Comput Econ (2011) 37:237–248 DOI 10.1007/s10614-011-9252-4
An Efficient Stochastic Simulation Algorithm for Bayesian Unit Root Testing in Stochastic Volatility Models Yong Li · Zhongxin Ni · Jie Zhang
Accepted: 3 January 2011 / Published online: 19 January 2011 © Springer Science+Business Media, LLC. 2011
Abstract In financial times series analysis, unit root test is one of the most important research issues. This paper is aimed to propose a new simple and efficient stochastic simulation algorithm for computing Bayes factor to detect the unit root of stochastic volatility models. The proposed algorithm is based on a classical thermodynamic integration technique named path sampling. Simulation studies show that the test procedure is efficient under moderate sample size. In the end, the performance of the proposed approach is investigated with a Monte Carlo simulation study and illustrated with a time series of S&P500 return data. Keywords Financial times series · Stochastic volatility models · Unit root testing · Bayes factor · Path sampling 1 Introduction In theoretic and empirical finance literature, the stochastic volatility (SV) models have been regarded as an efficient alternative for ARCH-type models to describe financial times series with time-varing volatility. Modern asset pricing theories have shown that the persistence in risk premium can be reflected by a unit root in volatility,
Y. Li School of Business, Sun Yat-Sen University, Guangzhou 510275, China Z. Ni (B) School of Economics, Shanghai University, Shanghai 200444, China e-mail:
[email protected] J. Zhang Institute of China’s Economic Reform Development, Renmin University of China, Beijing 100872, China
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see Chou (1988) and So and Li (1999). Hence, as to stochastic volatility models, if there is a unit root in the log-volatility process, it means that shocks to volatility do not decay rapidly. In other words, the effect of past shocks on current volatility will remain persistent for long periods. A change in risk premium today will have a significant effect on security prices. High persistency of shocks to volatility will increase the fluctuation in the volatility which causes the market to plunge, see Chou (1988) and Bollerslev and Engle (1993) who studied the volatility and its relationship with market fluctuations. Thus, it is very meaningful to develop some efficient methods to detect the unit root in stochastic volatility models, see So and Li (1999). In the literature, some asymptotically significant statistics such as the well-known DF and ADF test statistics (Dickey and Fuller 1979) may be used for testing the unit root for financial times series. When using these frequent unit root approaches for SV models, however, the unit root testing are generally tedious, even impossible. First, SV models involved latent volatilities such that their likelihood functions are very complicated and intractable. Hence the parameter estimates required by classical test statistics are extremely difficult to obtain. Second, as pointed by So and Li (1999), if the mean intercept or exogenous variables were incorporated into the SV models, the classical DF and ADF test statistics is no longer applicable. Finally, the value of the significant test statistics is only a measure of evidence against the null hypothesis, not a means of supporting the null hypothesis. For example, a test result of failing to reject the unit root hypothesis does not give any supportive evidence for existing a unit root in the time series. Fortunately, for SV models, Bayesian unit root inference does not have the aforementioned problems. Hence, it is an appealing substitute for the classical significant test statistics. Under Bayesian framework, the unit root test problem is formulated as a Bayesian model comparison problem in which two nonnested models, constructed respectively under the null and alternative hypothesis, are compared. So and Li (1999) firstly developed a Bayesian unit root testing approach based on the well-known model comparison statistic, i.e, Bayes factor (Kass and Raftery 1995) for SV models. This Bayes factor was approximately computed using the method developed by Chib (1995). However, this approach for calculating Bayes