BAYESIAN ANALYSIS OF A MARKOV SWITCHING
STOCHASTIC VOLATILITY MODEL
Mai Shibata* and Toshiaki Watanabe**
This article analyzes a Markov switching stochastic volatility (MSSV) model to accommodate the shift in the mean of log-volatility. Since it is difficult to estimate the parameters in this model based on the maximum likelihood method, a Bayesian Markov-chain Monte Carlo (MCMC) approach is adopted. A particle filter for the MSSV model, which is used for model comparison and diagnostics, is constructed. The estimation result, based on weekly returns of the TOPIX, confirms the finding by previous researchers that the estimate of the persistence parameter drops and the estimate of the error variance rises in the volatility equation of the MSSV model com-pared to those of the standard SV model. The model comparison provides evidence that the MSSV model is favored over the standard SV model. It is also found that the MSSV model passes the diagnostic tests based on the statistics obtained from the particle filter while the SV model does not.
Key words and phrases: Marginal likelihood, Markov-chain Monte Carlo, Markov switching, particle filter, stochastic volatility, TOPIX.
1. Introduction
A stochastic volatility (SV) model specifies the log-volatility as a linear au-toregressive process to capture the well-known phenomenon in financial markets of a high persistence in volatility. As pointed out by Lamoureux and Lastrapes (1990) and Hamilton and Susmel (1994), the high persistence in volatility may, however, be caused by a structural change in volatility. Some researchers such as So et al. (1998) and Kalimipalli and Susmel (2004) combined the SV model with the Markov switching (MS) model proposed by Hamilton (1989) to accommodate the shift in the mean of log-volatility. They applied the resulting model, which is called a Markov switching stochastic volatility (MSSV) model, to the S&P 500 and 3-month US T-bill data respectively, and document that the estimate of the persistence in volatility drops significantly in the MSSV model. This result indicates that the high persistence in volatility may partly be caused by a switch between the high- and low-volatility states.
It is difficult to estimate the parameters in the likelihood of the MSSV model as well as the SV model using the maximum likelihood method. Pre-vious researchers resort to a Bayesian method using Markov-chain Monte Carlo (MCMC) techniques. Bayesian MCMC methods for estimating the SV model (see Shephard and Pitt (1997) and Kim et al. (1998)) and the MS model (see
Received September 30, 2004. Revised January 25, 2005. Accepted June 23, 2005.
*Research Associate, Institute for Monetary and Economic Studies, Bank of Japan, 2-1-1 Nihonbashi-Hongokucho, Chuo-ku, Tokyo 103-8660, Japan.
**Senior Fellow, Institute for Monetary and Economic Studies, Bank of Japan, 2-1-1 Nihonbashi-Hongokucho, Chuo-ku, Tokyo 103-8660, Japan.
Chib (1996) and Kim and Nelson (1998, 1999)) are well developed, and hence the application of this method to the MSSV model, which is a synthesis of the SV model and the MS model, is straightforward.
The most important point in working with the MSSV model is to analyze whether a switch occurs or not, which is equivalent to comparing the MSSV model with the SV model. Classical test statistics such as the likelihood ratio statistics are not directly applicable to this analysis because the transition probabilities are not identified under the null hypothesis of no switch (see Hansen (1992) and Garcia (1998)). In a Bayesian framework, model comparisons are based on the posterior odds ratio, which does not cause any problem in analyzing whether a switch occurs. It is, however, not straightforward to evaluate the marginal likelihood, which is required for evaluating the posterior odds ratio, of the MSSV model. The marginal likelihood is decomposed into the three components: the likelihood, the prior density and the poster density. The prior density can easily be evaluated. Chib (1995) and Chib and Jeliazkov (2001, 2005) propose methods for evaluating the posterior density, which are applicable to the SV and MSSV models.
The problem in the SV and MSSV models is that the likelihood cannot be evaluated analytically. Methods for evaluating the likelihood of the SV model based on simulation are available. Danielsson (1994) and Danielsson and Richard (1993) propose a method using AGIS (accelerated Gaussian importance sampler), and Kim et al. (1998) propose a method using a particle filter. This article develops a particle filter for evaluating the likelihood of the MSSV model. This filter yields not only the likelihood but also the variable used for diagnostics of the MSSV model.
