Quasi-Maximum Likelihood (QML) estimates of the φ parameter of the volatility process to be 0.9912 (s.e. 0.007), 0.9646 (s.e. 0.021), 0.9948 (s.e. 0.005), and 0.9575 (s.e. 0.002), respectively along with the standard error (s.e.) of the estimates in parenthesis. They test the null hypothesis of a unit root (i.e. H0 : φ = 1) by applying Augmented Dickey–Fuller (ADF) unit-root test (see [19]) to the log-squared returns for all the four series. The hypothesis of a unit-root is strongly rejected at the 1% level for all the series, despite the fact that the estimates were very close to unity. Based on this result, they conclude that the reliability of the classical ADF test is questionable for SV models. We observe similar problems with some of the frequentist unit-root tests for SV models (not reported in this article, however see Section 7). Bayesian data analysis is becoming more and more appealing because of its flexibility in handling complex models that typically involve many parameters and/or missing observations. This is one of the reasons that we consider a Bayesian perspective for unit-root testing. It may be noted that the data generated from a SV model has the same number of unobserved volatilities as the sample size. So and Li [21] propose a Bayesian unit-root test procedure to test for the unit root in SV models. Their rejection criterion is to reject the null hypothesis if the posterior odds in favor of H0 is less than or equal to one. Based on their simulation results, they argue that one can get reliable results from their Bayesian unitroot testing procedure. However, using a prior density that is defined on an interval that excludes the point one is not suitable for statistical inference where the main goal is testing for a unit root. Also the MCMC computation method proposed by So and Li [21] is based on approximating the error process of log-volatilities by discrete mixture of several (seven) normals. Their method is computationally inefficient as the efficiency is lost in estimating several nuisance parameters of the mixing normals. In contrast, we develop methods that do not require such approximation, use a flexible class of prior distributions and make use of a computation method that is easy to implement in practice. Z.I. Kalaylıoğlu, S.K. Ghosh / Statistical Methodology 6 (2009) 189–201 191 Prior densities in Bayesian inference play a crucial role. Suitably chosen non-informative priors often yields estimators with good frequentist properties. We find the model building procedure to be quite flexible from a Bayesian perspective, but find it more convenient to validate its use from a frequentist perspective. In this article, we address the problems associated with employing different prior densities for the φ parameter of the SV models. In particular, we suggest the use of a noninformative prior that is supported on a space that includes one with positive probability. We also illustrate how our proposed method can be implemented in practice using a public domain software like WinBUGS. The rest of the article is organized as follows; Section 2 describes the SV model, and some alternative assumptions concerning this model. Section 3 presents the motivation for our choice of prior densities for the parameters and Bayesian unit-root testing methods that we suggest for SV models. Section 4 describes the proposed Gibbs sampling algorithm, and how we implement it in WinBUGS. Extensive simulation studies are presented in Section 5 to validate the use of our proposed method from a frequentist perspective. We illustrate our proposed method by applying the testing procedure to a real data set on exchange rates in Section 6. Finally, in Section 7, we indicate few limitations of our approach and some directions for further research in SV models. 2. The Stochastic Volatility (SV) model The widely used SV models can be described in its simplest form as follows. Let rt denote the mean corrected return observed at time t. As an example of return data rt , first consider pt which denotes, say, the Turkish Lira/U.S. Dollar exchangeP rate at time t = 1, 2, . . . , n. Then mean corrected return, n pi p ). We analyze such data sets in Section 6. Let rt , can be computed as rt = log p t − 1n i=1 (log p t −1 i−1 Vt = Var(rt ) be the unobserved volatility at time t. In SV models it is customary to model the logvolatilities, ht = log Vt in contrast to ARCH and GARCH models. The SV model is represented as, 1 rt = ut e 2 ht ht − µ = φ(ht −1 − µ) + σ ηt , t = 2, . . . , n (1) h1 = µ1 + σ1 η1 , where ηt ’s and ut ’s are assumed to be