factor requires the marginal likelihood, a marginalization over the unknown parameters and latent volatility under each model. In SV models, the number of unknown parameters and latent volatilities is very large (exceeding the number of observations), hence, obviously, the computational burden of the marginal likelihood is formidable. In addition, So and Li (1999) did not consider the exogenous variables in SV models which were very common in practice. For SV models with exogenous variables, it is also very tedious to compute the Bayes factor for unit root testing. Instead of using the method of Chib (1995), we propose a new simple numerical procedure for computing Bayes factor to detect unit root. This procedure is free of complex marginalization and developed on the basis of an efficient classical thermodynamic integration technique named path sampling (Gelman and Meng 1998) which is a dependable tool for computing the ratio of normalizing constants of probability models. We will show that the proposed Bayesian procedure for unit root test is very simple by constructing an intermediate model to link the two nonnested models to be compared. Its main computational effort lies on drawing random observations from the posterior distributions. Therefore the
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implementation is relatively easier compared with the approach of Chib (1995). The efficiency of the proposed procedure is illustrated with simulation studies and a real data example. The remainder of this paper is organized as follows. Section 2 gives a simple description of SV models with exogenous variables. The Bayesian approach for analyzing SV models is also presented in this section. The procedure for computing Bayes factor based on path sampling for SV models is established in Sect. 3. Section 4 reports the simulation study results of the established model comparison approach in Sect. 3. The S&P500 return data is investigated in Sect. 5. Finally, Sect. 6 summarizes this paper. 2 Bayesian Analysis for Stochastic Volatility Models In this section, we give a simple description about Bayesian analysis for SV models with the exogenous variables. Given a time series of return yt , t = 1, 2, . . . , n, the basic formulation of SV models with the exogenous variables can be generally expressed as follows: yt = x tT β + t , t = exp(h t /2)u t , u t ∼ N (0, 1), t = 1, 2, . . . , n
(1)
where {x t } is the exogenous variable, {t } is random error, {h t } is the unobserved state, and {u t } is a sequence of independently Gaussian white noise. The unknown states {h t } are assumed to follow a Markov process and are described as: h t = τ + φ(h t−1 − τ ) + σ ηt , ηt ∼ N (0, 1), t = 1, 2, . . . , n
(2)
where state h 0 ∼ N (τ, σ 2 ), ηt is also independently Gaussian white noise and is uncorrelated with u t for all t. In this paper, our main concern is to provide an efficient unit-root testing procedure, i.e, to test the hypothesis H0 : φ = 1 vs H1 : |φ| < 1. Since positive estimates of φ were observed in all the stochastic volatility literature, we now only consider the hypothesis H0 : φ = 1, H1 : 0 < φ < 1. For the SV models, Bayesian analysis is based on posterior distribution of unobservable components, that is, parameters and latent states given the data when the prior distribution of parameter are specified. Let p(θ) be the joint prior distribution of θ = (β, τ, φ, σ 2 ) in formula (1) and (2). Denote the observable vector y = (y0 , y1 , . . . , yn ) and latent state vector h = (h 0 , h 1 , . . . , h n ). Thus, the complete likelihood of (θ , h, y) can be expressed as follows: p(θ , h, y) = p(θ , h) p( y|θ, h) = p(θ) p(h 0 |τ, σ 2 )
n t=1
p(h t |h t−1 , θ )
n
p(yt |θ, h)
t=1
(3) By Bayes’s theorem, the posterior distribution of (θ, h) given by the data y is proportional to the complete likelihood, that is,
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p(θ , h, y) ∝ p(θ, h, y) p( y) n n 2 p(h t |h t−1 , θ ) p(yt |θ, h) = p(θ ) p(h 0 |τ, σ )
p(θ, h| y) =
t=1
(4)
t=1