Using the Bayesian MCMC method, we fit the MSSV model to the weekly returns of the Tokyo stock price index (TOPIX). Our estimation result confirms the finding by So et al. (1998) and Kalimipalli and Susmel (2004) that the es-timate of persistence in volatility drops significantly in the MSSV model. We also evaluate the marginal likelihood and the diagnostic statistics of the MSSV model using our method and those of the SV model using the method by Kim et al. (1998). The model comparison based on the marginal likelihood provides evidence that the MSSV model is favored over the SV model, indicating that a shift in the mean of log-volatility occurs. Moreover, the MSSV model passes all diagnostic tests while the SV model does not.
The rest of this article is organized as follows. Sections 2 and 3 review the MSSV model and its Bayesian MCMC estimation respectively. Section 4 develops a method for model comparison and diagnostics of the MSSV model. Section 5 applies the MSSV model to the TOPIX weekly return series and summaries the results. Conclusions are given in Section 6.
2. The MSSV model
In this article, we analyze a simple MSSV model where there are only two states, high- and low-volatility states, and it is only the mean of log-volatility
that may shift depending on the state. It is, however, straightforward to extend to more general models where the number of states is more than two and the other parameters may also shift. So et al. (1998) analyze a model where there are three states: high-, medium- and low-volatility states.
Suppose that we have a financial return series (y1, . . . , yT), from which the
mean and the autocorrelation are subtracted, and define St as a latent variable
that takes one in the high-volatility state and zero in the low-volatility state. Then, the MSSV model analyzed in this article is represented by
yt= exp(ht/2)t, t∼ i.i.d.N(0, 1), (2.1) ht= µst+ φ(ht−1− µst−1) + ηt, ηt∼ i.i.d.N(0, σ2η), (2.2) µst = µ0+ µ1St, µ1 > 0, (2.3)
where exp(ht/2) is the volatility of ytand hence htis the log of squared volatility.
µst is the mean of ht, which may shift depending on the state. The reason to
assume that µ1 > 0 is that the mean of ht is greater in the high-volatility state
(St = 1) than that in the low-volatility state (St = 0). This model collapses to
the standard SV model if µ1= 0.
St is assumed to followa first-order Markov process with transition
proba-bilities
P (St= 1| St−1= 1) = p, P (St= 0| St−1= 1) = 1− p,
P (St= 1| St−1= 0) = 1− q, P (St= 0| St−1= 0) = q.
Then, equations (2.2) and (2.3) constitute a Markov switching model proposed by Hamilton (1989) once (h1, . . . , hT) is provided.
3. Bayesian estimation
As is well known, it is difficult to evaluate the likelihood of the SV and MSSV models and hence to estimate the parameters of those models using the classical maximum likelihood method. Following previous researchers, we use a Bayesian MCMC method. Specifically, we sample the latent variables (h1, . . . , hT) and
(S1, . . . , ST) as well as the parameters (µ0, µ1, φ, ση2, p, q) from their joint posterior
distribution using the Gibbs sampler, which is a method for sampling from the joint distribution by sampling sequentially from the conditional distributions.
For the unknown parameters in the MSSV model, we work with the following prior distributions.
[µ0, µ1]∼ N(M, V )I[µ1 > 0], (φ + 1)/2∼ beta(φ1, φ2),
σ2η ∼ IG(σr/2, Sσ/2), p∼ beta(u11, u10), q ∼ beta(u00, u01),
where I[·] is the indicator function that takes one if the condition in the bracket is satisfied and zero otherwise. Thus, N (M, V )I[µ1 > 0] is the truncated normal distribution. beta(·, ·) and IG(·, ·) represent the beta and inverse gamma distri-butions respectively. Beta prior for (φ + 1)/2 satisfies the stationarity condition
of ht, that is, |φ| < 1. In the empirical application in Section 5, we also estimate
the SV model to compare with the MSSV model. For µ in the SV model, we use the prior µ∼ N(m, v), and for φ and ση2, we use the same priors as those in
the MSSV model with different hyperparameter values (see Section 5.2 for the hyperparameter values).