a sequence of iid realizations from a distribution with mean zero and variance one. The parameters µ1 and σ1 of the initial distribution are chosen (as a function of φ and σ ) appropriately, so as make the process ht stationary when |φ| < 1. When φ = 1, we set µ1 = µ and σ1 = σ . Note that stationarity of the process ht implies stationarity of the process rt . In this article the ut ’s and ηt ’s are also assumed to be independent. However several extensions of the above model in (1), are possible. For instance, Sandmann and Koopman [20] consider a SV model where ut ’s and ηt ’s are assumed to be correlated. It is quite straightforward to extend the AR(1) process of the latent ht ’s to a general AR(p) process. The main parameter of interest in this model is the persistence parameter, φ and our focus is to test the unit-root hypothesis, H0 : φ = 1 against the alternative Ha : |φ| < 1. When there is a unit root in the log-volatility process, shocks to volatility do not decay rapidly. That is, the effect of past shocks on current volatility (conditional variance of returns) remain persistent for long periods. Persistence of shocks to volatility has important implications in economics and finance. High persistence of shocks to volatility increases the fluctuation in the volatility (change in the return over time). As a result, the business environment becomes more uncertain. That causes the market to plunge. See Pyndick [16,17], Poterba and Summers [15], Chou [3] and Bollerslev and Engle [2], who studied the volatility and its relationship with market fluctuations. Tests for a unit-root in SV models continue to receive much attention. Jacquier et al. [10] suggested several methods to estimate the parameter, θ = (µ, φ, σ 2 ) of model (1). From a Bayesian perspective, the above model can be viewed as a three-level hierarchical model. In the first level of hierarchy, the conditional distribution of the data (viz., rt ’s) is specified given the unobserved volatility (ht ’s). At the second level, the conditional distribution of the unobserved volatility is specified given the parameter θ . Finally at the last level, prior distributions are specified. 192 Z.I. Kalaylıoğlu, S.K. Ghosh / Statistical Methodology 6 (2009) 189–201 Jacquier et al. [10] obtained the Bayes estimates of θ under conjugate prior specification and more importantly under the assumption that Pr(0 < φ < 1) = 1. We argue that such priors may not be suitable for testing the unit-root hypothesis. [13] also obtained parameter estimates under model (1) using priors that puts a zero mass at φ = 1. Kim et al. [13] provide three efficient MCMC methods to obtain samples from the posterior distribution of θ , using state-space representation of the above model. So and Li [21] proposed a Bayesian unit-root test procedure to test for the unit root in the SV model in (1). They consider four different prior densities for the persistence parameter φ . The first one is the uniform prior on the interval (0, 1) and they claimed this prior to be ‘‘non-informative’’. The second prior that they used is the truncated Normal density on the interval (0, 1). The other two priors were Beta(10, 1) and Beta(20, 2) distributions. They treat the testing problem as comparing two models under the null hypothesis and the alternative hypothesis. They select the model based on the value of the posterior odds ratio. We compare our methods to their approach in Section 5. However to our knowledge, none of the previous methods have used a prior that puts a positive mass at unity, and hence the use of such priors as ‘‘non-informative’’ is questionable. We address this issue specifically in Section 3. 3. Prior distributions We consider several prior densities for the persistence parameter, φ , of the SV model in (1). The main reason to consider more than one prior density is to perform a sensitivity analysis of our Bayesian procedures. In the literature, continuous densities such as uniform, truncated normal, and beta distributions that are defined over the interval (0,1) have been used as prior densities for φ [10,21]. For estimation purpose such densities might be useful, but for testing the unit-root hypothesis H0 : φ = 1, any continuous distribution with support on only (0, 1) interval can produce results biased in favor of stationarity. For this reason, we broaden the support, and consider prior densities that either assign a positive mass at unity or a density that is defined on an interval that includes the unity (say (−a, a), with a > 1). Specifically, we propose to use two new types of prior densities