As to the posterior distribution p(θ , h| y), the Bayesian posterior computation can be performed via Markov chain Monte Carlo (MCMC) techniques employing the idea of data-augmentation strategy (Tanner and Wong 1987). Bayesian estimates of θ and the latent volatilities h can be obtained easily via the corresponding sampling means of the generated observations. Specifically, let {θ ( j) , h( j) , j = 1, 2, . . . , J } be the efficient random observations generated from the joint posterior distribution p(θ , h| y) after discarding some length of burn-in samples. Then the joint Bayesian estimates of θ , h can be obtained, respectively, as follows: J 1 ( j) θ , θˆ = J j=1
V ar (θ | y) =
J 1 ( j) hˆ = h , J j=1
1 ( j) ˆ ( j) ˆ T (θ − θ )(θ − θ ) , J −1 J
j=1
V ar (h| y) =
1 ( j) ˆ ( j) ˆ T (h − h)(h − h) . J −1 J
j=1
3 Stochastic Simulation Algorithm for Bayesian Unit Root Testing Statistic Under Bayesian framework, hypothesis testing can be performed via Bayes factor (Kass and Raftery 1995) which is a well-known statistic for the purpose of model comparison. Consider the hypothesis H0 : φ = 1 vs H1 : 0 < φ < 1. Let M0 and M1 be the model formulated under the null hypothesis and under the alternative hypothesis, respectively. Testing the null hypothesis versus alternative hypothesis is equivalent to a model comparison between model M0 and model M1 . According to Kass and Raftery (1995), Bayes factor can be defined as follows: B10 =
p( y|M1 ) , p( y|M0 )
(5)
where p( y|Mk ) is generally called marginal likelihood of model Mk , k = 0, 1, and can be obtained by integrating over the corresponding parameter space, that is, p( y|θ k , Mk ) p(θ k |Mk )dθ k , θ k ∈ k , k = 0, 1. (6) p( y|Mk ) = It is well-known that the computation of Bayes factor is non-trivial (see Chib 1995; Chib and Jeliazkov 2001, among others). Path sampling is a classical thermodynamic
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integration technique proposed by Gelman and Meng (1998) which is applied to compute the normalization constant. It has been shown that path sampling algorithm enjoys the following advantages (Gelman and Meng 1998; Lee 2007): (i) it is a generalization of the importance sampling and bridge sampling (Meng 1996), hence more accurate results can be obtained; (ii) instead of computing the ratio, it computes the logarithm scale of the ratio, hence the implementation is generally more stable; and (iii) its main computational effort is to draw random observations from the posterior distributions, hence its implementation is simple. In what follows, we first show that the Bayes factor for unit root testing is a special normalization constant, and then path sampling can be employed to compute the Bayes factor. In addition, our path sampling algorithm does not involve the marginal likelihood functions, hence can be simply implemented for Bayesian unit root testing problem. However, we should notice that although path sampling is very efficient, it can not be directly used to compute out Bayes factor for unit root testing, because these two competitive models are defined over different dimensional parameter space (Chen et al. 2000). To overcome this difficulty, we present two theorems as follows. Theorem 1 Let θ 0 = (β, τ, σ 2 )T be the parameter vector under M0 , θ 1 = (β, τ, φ, σ 2 )T be the parameter vector under M1 , k be the corresponding parameter space of θ k , k = 0, 1, and h be the support space of the latent state h. Then there must exist a class of probability densities, q(θ 1 , h|b), such that
q(θ 1 , h|1) = p( y|M1 ),
1 ∪h
q(θ 1 , h|0) = p( y|M0 ). 1 ∪h
Proof First, we define two probability functions as follows: q0 (θ 1 , h) = p( y|h, θ 0 , M0 ) p(h|θ 0 , M0 ) p(θ 0 |M0 )w(φ|θ 0 ) = p(θ 0 , h, y|M0 )w(φ|θ 0 ), q1 (θ 1 , h) = p( y|h, θ 1 , M1 ) p(h|θ 1 , M1 ) p(θ 1 |M1 ) = p(θ 1 , h, y|M1 ), where w(φ|θ 0 ) is an arbitrary completely known appropriate probability density function. Then we can easily get that
q0 (θ 1 , h)dhdθ 1 = p( y|M0 ), 1 ∪h
q1 (θ 1 , h)dhdθ 1 = p( y|M1 ). 1 ∪h
Consider a class of probability densities, q(θ 1 , h|b), such that q(θ 1 , h|1) = q1 (θ 1 , h) and q(θ 1 , h|0) = q0 (θ 1 , h), then q(θ 1 , h|b) is one probability function satisfying the requirements. Hence the theorem is proved. Theorem 2 For SV models, there exists a class of probability densities, p(θ 1 , h, y|b), defined on the support space 1 ∪ h , with a continuous path parameter b ∈ [0, 1], such that
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R = log B10
p( y|M1 ) = log = p( y|M0 )