Under these priors, it is straightforward to obtain the full conditional distri-butions of the parameters and sample from them except for φ, which is sampled using the Metropolis-Hastings algorithm following Chib and Greenberg (1994) and Kim et al. (1998).
There are two efficient methods available for sampling (h1, . . . , hT) from their
full conditional distribution. One is the mixture sampler proposed by Kim et al. (1998) and the other is the multi-move sampler proposed by Shephard and Pitt (1997) and modified by Watanabe and Omori (2004). Previous researchers such as So et al. (1998) and Kalimipalli and Susmel (2004) use the former method where (h1, . . . , hT) are sampled from the approximate distribution instead of the
true distribution. We use the latter method where they are sampled from the true distribution. We use the multi-move sampler proposed by Carter and Kohn (1994) and Kim and Nelson (1998, 1999) to sample (S1, . . . , ST) from their full
conditional distribution.
4. Model comparison and diagnostics 4.1. Marginal likelihood
It is important to examine whether the mean of the log-volatility shifts de-pending on the state. This is equivalent to comparing the MSSV model with the standard SV model. Model comparison in a Bayesian framework can be per-formed using the posterior odds ratio. Let yT denote (y1, . . . , yT). Then, the
posterior odds ratio, which is denoted by POR, between model i, Mi, and model
j, Mj, is given by POR = f (Mi| yT) f (Mj | yT) = f (yT | Mi) f (yT | Mj) f (Mi) f (Mj) ,
where f (yT|Mi)/f (yT|Mj) and f (Mi)/f (Mj) are called Bayes factor and prior
odds ratio respectively. If POR is greater than one, Mi is favored over Mj.
The prior odds ratio is usually set to be one, so that the posterior odds ratio is equal to the Bayes factor. To evaluate the Bayes factor, we must calculate f (yT | Mi) and f (yT | Mj) called marginal likelihood. Let θi denote the set of
unknown parameters in model Mi. From the Bayes theorem
f (θi| Mi, yT) =
f (yT | Mi, θi)f (θi| Mi)
f (yT | Mi)
, we can write the marginal likelihood of model Mi as
f (yT | Mi) =
f (yT | Mi, θi)f (θi| Mi)
f (θi | Mi, yT)
where f (yT | Mi, θi) is likelihood, f (θi| Mi) is prior density and f (θi | Mi, yT)
is posterior density. The above identity holds for any value of θi, but following
Chib (1995), we set θi equal to its posterior mean ˆθi.
It is straightforward to evaluate the prior density f (ˆθi | Mi). Chib (1995)
proposes a method for evaluating the posterior density f (ˆθi | Mi, yT) using a
Gibbs sampler, which requires the posterior density to be known up to the nor-malizing constant. The nornor-malizing constant of the posterior density of φ in the SV and MSSV models is not known, but φ can be sampled using the Metropolis-Hastings algorithm. In such a case, we can evaluate the posterior density using a method proposed by Chib and Jeliazkov (2001) (If we sample φ using the accept-reject Metropolis-Hastings algorithm proposed by Tierney (1994) instead of the simple Metropolis-Hastings algorithm, we must use the method proposed by Chib and Jeliazkov (2005) to evaluate its posterior density). We combined their method with the method proposed by Chib (1995) to evaluate the posterior density. The problem is likelihood because it is difficult to evaluate the likelihood of the MSSV model analytically.
4.2. Likelihood
Kim et al. (1998) propose a method for evaluating the likelihood of the SV model using a particle filter. We extend their particle filter to evaluate the likeli-hood of the MSSV model by combining it with the particle filter for the dynamic Markov switching factor model proposed by Kaufmann (2000) and Watanabe (2003). Lopes (2002), Lopes and Marinho (2002) and Marinho and Lopes (2002) have proposed a different particle filter for the MSSV model. Their method is based on the basic sampling/importance resampling (SIR)-based auxiliary parti-cle filter proposed by Pitt and Shephard (1999) while our method is an extension of the rejection-based fully adapted particle filter proposed by Kim et al. (1998). Pitt and Shephard (1999) showthat the rejection-based fully adapted particle filter is considerably more accurate than the basic SIR-based auxiliary particle filter using a Monte Carlo experiment.