for φ . (i) The mixed priors. Let B be a Bernoulli random variable with success probability p = P (B = 1). In addition, let U be a continuous random variable having support on the open interval (0, 1). U is assumed to be independent of B. We set the persistence parameter φ = B + (1 − B)U. Here, p can be viewed as the mixing probability. The mixing probability can be assumed to be a known constant or an unknown parameter. When p is assumed to be a constant, the marginal prior density of φ is fφ (φ) = pI (φ = 1) + (1 − p)fU (φ)I (0 < φ < 1), (2) where I (A) = 1 if A is true, otherwise I (A) = 0. However when p is assumed to be a parameter with hyper-prior density say fp (·), then, f (φ|p) = pI (φ = 1) + (1 − p)fU (φ)I (0 < φ < 1) and fφ (φ) = Z f (φ|p)fp (p) dp = E [p]I (φ = 1) + (1 − E [p])fU (φ)I (0 < φ < 1). That is, the marginal distribution of φ is also a mixture distribution as given in (2) with p = E [p], where E [p] denotes the expectation of p w.r.t. fp (·). Here, U can be assumed to have a Beta distribution defined over (0, 1) with parameters αU and βU . In this article, we set αU = βU = 1. That is, we take U ∼ U (0, 1), and denote this mixed uniform (MU) density as MU (p). In our simulations (see Section 5) we study the sensitivity of the constant p on the posterior distribution of φ . Z.I. Kalaylıoğlu, S.K. Ghosh / Statistical Methodology 6 (2009) 189–201 193 (ii) The flat priors. In addition to the above mixed priors, we suggest using flat priors that are defined on a region that includes the point one. We suggest the use of priors in the form of U (a, b) where b > 1. For example, U (−2, 2) and U (0, 1 + ǫ). In the latter case, ǫ > 0 can be chosen arbitrarily. We choose ǫ such that the expected value of φ having this density is equal to the expected value of φ having a mixed density with mixing probability p as in MU (p). Another alternative is to choose ǫ such that P (1 ≤ φ < 1 + ǫ) = p, where the probability is taken with respect to the U(0,1+ǫ ) distribution. p In this case, ǫ = 1−p . Finally to complete the full model specification, we assume that apriori the parameters φ, µ and σ are independent. Note that prior independence does not necessarily imply posterior independence and certainly not for our version of SV models. The prior we use for µ, the population mean of the log-volatilities, is N (µ0 , σ02 ) where µ0 and σ02 are assumed to be known quantities. We choose the hyper-parameters µ0 and σ0 so as to make the prior belief have a minimal influence on the posterior distribution. For instance, we can set σ02 = 106 to approximate a diffuse prior. For the parameter σ 2 , we use an Inverse-Gamma (IG) distribution, IG(a0 , b0 ), with mean b0 /(a0 − 1) and variance b20 (a0 −1)2 (a0 −2) , respectively. Again to have minimal prior influence we can set a0 = 2 + 10−10 and b0 = 0.1 (also suggested by So and Li [21]). We study the sensitivity of these priors in Section 5. 4. Testing the unit-root hypothesis We use two approaches to test the unit-root hypothesis H0 : φ = 1 against the alternative Ha : |φ| < 1. First one is based on constructing an interval estimate for φ based on its marginal posterior distribution. The criteria are to reject the null hypothesis if the number one is not included in the posterior interval. For all our simulation studies, we have used a 95% equal tail credible interval for φ . The second approach is based on the popular Bayes Factor (BF) criteria, which reject the null hypothesis if log10 (BF ) < 0. It is to be noted that our proposed method of computing the BF is different from that of [21] in the sense that we use an exact SV model to obtain the predictive distribution, instead of approximating the errors of volatility processes (ηt ’s) by a discrete mixture of normals. In our simulation studies (see Section 5) we find that the posterior interval method is easier to implement and maintains good frequentist power as compared with the BF criteria. We found the BF for SV models can be slightly more intensive to compute, as it requires evaluation of the conditional density of the returns given the volatilities which are of dimension as the sample n and hence grow with the sample size. We briefly explain the model fitting techniques that we have used for our computations. 