1 E θ1 ,h [U(θ 1 , h, y, b)]db.
(7)
0
where E θ1 denotes the expectation about the probability distribution p(θ, h| y, b), with p(θ , h| y, b) =
p(θ 1 , h, y|b) , p( y|b)
p( y|b) =
p(θ 1 , h, y|b)dhdθ 1 1 ∪h
d log p(θ 1 , h, y|b) U(θ, h, y, b) = db Proof In practice, p(φ|θ , M1 ) is an appropriate prior distribution specified for parameter φ under Model M1 . According to Theorem 1, we can construct the following two probability density functions: p(θ 1 , h, y|0) = p(θ 0 , h, y|M0 ) p(φ|θ 0 , M1 ) = p(h, y|M0 ) p(θ 0 |M0 ) p(φ|θ 0 , M1 ), p(θ 1 , h, y|1) = p(θ 1 , h, y|M1 ) = p(h, y|M1 ) p(θ 1 |M1 ) = p(h, y|M1 ) p(θ 0 |M1 ) p(φ|θ 0 , M1 ), such that
p(θ 1 , h, y|1)) = p( y|M1 ),
1 ∪h
p(θ 1 , h, y|0)) = p( y|M0 ). 1 ∪h
According to Gelman and Meng (1998), we now consider a class of probability density functions, p(θ 1 , h, y|b), on the support space 1 ∪ h with a continuous path parameter b ∈ [0, 1]. We can get p(θ, h| y, b) =
1 p(θ , h, y|b), z(b)
where z(b) = p( y|b) such that z(1) = p( y|1) = p( y|M1 ), z(0) = p( y|0) = p( y|M0 ). Taking logarithm on z(b), we can get: ∂logz(b) = ∂b
1 ∪h
∂ 1 ∂ p(θ, h, y|b)dhdθ 1 = E θ,h log p(θ , h, y|b) z(b) ∂b ∂b
where the expectation E θ,h is taken with respect to the distribution p(θ , h| y, b). Let U(θ , h, y, b) =
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then z(1) = R = log z(0)
1 E h,θ [U(θ, h, y, b)]db. 0
Thus, the theorem is proved. Note that formula (7) generally does not enjoy a closed-form solution, some numerical integration technique have to be used for its estimated value. To approximate R, we follow the idea of Gelman and Meng (1998) to numerically evaluate the integral S is an equipartition of the interval [0, 1] over b. To be specific, suppose that {b(s) }s=0 such that b0 = 0 < b(1) < b(2) < · · · < b(S) < b(S+1) = 1, then we can estimate R by = 1 R (b(s+1) − b(s) )(U (s+1) + U (s) ), 2 S
(8)
s=0
where U (s) = J −1
J
U(θ ( j) , h( j) , y, b(s) ),
(9)
j=1
here {θ ( j) , j = 1, 2, . . . , J } are efficient random draws simulated from p(θ | y, b(s) ) after discarding some burn-in samples. In practice, the prior distributions of the common parameter vector θ 0 under the two competitive models are often specified to be the same, i.e., p(θ 0 |M0 ) = p(θ 0 |M1 ). Thus it can be simply derived d log p(θ 1 , h, y|b) db d d log[ p(h, y|θ 1 , b) p(θ 1 |M1 )] = log p(h, y|θ 1 , b) = db db
U(θ, h, y, b) =
which means that U(·) is independent with the prior distribution of the parameters. Thus, the continuous parameter b links the likelihood functions of these two competitive models by an intermediate model. Without loss of generality, a simple intermediate model for the test of unit root hypothesis for SV models can be chosen as: Mb∗ : yt = x tT β + t , t = exp(h t /2)u t , h t = τ + (1 − b + bφ)(h t−1 − τ ) + σ ηt , ηt ∼ N (0, 1), t = 1, 2, . . . , n (10)
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It is obvious that M0∗ = M0 , M1∗ = M1 . Further, a simple algebra shows that U(θ, h, y, b) = σ
−2
n {(φ − 1)(h t−1 − τ )[h t − τ − (1 − b + bφ)]}.