For simplicity, we omit Miand θi in what follows. Let ytdenote (y1, . . . , yt).
Then, the likelihood can be expressed as
f (yT) = f (y1)
T−1
t=1
f (yt+1| yt).
As for the MSSV model, f (yt+1| yt) is written as
f (yt+1| yt) =
f (yt+1 | ht+1)f (ht+1| St+1, ht)
(4.1)
× P (St+1| St)f (St, ht| yt)dht+1dhtdSt+1dSt,
where strictly speaking, integration with respect to St+1and Stmust be replaced
by summation because they are discrete variables that take 0 or 1, but we use integration for simplicity following Kaufmann (2000) and Watanabe (2003).
Suppose that we have M draws (h(m)t|t , St(m)|t ) (m = 1, . . . , M ) sampled from the filtering density f (ht, St | yt). Then, we can sample S
(m)
t+1|t (m = 1, . . . , M )
using the transition probability P (St+1| St(m)|t ) and h(m)t+1|t (m = 1, . . . , M ) from
f (ht+1| St+1(m)|t, h(m)t|t , St(m)|t ), which is the normal density with mean µ0+µ1St+1(m)|t+
φ(h(m)t|t − µ0− µ1St(m)|t ) and variance ση2. Using h
(m) t+1|t (m = 1, . . . , M ), equation (4.1) can be evaluated as f (yt+1| yt)≈ 1 M M m=1 f (yt+1| h(m)t+1|t), where f (yt+1| h(m)t+1|t) = 1 2π exp(h(m)t+1|t) exp − y2t+1 2 exp(h(m)t+1|t) .
The remaining problem is howto sample from the filtering density f (ht, St|
yt). To sample from this density, we use a particle filter, which is an algorithm to sample from the filtering density sequentially starting from t = 0 (see Pitt and Shephard (1999)). We develop a particle filter for the MSSV model. Suppose that we have M draws (h(m)t−1|t−1, St(m)−1|t−1) (m = 1, . . . , M ) sampled from f (ht−1, St−1|
yt−1). Then, the filtering density of the MSSV model can be written as
f (ht, St| yt) (4.2) ∝ f(yt| ht) f (ht| St, ht−1, St−1) × P (St| St−1)f (ht−1, St−1 | yt−1)dht−1dSt−1, ≈ f(yt| ht) 1 M M m=1 f (ht| St, h (m) t−1|t−1, S (m) t−1|t−1)P (St| S (m) t−1|t−1), where ln f (yt| ht) = const− 1 2ht− yt2 2 exp(−ht). Define ln f∗(ht) =−12ht−y 2 t
2 exp(−ht). Since this function is concave, applying the first-order Taylor expansion to ln f∗(ht) around ht= ˆht yields the following
inequality (see Section 5.2 for the selection of ˆht).
ln f∗(ht) = − 1 2ht− y2t 2 exp(−ht), ≤ −1 2ht− y2t 2 exp(−ˆht)(1 + ˆht− ht), = ln g∗(ht).
The product of g∗(ht) and f (ht | St, h(m)t−1|t−1, St(m)−1|t−1)P (St | St(m)−1|t−1) appeared
in equation (4.2) can be expressed as
g∗(ht)f (ht| St, h(m)t−1|t−1, St(m)−1|t−1)P (St| St(m)−1|t−1)
∝ π(St, m)fN(ht| h∗t(St, m), ση2),
where fN(ht | h∗t(St, m), ση2) is the normal density with mean h∗t(St, m) and
variance ση2, and h∗t(St, m) = µ0+ µ1St+ φ(h(m)t−1|t−1− µ0− µ1St(m)−1|t−1) + ση2 2 {y 2 texp(−ˆht)− 1}, π(St, m) = exp − 1 2σ2 η [{µ0+ µ1St+ φ(h(m)t−1|t−1− µ0− µ1St(m)−1|t−1)}2 − {h∗t(St, m)}2] × P (St| St(m)−1|t−1).