4.1. The Gibbs sampling algorithm for SV models Deriving the joint posterior density of the SVM parameters θ = (µ, φ, σ 2 ) amounts to integrating out the unobserved log-volatilities h, which is difficult to perform both analytically and numerically, as the number of unobserved log-volatilities is equal to the sample size. This makes obtaining the posterior inference about the parameters very difficult. At this point, sampling based techniques provide an alternative approach to obtain posterior distribution of the parameters. One class of such techniques is known as the Markov Chain Monte Carlo (MCMC) methods. MCMC methods are being increasingly used for cases where marginal distribution of the random parameters cannot be obtained either analytically, or by numerical integration. MCMC methods consist of sampling random variates from a Markov Chain, such that its stationary distribution is the posterior distribution of the parameter of interest (see [7] and [26] for an excellent overview of the theoretical properties of MCMC). That means, under some regularity conditions, the realizations of this Markov Chain (after some ‘‘burn-in’’ time) can be thought of as realizations sampled from the desired posterior distribution. We use a Gibbs sampler [5], a widely used MCMC method, to obtain dependent samples from the posterior distribution p(θ |r ). To implement the Gibbs sampler, one obtains the full conditional densities, i.e. the conditional density of a component of the vector of parameters given the other components and the observed data. Specifically, we derive the conditional densities, 194 Z.I. Kalaylıoğlu, S.K. Ghosh / Statistical Methodology 6 (2009) 189–201 f (µ|φ, σ 2 , h, r ), f (σ 2 |µ, φ, h, r ), f (φ|µ, σ 2 , h, r ), and f (ht |µ, φ, σ 2 , h−t , r ) as the full conditional densities of µ, σ 2 , φ , and ht in model (1) respectively. Here h−t is the vector of unobserved logvolatilities excluding the one at time t. The full conditional densities of µ and σ 2 are Normal and Inverse Gamma densities respectively. The full conditional density of ht is obtained as, f (ht |µ, φ, ση2 , h−t , r ) ∝ f (rt |ht )f (ht |ht −1 , µ, ση2 , φ)f (ht +1 |ht , µ, ση2 , φ). (3) In this expression, f (rt |ht ) and f (ht |ht −1 , µ, σ 2 , φ) are the probability density functions of N (0, eht ) and N (µ(1 − φ) + φ ht , σ 2 ) respectively. When we use a flat prior distribution, which is in the form of U (a, b) with b > 1 for the persistence parameter φ , the full conditional density of φ is given by, f (φ|µ, σ 2 , h, r ) ∝ p − 1 − φ2e 1 2σ 2 " φ 2 {(h1 −µ)2 + n P t =2 (ht −1 −µ)2 }−2φ n P (ht −µ)(ht −1 −µ) t =2 # . (4) When we use a mixed prior for φ with a constant mixing probability p, full conditional density of this parameter is given by, # " n Y  2   f (ht |ht −1 , µ, σ ) f (h1 |µ, σh21 )p   t = 2 # f (φ|µ, σ 2 , h, r , b) ∝ " n  Y  2   f (ht |ht −1 , µ, φ, σ ) f (h1 |µ, φ, σh21 )(1 − p)  if φ = 1 if φ < 1 t =2 where σh21 = σ 2 for φ = 1 and it is σ 2 /(1 − φ 2 ) for φ < 1. For the model under the null hypothesis, the full conditional densities are obtained in a similar manner, along with φ = 1. Given an arbitrary choice of starting values for the parameters at the beginning of the algorithm, say, µ(0) , φ (0) and σ 2(0) , Gibbs Sampling algorithm for SV model is given by Initialize µ(0) , φ (0) , and σ 2(0) . For k = 1, 2, . . . , m + M, (k) 1. Draw ht from f (ht |µ(k−1) , φ (k−1) , σ 2(k−1) , h(−kt−1) , r ) for t = 1, . . . , n. 2. Draw µ(k) from f (µ|φ (k−1) , σ 2(k−1) , h(k) , r ). 3. Draw σ 2(k) from f (σ 2 |µ(k) , φ (k−1) , h(k) , r ). 4. Draw φ (k) from f (φ|µ(k) , σ 2(k) , h(k) , r ). Repeating the above sampling steps, we obtain a discrete-time Markov Chain {(µ(k) , σ 2(k) , φ (k) , h ), k = 1, 2, . . .} whose stationary distribution is the joint posterior density of the parameters i.e., the conditional density of (θ , h) given r, under the corresponding model. It follows that the marginal samples θ (k) = (µ(k) , σ 2(k) , φ (k) ) converge in distribution to our target posterior density of (µ, σ 2 , φ) given r. We carry out the Gibbs sampling algorithm as proposed above by means of a software package known as WinBUGS [25]. For our simulation study, we use the unix version of the software called BUGS [8]. We suggest reading [22–24], to get a satisfactory insight to the use of BUGS. One interesting feature of applying our Gibbs sampling scheme via BUGS or WinBUGS for SV models merits a mention. Although the algorithm samples the log-volatilities one at a time (as univariate updates), the software performs this operation outstandingly fast. In fact, to obtain a sample of M = 5000 samples after a burn-in of m = 5000 from a parallel chain of three starting values, it takes only about 65 s. This makes the use of our proposed method much simpler for end-users. The code we have used for our data analysis is available upon request from the first