(11)
t=1
We should point out that all the Bayesian numerical computation procedures in this paper are implemented using a free and reliable software package named WinBUGS (Spiegelhalter et al. 2003). The latest version of BUGS is WinBUGS 1.4 which is developed by the medical Research Council(MRC) Biostatistics Unit (Cambridge, UK) and the department of Epidemiology and Public Health of the Imperial College School of Medicine at St Mary’s Hospital (London). Meyer and Yu (2000) illustrated the easy use for stochastic volatility models. Unfortunately, the WinBUGS language is relatively limited and hard to use for effective simulations studies involving repeated calls for WinBUGS for our test purpose. R2WinBUGS is a R package that calls WinBUGS 1.4 and exports the results into R (Sturtz et al. 2005). We use this package to implement our developed Bayesian approach. The Bayes factor based on path sampling can be conveniently obtained without too much extra efforts. Thus, our developed Bayesian model comparison procedure also avoids lengthy codes of MCMC simulation algorithms using complicated programming language such as C or C++. For more details on R2WinBUGS, please refer to Sturtz et al. (2005). For interested readers, the programming code using R2WinBUGS under R and WinBUGS can be obtained by sending a request to the corresponding author. 4 Simulation Study This section investigates the effectiveness of the Bayesian unit root testing procedure with a MCMC simulation approach. We consider three true values of φ, 1, 0.98, 0.95, which respectively represent the nonstationary case, a nearly nonstationary case and a stationary case. The explanatory variables is generated from standard uniform distribution, and the parameters are specified as follows: β = 0.5, τ = 0.0, σ 2 = 0.1, t = 1, 2, . . . , n Time series of different lengths, n = 300, 500, 1000, were considered for each φ. The whole simulation study is based on 100 replications. For Bayesian analysis of SV models, the appropriate prior distribution requires to be specified. According to Kim et al. (1998) and Meyer and Yu (2000), we may specify the prior distributions as follows: τ ∼ N (a0 , b0 ), φ0 ∼ N (c0 , d0 ), φ ∼ Beta( p0 , q0 ),
1 ∼ Gamma(m 0 , n 0 ) σ2
To demonstrate the efficiency of the unit root testing method, we record the proportion of correct decisions for various φ and n over the 100 replications. The decisions are based on whether the Bayes factor is greater than 1. The hyper-parameters may be
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Table 1 The average of the posterior mean and standard error of φ and the value of test statistic under 100 replications φ=1
φ = 0.98
φ = 0.95
n
Statistic
500
φˆ
0.9910
0.9778
0.9493
0.0054
0.0106
0.0164
750
ˆ S E(φ) φˆ
0.9942
0.9767
0.9509
0.0036
0.0091
0.0131
1000
ˆ S E(φ) φˆ
0.9956
0.9791
0.9502
ˆ S E(φ)
0.0027
0.0074
0.0114
Table 2 The proportions of correct decisions among 100 replications under different sample size φ 1.00 0.98 0.95
Statistics
n = 500
n = 750
n = 1000
BF
0.89
0.95
0.95
DIC
0.44
0.50
0.50
BF
0.57
0.86
0.96
DIC
0.76
0.94
0.91
BF
1.00
1.00
1.00
DIC
0.91
1.00
1.00
specified as follows: a0 and c0 are fixed at 0 while other parameters are specified as b0 = 100, d0 = 100, p0 = 20, q0 = 1, m 0 = 0.001, n 0 = 0.001. This type of prior setting may be regarded as a subjective prior. We point out that they are used here only for the purpose of illustration, and we would not routinely recommend this method to obtain some hyper-parameter’s values for Bayesian analysis. In the replications, we simulated 15000 random samples of which 10000 are discarded as burn-in samples. The burn-in cutoff point can be determined by plotting a convergence graph from three different initial values or using CODA (Convergence Diagnostic and Output Analysis Software for Gibbs sampling output) under R software in one test run, see Best et al. (1995). Here, after convergence diagnostics, we take 20 grids in [0, 1] and the remaining 5000 effective random samples were used to obtain the Bayesian results. The simulation results are reported in Tables 1 and 2. From these tables, we can easily draw the following conclusions. First, the estimates of φ are always close to the true value and the SEs are always small, suggesting MCMC provides reliable estimates on φ. Second, the behavior of estimates improves (smaller Bias and SE) when the sample size increases. Third, the numbers of correct decisions increased with the sample size for all three cases. Fourth, Berg et al. (2004) suggested deviation information criterion(DIC) to replace Bayes factor based on marginal likelihood method for model comparison. We compared DIC with our proposed approach for testing the unit root. We reported the simulation results in Table 2. We can find that these results using DIC are no way satisfactory because for the nonstationary case (φ = 1.0), the number of correct decisions among 100 replications is not in acceptable range, for example, under n = 1000, the number is only about 50. However, our proposed approach can achieve good finite sample behaviors. At last, in summary,