Thus, equation (4.2) may be written as
f (ht, St| yt) ∝ f∗(ht) 1 M M m=1 f (ht| St, h(m)t−1|t−1, St(m)−1|t−1)P (St| St(m)−1|t−1), ≤ g∗(ht) 1 M M m=1 f (ht| St, h(m)t−1|t−1, St(m)−1|t−1)P (St| St(m)−1|t−1), ∝ 1 M M m=1 π(St, m)fN(ht| h∗t(St, m), ση2).
Therefore, we can sample from the filtering density f (ht|yt) using the
accept-reject algorithm. Fisrt, we draw a proposal (ht, St) from the mixture of M normal
densities M m=1 π∗(St, m)fN(ht| h∗t(St, m), ση2), where π∗(St, m) = π(St, m)/ M
m=1π(St, m). We can sample from this mixture
distribution by first selecting the indices (St, m) with probability
π(St, m) 1 St=0 M m=1π(St, m) ,
and then sampling from fN(ht| h∗t(St, m), σ2η). Second, we accept it with
prob-ability f∗(ht)/g∗(ht). If rejected, we return to the first step and draw a new
4.3. Diagnostics
Draws h(m)t+1|t (m = 1, . . . , M ; t = 0, . . . , T − 1) obtained in the above proce-dure can be used also for diagnostics of the MSSV model. Let yt+1o denote the
observation of yt+1. The probability that y2t+1 will be less than yo2t+1 conditional
on yt can be written as P (y2t+1≤ yt+1o2 | yt) = P (y2t+1≤ yt+1o2 | ht+1)f (ht+1| yt)dht+1, ≈ 1 M M m=1 P (yt+12 ≤ yo2t+1| h(m)t+1|t).
The distribution of yt+1 conditional on ht+1 is the normal with mean 0 and
variance exp(ht+1), so that it is straightforward to evaluate the above probability.
Let uMt+1 = M1 M
m=1P (yt+12 ≤ yo2t+1 | h
(m)
t+1|t). Under the null hypothesis
of a correctly specified model, uMt converges in distribution to independently
and identically distributed uniform random variables as M → ∞ (see Rosenblatt (1952) and Kim et al. (1998)). This provides a valid basis for diagnostic checking. These variables can be mapped into the normal distribution, by using the inverse of the normal distribution function nMt = F−1(uMt ) to give a standard sequence
of independently and identically distributed normal variables. Therefore, the diagnostics of the MSSV model can be done by testing whether nMt follows the
independent standard normal distribution.
5. Empirical application 5.1. Data description
We illustrate our method using weekly returns of the TOPIX for the pe-riod 01/06/1971–08/25/2004. Wednesday-to-Wednesday returns are used, and if Wednesday is a trading holiday, then Tuesday’s prices are substituted. The sample size is 1,755.
The descriptive statistics are summarized in Table 1. The statistics reported are the mean, the standard deviation, the skewness, the kurtosis, and the Ljung-Box (LB) statistics for 10 lags corrected for heteroskedasticity following Diebold (1988). The LB statistics indicate that the TOPIX return is serially uncorrelated. Hence, as for (y1, . . . , yT), we use the return series from which only the mean is
subtracted.
Table 1. Descriptive statistics of weekly returns (%) for the TOPIX.
Sample size Mean S.D. Skewness Kurtosis LB(10)
1755 0.1158 2.4059 −0.3084 5.9214 13.75
(0.0574) (0.0585) (0.1169)
NOTE: The figures in brackets are standard errors. LB(10) is the heteroskedasticity-corrected Ljung-Box statistic including 10 lags. The corrected Ljung-Box statistic is calculated following Diebold (1988). The critical values for LB(10) are: 15.99 (10%), 18.31 (5%), 23.21 (1%).
5.2. Details
For the hyperparameters in the prior distributions, we set m = 1, v = 1, φ1 = 20, φ2 = 1.5, σr = 5, Sσ = 0.25 for the SV model and M = [0.5, 1.5],
V = I, φ1 = 20, φ2 = 4, σr = 5, Sσ = 1.5, u11 = u00= 200, u10= u01= 1.5 for
the MSSV model where I is the 2× 2 identity matrix.