author. (k) Z.I. Kalaylıoğlu, S.K. Ghosh / Statistical Methodology 6 (2009) 189–201 195 4.2. Posterior interval method to test the unit-root hypothesis In frequentist analysis, one way to perform hypothesis testing is to use confidence intervals as acceptance regions for the corresponding testing problems. Similarly, one can employ the posterior interval of the parameter of interest to conduct hypothesis testing in Bayesian analysis. It may be noted that this method is applicable in our context, only if we use those prior densities for φ that includes 1 in their support. For instance, the usual Beta densities (or any other continuous densities with support (0, 1)) as priors for φ are not suitable for this posterior interval method as the posterior probability Pr(φ ≥ 1|r ) = 0 for all such priors. One can obtain a credible interval based on the posterior distribution of φ from the Markov chain that we construct via the Gibbs sampling algorithm. For instance, a 95% equal-tail posterior interval for φ can be arbitrarily approximated by computing the 2.5th and 97.5th percentiles of the chain for φ (k) , k = m + 1, . . . , m + M, for large values of m and M. In Sections 5 and 6 we provide the specific values of m and M that have been used for our simulation studies and data analysis. In order to test the unit-root hypothesis H0 : φ = 1, we may simply use the rule of rejecting the null hypothesis, if φ = 1 is not included in the posterior interval. From our simulation studies, we find that this simple rule actually performs remarkably well in maintaining good frequentist properties, such as high power or equivalently low total (Type I plus Type II) error rates. Besides maintaining good frequentist properties, this rule is very easy to implement by constructing posterior intervals based on MCMC methods. 4.3. Bayes factor to test the unit-root hypothesis Testing the null hypothesis H0 : φ = 1 versus the alternative hypothesis H1 : |φ| < 1, can be thought of as comparing the two competing models. In this sense the ratio of the posterior Pr (H |r ) probabilities of the models, Pr (H0 |r ) , is considered as a suitable test statistic. The Bayes Factor (BF) is 1 defined as the ratio of posterior odds to prior odds. The BF quantifies the amount of information gained from the data in support of one model with respect to that from prior information. In most of the data analyses, prior odds are usually taken to be 1 to indicate prior ignorance. In calculating the Bayes Factor for SV Models, we use the exact distribution of {log u2t }, instead of using an approximation that is obtained by a mixture of several normal densities. This is in sharp contrast to the method proposed by [21] that writes the model in a linear state space form and approximate the exact distribution of log u2t , by a mixture of seven normal distributions. Note that the Bayes Factor is defined as, BF = m0 (r ) m1 (r ) where mi (r ) is the marginal likelihood of the mean corrected returns under the model i, where i = 0, 1 correspond to H0 and H1 , respectively. Denoting the parameter vector of model i by θ i , one can write mi (r ) as mi (r ) = f (r |h)f (h|θ i )f (θ i ) f (h, θ i |r ) which leads to 1 mi (r ) = Eh|r  1 f (r |h) . Using Monte Carlo integration estimate for the denominator in the above expression, we obtain an estimator of the marginal likelihood of the data. That is, 1 m̂i (r ) = 1 M M P l= 1 1 f (r |h(l) ) (5) 196 Z.I. Kalaylıoğlu, S.K. Ghosh / Statistical Methodology 6 (2009) 189–201 Table 1 Proportion of correct decisions based on posterior interval (PI) method and BF method for small sample size (n = 100, 500 MC replications) Prior for φ True φ PI method BF method MU (p = 0.95) 1 0.98 0.95 0.70 0.39 0.60 0.62 0.37 0.32 MU (p = 0.5) 1 0.98 0.95 0.41 0.78 0.94 0.57 0.32 0.31 U (−2, 2) 1 0.98 0.95 0.87 0.19 0.87 0.26 0.89 0.98 U (0, 1) 1 0.98 0.95 0.00 1.00 1.00 0.75 0.24 0.17 can be used to estimate mi (r ). It is well known that the above form of (harmonic mean) Monte Carlo estimation for BF does not satisfy the the CLT for Monte Carlo sampling (see [12]). However, we did not find any problem with convergence. But we found the method to be extremely inefficient computationally, as it requires the evaluation of the high-dimensional densities. After we obtain the BF using above steps, we use the rule of rejecting the null hypothesis if log10 (BF ) < 0 (also suggested by So and Li [21]). It may be noted that other cut off values than 0 can be used (see [12]). 