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under moderate sample size, say n ≥ 1000, we can say that the proposed test statistic obtains good test power more than 0.9, which shows that the proposed approach is efficient for detecting unit root. 5 An Empirical Study For illustration, we now apply the proposed method to the monthly log-return data of S&P 500 index from January 1962 to December 1999, which was freely obtained from Tsai (2002). This data set is composed of 456 observations which can be fitted using the following SV model yt = μ + t , t = exp(h t /2)u t , u t ∼ N (0, 1) h t = τ + φ(h t−1 − τ ) + σ ηt , ηt ∼ N (0, 1), t = 1, 2, . . . , n. First, we need to specify hyper-parameter values in the proper conjugate prior distributions. Since this data set has been analyzed by Tsai (2002), the results reported there can be regarded as prior information and then incorporated into our prior distribution. According to Tsai (2002) and Kim et al. (1998), we may specify the prior distributions as follows: μ ∼ N (0.66, 100), τ ∼ N (0.0, 100), φ ∼ Beta(1.0, 1.0),
1 ∼ χ52 σ2
This type of prior specification may be regarded as one kind of vague prior distributions. To detect whether there is a unit root, we apply the developed numerical procedure for computing the Bayes factor to compare the null model M0 and the alternative model M1 . Here, after convergence diagnostics, we take 10 grids in [0, 1] and J = 2000 (8000 of total random samples are burn-in samples). According to formula (8), (9), (10) and (11) in Sect. 3, we can easily compute out the estimated value of Bayes factor, that is, log B10 = 6.24. Thus, according to the criterion given in Kass and Raftery (1995), the Bayes factor showed strong evidence to reject the unit root hypothesis. To give some idea about the sensitivity of Bayes factor with respect to the prior inputs, following the suggestion of Kass and Raftery (1995), we can perturb the hyperparameter values of the prior inputs. As to the interest parameter φ, we consider another two Beta distributions, that is, Beta(10,1) and Beta(20,1). Then the data are reanalyzed
Table 3 Empirical results under different priors distribution Prior
μ
τ
φ
σ2
log B10
Beta(1,1)
0.843(0.176)
2.669(0.144)
0.819(0.080)
0.080(0.020)
6.240
Beta(10,1)
0.841(0.177)
2.685(0.183)
0.879(0.044)
0.081(0.023)
6.556
Beta(20,1)
0.829(0.172)
2.698(0.219)
0.899(0.041)
0.084(0.023)
6.288
The numbers in parentheses are the corresponding standard errors
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via the proposed procedures. The estimation of the parameters and the log-Bayes factor are reported in Table 3. Although there is some difference in different prior inputs, the estimated logarithm of Bayes factors all support the same conclusion—choose model M1 . From the results of parameter estimation, obviously, we can obtain the same findings. Thus, to some extent, it seems that there is no unit root in the SV models fitting this data set.
6 Conclusion and Discussion Unit root testing is one of the important issues in the analysis of SV models. The main contribution of this paper is to propose an efficient numerical integration procedure based on powerful path sampling to compute Bayes factor for unit root testing in SV models. The efficacy of the developed Bayesian test approach is illustrated with a simulation study and a real data example. In contrast to the approach used in So and Li (1999), our proposed approach does not require a complex marginalization over the parameters and latent volatilities. Its main computational effort is only to draw random observations from the posterior distributions. Thus, for practical statisticians or econometricians, the advantage of our proposed method is its easy implementation. However, it is still ambitious to say that path sampling is better than other methods such as the marginal likelihood method developed Chib and Jeliazkov (2001). To answer this question, further investigations into different methods for estimating Bayes factor are necessary, which can be regarded as a future research topic. Acknowledgements The first author is supported by the Chinese National Science fund (70901077) and the Chinese Education Ministry Social Science fund (09YJC70266). The second author is partially supported by the National Science Fund of China (71001061), Innovation Program of Shanghai Municipal Education Commission (10YZ24) and Shanghai Pujiang Program. We would like to thank the editor and an anonymous referee for numerous constructive comments that improved the quality of the paper.
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