The number of blocks in the multi-move sampler to sample the latent variable (h1, . . . , hT) is set equal to 20 and blocks are selected randomly (see Shephard and
Pitt (1997) and Watanabe and Omori (2004)). For the MSSV (SV) model, we conduct an MCMC simulation with 20,000 (15,000) iterations. The first 10,000 (5,000) draws are discarded and then the next 10,000 are recorded. Using these 10,000 draws for each of the parameters, we calculate the posterior means, the standard errors of the posterior means, the 95% intervals and the convergence diagnostic (CD) statistics proposed by Geweke (1992). The posterior means are computed by averaging the simulated draws. The standard errors of the posterior means are computed using a Parzen window with a bandwidth of 1,000 (see Shephard and Pitt (1997) and Kim et al. (1998)). The 95% intervals are calculated using the 2.5th and 97.5th percentiles of the simulated draws. Geweke (1992) suggests assessing the convergence of the MCMC by comparing values early in the sequence with those late in the sequence. Let X(i) be the ith draw of a parameter in the recorded 10,000 draws, and let ¯XA = n1A
nA
i=1X(i) and
¯
XB = n1B
10,000
i=10,001−nBX
(i). Using these values, Geweke (1992) proposes the following statistics called convergence diagnostics (CD).
CD = X¯A− ¯XB ˆ σ2 A/nA+ ˆσ2B/nB , where ˆ σ2 A/nAand ˆ σ2
B/nB are standard errors of ¯XAand ¯XB. If the sequence
of X(i) is stationary, it converges in distribution to the standard normal. We set nA= 1, 000 and nB = 5,000 and compute ˆσA2 and ˆσ2Busing Parzen windows with
bandwidth of 100 and 500 respectively.
We evaluate the posterior density of φ following Chib and Jeliazkov (2001) and all other parameters following Chib (1995). In the both procedures, we set the number of iterations equal to 5,000. In the particle filter to evaluate the likelihood, we set the number of simulations M equal to 2,500. Kim et al. (1998) suggest selecting ˆht in the particle filter as the forecast of ht from the previous
period, that is, µ + φ(M1 Mm=1h
(m)
t−1|t−1− µ) for the SV model. We find that
this selection sometimes makes the particle filter stuck by an excessive amount of rejections in the accept-reject algorithm. We select ˆhtas the posterior mean of
ht, which does not cause this problem. When ˆht is set to the posterior mean of
ht, the acceptance rates in the accept-reject algorithm are 90% for the SV model
and 87% for the MSSV model.
5.3. Results
Estimation results are summarized in Table 2. According to the CD values, the null hypothesis that the sequence of 10,000 draws is stationary is accepted
Table 2. Estimation results.
Mean S.E. 95% Interval CD
SV model φ 0.9636 0.0012 [0.9399, 0.9817] −1.25 ση 0.2524 0.0047 [0.1951, 0.3273] 1.29 µ 1.3343 0.0020 [0.9846, 1.6851] −0.78 MSSV model φ 0.6955 0.0041 [0.5591, 0.8035] 0.81 ση 0.5704 0.0036 [0.4703, 0.6734] −1.27 µ0 0.4574 0.0036 [0.2373, 0.7021] 0.62 µ1 1.3045 0.0053 [0.9949, 1.5755] −0.69 p 0.9950 0.0000 [0.9884, 0.9988] 0.76 q 0.9965 0.0000 [0.9916, 0.9992] 0.46
NOTE: For the MSSV (SV) model, the first 10,000 (5,000) draws are discarded and then the next 10,000 are used for calculating the posterior means, the standard errors of the posterior means, the 95% intervals and the convergence diagnostic (CD) statistics proposed by Geweke (1992).
at the 5% significance level for all parameters in the both models.
The top of Table 2 reports the estimation results of the SV model. The pos-terior mean and the 95% interval of φ are 0.9636 and [0.9399, 0.9817] respectively, exhibiting a high persistence in return volatility typical of estimates in the SV literature.