5. Simulation studies We have conducted extensive simulations to explore the frequentist properties of the posterior interval method, and the BF method to test the unit root hypothesis. We present some of the significant findings of our simulation experiments. In order to perform the simulation experiments, we simulate the data from the model (1), by using the same parameter values as in the power study of So and Li [21]. In other words, for each data set throughout our simulation experiments, we fix the true value of the mean, µ = −9 and variance of the shocks to volatility σ 2 = 0.1. Several test values (e.g. φ = 1, 0.98 and 0.95) have been used for φ to test the null hypothesis H0 : φ = 1 vs. Ha : |φ| < 1. We use µ ∼ N (0, 106 ) and N (0, 10) and σ 2 ∼ IG(2 + 10−10 , 0.1) and IG(5, 0.1) as diffuse and informative priors, respectively, for µ and σ 2 . However we did not find any significant difference between the posterior inference based on these priors. So to save space, we report the results based on only the non-informative priors. For φ , we consider a wide class of prior densities, see Tables 2 and 3 for more details. In order to compare the methods, we compute the proportion of correct decisions under different point null alternatives. The proportion of correct decisions is the ratio of number of correct decisions to the total number of Monte Carlo replications. For the test based on the posterior interval, the number of correct decisions is the number of samples for which the value of 1 is covered by the posterior interval of φ when the true value of φ is 1. When the true value of φ is 0.98 or 0.95, the number of correct decisions is the number of posterior intervals that do not include 1. For the test based on the log10 (BF ), the number of correct decision is the number of the samples for which log10 (BF ) ≥ 0 when the true value of φ is 1. On the other hand, it is the number of samples for which log10 (BF ) < 0 when the true value of φ is a point under the alternative hypothesis. In order to determine suitable values of m and M to be used throughout our simulations, we used some initial plots (e.g. trace and autocorrelation plots) of the chains. We also checked the numerical summary values as suggested by Gelman and Rubin [6] and Raftery and Lewis [18] to diagnose the convergence. All these diagnostics were performed using the R version of CODA (see [1] 197 Z.I. Kalaylıoğlu, S.K. Ghosh / Statistical Methodology 6 (2009) 189–201 Table 2 Proportion of correct decisions for moderate to large samples (based 100 MC replications) True φ Prior density n = 500 n = 1000 1 U (0, 1) TN (0.9, 0.052 ) Beta(10, 1) Beta(20, 2) U(0, 1.95) U(−2, 2) MU(p = 0.95) MU with p ∼ Beta(0.5, 0.5) 0.96 0.88 0.86 0.85 0.93 0.93 0.94 0.79 0.90 0.94 0.75 0.84 0.93 0.94 0.98 0.86 0.98 U (0, 1) TN (0.9, 0.052 ) Beta(10, 1) Beta(20, 2) U(0, 1.95) U(−2, 2) MU(p = 0.95) MU with p ∼ Beta(0.5, 0.5) 0.36 0.44 0.60 0.50 0.43 0.47 0.41 0.77 0.64 0.67 0.66 0.80 0.86 0.87 0.77 0.93 0.95 U (0, 1) TN (0.9, 0.052 ) Beta(10, 1) Beta(20, 2) U(0, 1.95) U(−2, 2) MU(p = 0.95) MU with p ∼ Beta(0.5, 0.5) 0.82 0.87 0.89 0.93 0.94 0.94 0.89 0.97 0.98 0.98 0.97 0.98 0.99 0.99 0.98 0.99 Bold numbers are based on proposed PI method, while other numbers are based on So and Li’s BF method. Table 3 Total Error Rates using PI’s for sample sizes n = 500 and 1000 Method Posterior interval (φ ∼ MU (p = 0.95)) Posterior interval (φ ∼ U (01.95)) Posterior interval (φ ∼ U (−22)) BF of So and Li (φ ∼ U (0, 1)) φ = 0.98 n = 500 n = 1000 φ = 0.95 n = 500 n = 1000 0.65 0.25 0.17 0.04 0.64 0.21 0.13 0.08 0.60 0.19 0.13 0.07 0.68 0.46 0.22 0.12 for convergence diagnostics in CODA). Based on these initial plots and diagnostics we decided to use m = 1000 and M = 5000 for all our simulations. In Table 1, we present the proportion of correct decisions made by the posterior interval (PI) method of testing (in column 3), as well as the test based on the log10 (BF ) (in column 4) for testing φ = 1 using our SV model. For this study, we used a small sample of size n = 100, and repeated the procedures 500 times. In this table, we observe that the posterior interval based test is quite sensitive to the choice of priors. For instance, the prior U (−2, 2) performs very well in detecting the presence of unit-root under null compared to other prior and BF. However, it loses power when φ moves below one. It appears that MU priors perform reasonably well in terms of posterior interval method compared to the BF method. It is obvious that using U (0, 1) as a prior for φ makes the posterior interval test incapable of detecting the unit root. When a mixed density is used, the probability of committing a Type I error by the posterior interval test is quite high but not as severe as the case when U(0,1) is used as a prior. In general, the proportion of correct decisions made by the posterior interval test is higher than that of the test based on the BF. One nice feature of the BF test is that it is not as severely affected by the use of U (0, 1) as compared to interval method. 198 Z.I. Kalaylıoğlu, S.K. Ghosh / Statistical Methodology 6 (2009) 189–201 In Table 2, we compare the power of the Bayesian unit root testing criteria of [21] (based on BF) to that of our test that is based on the posterior interval of φ . In this table the first four priors were used by So and Li [21], and the proportion of correct decisions are reproduced from their article that they obtained based on their BF test. Our results have been indicated by bold numbers. We observe that, under H0 , the proportion of correct decisions based on our tests are higher in most cases than that of So and Li’s test based on the BF. This clearly suggests that one can get improved results from posterior interval test by using the prior densities we suggested for φ when applied to SV models. In addition, we observe that under H0 , proportion of correct decisions made by the test criteria of So and Li decreases as the sample size increases in all the cases except for the Truncated Normal (TN) prior. We do not see such a surprising phenomenon with our testing method. As expected, the mixed priors which have a high probability on the point φ = 1, seem to perform better as compared with the ones that have a lower probability on the point φ = 1 in terms of the probability of committing a Type 1 error. However the other priors that do not put high mass at unity a priori also perform reasonably well. Based on our simulation study, we suggest that practitioners should use both types of priors to see the sensitivity on the posterior inference. From Table 2 we also observe that under the alternative, our proposed testing method maintains a reasonably higher proportion of correct decisions as compared with that of [21]’s method. In order to compare the priors for this testing problem, we present the total (Type I plus Type II) error rates in Table 3. It appears that the U (−2, 2) performs reasonably well as compared with other priors in maintaining low total error rates. So in practice, we would recommend this prior as a default choice for testing unit root for SV models. Table 2 demonstrates the superiority of our test over the existing unit root test criteria for SV models. From Table 2, we see that, in general, So and Li’s test yields high power at the cost of high Type I Error. On the other hand, our proposed testing method maintains reasonably low total error rates as observed in Table 3. The PI methods performs much better in terms of detecting the absence of unit root, when the true value of φ is far away from 1 and when sample size increases. So we did not report such findings here for space limitations. For instance in Table 3 we observe that the total error rates drops to about only 19% even when φ = 0.98 as compared to that of BF method yielding a total error rate of 46%. 6. Application to exchange rate data The data set for our application consist of the following four exchange rates: Pound/Dollar, Deutschmark(DM)/Dollar, Yen/Dollar, and Swiss-Frank(SF)/Dollar. The data are closed exchange rates from October 1st, 1981 to June 28th, 1985 on the weekdays. The sample size for four series is n = 946. Let us denote the exchange rate for the i-th series on t-th day by pit . Then, for each series i, the mean corrected return data is calculated as rit = 1 log pit − n 1X 1 log pit t = 1, . . . , n n i=1 where 1 is the difference operator, i.e., 1 log pit = log pit − log pi,t −1 . Here, i = 1, 2, 3, or 4 corresponds to Pound/Dollar, Deutschmark/Dollar, Yen/Dollar, or Swiss-Frank/Dollar, respectively. The time series plots of the returns are displayed in Fig. 1. Visually, it is hard to tell from the plots whether the affect of the random shocks to volatilities are persistent or not. So we use our testing method to see if φ = 1 assuming that the series of returns can be approximately described by a SV model. For each of the four series, we fitted an ARMA (1, 1) model to the data in the form of log rit2 and checked the autocorrelations of the residual from these fits by using PROC ARIMA in SAS. The test result is that residuals are white noise implying that ARMA(1, 1) is an adequate fit to log rit2 for each series, i = 1, . . . , 4 which in turn implies that a SV model is a reasonable fit to the mean corrected exchange rate returns. Based on our finding from simulation studies, we use only the posterior interval method to test the null hypotheses for these four series on exchange rates. We decided to use same set of flat prior distributions for µ and σ 2 that were used in our simulations. For the parameter φ we use MU (0.95), U (0, 1.95) and U (−2, 2) priors. Z.I. Kalaylıoğlu, S.K. Ghosh / Statistical Methodology 6 (2009) 189–201 199 Fig. 1. Plot of the mean corrected returns rit versus time t. To implement the Gibbs sampling, we used three dispersed initial values for all the parameters. We execute the Gibbs Sampling (in WinBUGS) to obtain 10,000 samples from each of the three parallel chains resulting into 30,000 samples. Using plots and numerical summary measures for convergence diagnostics (available in CODA) we decided to throw out first 5000 samples from each of the three parallel chains. Thus we used 15,000 final samples to obtain summary values of the posterior distribution of the parameters. The results based on different priors for φ are presented in Table 4. When we use a mixed prior with p = 0.95, in addition to the posterior inference given in the table, the posterior median of φ turns out to be one for each series. It appears that, for each series, there is more than 50% probability that φ is at least 1 based on the data. The posterior intervals cover the value of one implying that null hypothesis can not be rejected for any of the four series. However, when we use φ ∼ U (0, 1.95) and φ ∼ U (−2, 2), the test based on the posterior interval does not find sufficient evidence in favor of the unit root for all series, but for the Yen/Dollar series. This makes it harder to conclude, but for the Yen/Dollar series. However given the superior performance of the test based on U (−2, 2) prior from our simulation study, for this data we would conclude that only for the Yen/Dollar series a unit-root hypothesis can be accepted and for the other three series there is not enough evidence to believe the presence of a unit root. 7. Conclusions In this article, we have developed Bayesian unit root tests for Stochastic Volatility (SV) Models, where the log-squared volatilities are assumed to have a first order autoregressive model. We have introduced a mixed prior density that puts a nonzero mass on the null hypothesis value of the autoregressive parameter. We also suggested to use continuous densities that include the nonstationary values of φ as prior densities. By using such densities on φ , one can use the posterior 200 Z.I. Kalaylıoğlu, S.K. Ghosh / Statistical Methodology 6 (2009) 189–201 Table 4 Posterior Inference for φ for three different prior distributions Exchange rate Estimated posterior mean Standard error of the estimate Estimated 95% posterior interval 0.0013 0.0018 0.0016 0.0033 [0.9528, 1.0] [0.9363, 1.0] [0.9437, 1.0] [0.8925, 1.0] 0.0007 0.0013 0.0007 0.0017 [0.9462, 0.9971] [0.9167, 0.9978] [0.9557, 1.0000] [0.8739, 0.9869] 0.0006 0.0009 0.0007 0.0014 [0.9415, 0.9972] [0.9206, 0.9978] [0.9430, 1.0000] [0.8670, 0.9830] MU (p = 0.95) Pound/$ DM/$ Yen/$ SF/$ 0.9920 0.9881 0.9898 0.9791 Pound/$ DM/$ Yen/$ SF/$ 0.9769 0.9642 0.9868 0.9405 Pound/$ DM/$ Yen/$ SF/$ 0.9761 0.9679 0.9796 0.9343 U (0, 1.95) U (−2, 2) interval of φ to test for a unit root in a SV model. We showed that, applying a test based on the Bayes Factor in SV models is not practical as computing the Bayes Factor is a computationally intensive process due to the high-dimension of the missing observations (in this case, volatilities). On the other hand, calculating the posterior interval is very simple, and can be performed easily in WinBUGS. Using a Monte Carlo study, we have shown that the posterior interval unit root test in SV Models is reliable. In addition to the Bayesian methods presented in this article, we also compared the total error rates of posterior interval tests to that of the commonly used frequentist tests (not shown here, however see [11]). The results from several frequentist procedures will be reported in a separated article elsewhere. From our studies, we found that our Bayesian methods based on posterior interval method perform relatively better in terms of having low total error rates as compared with the frequentist counterparts. In conclusion we would recommend users to use the Bayesian unit root test, based on the posterior interval of φ as it is a simple, practical, and reliable solution to the problem of seeking evidence for a unit root in SV models. 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