Next, we turn to the estimation results of the MSSV model. Previous re-searchers such as So et al. (1998) and Kalimipalli and Susmel (2004) document that the estimate of φ is much smaller and the estimate of ση is much larger in
the MSSV model than those in the SV model. This is true also for our result where the posterior mean and the 95% interval of φ drop significantly to 0.6955 and [0.5591, 0.8035]. The posterior mean and the 95% interval of ση are 0.5704
and [0.4703, 0.6734], which are much larger than those in the SV model. The estimates of transition probabilities p and q are very close to one, indicating that the probability of switching between the high- and low-volatility states is quite low. Since the switch between the two states is a rare event, we may use the stochastic volatility jump (SVJ) model instead of the MSSV model. Eraker et al. (2003) and Eraker (2004) document that the estimate of φ rises and the estimate of ση drops in the SVJ model compared to those in the standard SV
model. This difference comes from the fact that a Markov switching causes an autocorrelation while a Poisson jump does not. It is important to analyze which model is better fitted to the data between the MSSV and SVJ models, but we will leave this for future research.
Table 3 reports the estimates of the log likelihood and the log marginal likelihood with their numerical standard errors in brackets for the both models. The log marginal likelihood of the MSSV model of −3794.18 is larger than that of the SV model of −3796.04 significantly at the 5% level. We may conclude
Table 3. Likelihood and marginal likelihood.
SV MSSV
Log likelihood −3791.00 −3785.65 (0.5077) (0.4338) Log marginal likelihood −3796.04 −3794.18
(0.5188) (0.4436)
NOTE: The figures in brackets are numerical standard errors.
Table 4. Diagnostic statistics.
Mean S.D. Skewness Kurtosis LB(10)
SV −0.0054 0.9969 0.1624 3.0594 8.45
(0.0238) (0.0585) (0.1169)
MSSV 0.0400 0.9688 0.0086 2.9262 10.83
(0.0231) (0.0585) (0.1169)
NOTE: The figures in brackets are standard errors. LB(10) is the heteroskedasticity-corrected Ljung-Box statistic including 10 lags. The corrected Ljung-Box statistic is calculated following Diebold (1988). The critical values for LB(10) are: 15.99 (10%), 18.31 (5%), 23.21 (1%). 㪇 㪇㪅㪈 㪇㪅㪉 㪇㪅㪊 㪇㪅㪋 㪇㪅㪌 㪇㪅㪍 㪇㪅㪎 㪇㪅㪏 㪇㪅㪐 㪈 㪈㪐㪎㪈㪆㪈 㪈㪐㪎㪍㪆㪍 㪈㪐㪏㪈㪆㪈㪉 㪈㪐㪏㪎㪆㪍 㪈㪐㪐㪉㪆㪈㪉 㪈㪐㪐㪏㪆㪌 㪉㪇㪇㪊㪆㪈㪈
Figure 1. Posterior probabilities of high-volatility state (St= 1).
that the MSSV model is favorable over the SV model and that a switch occurs in the mean of log-volatility. The likelihood of the SV model can also be evaluated by the AGIS proposed by Danielsson (1994) and Danielsson and Richard (1993). We also use this method to evaluate the likelihood of the SV model, but the difference is marginal.
Table 4 shows the results of diagnostic checking based on variables nMt
ex-plained in the previous section. The table shows the mean, the standard devi-ation, the skewness, the kurtosis and the Ljung-Box statistics used to test the null hypothesis of no serial correlation for 10 lags, where the figures in brackets
㪇 㪈 㪉 㪊 㪋 㪌 㪍 㪎 㪈㪐㪎㪈㪆㪈 㪈㪐㪎㪍㪆㪎 㪈㪐㪏㪈㪆㪈㪉 㪈㪐㪏㪎㪆㪍 㪈㪐㪐㪉㪆㪈㪉 㪈㪐㪐㪏㪆㪌 㪉㪇㪇㪊㪆㪈㪈 (a) SV model 㪇 㪈 㪉 㪊 㪋 㪌 㪍 㪎 㪈㪐㪎㪈㪆㪈 㪈㪐㪎㪍㪆㪎 㪈㪐㪏㪈㪆㪈㪉 㪈㪐㪏㪎㪆㪍 㪈㪐㪐㪉㪆㪈㪉 㪈㪐㪐㪏㪆㪌 㪉㪇㪇㪊㪆㪈㪈 (b) MSSV model
Figure 2. Posterior means of volatility exp(ht/2).
showthe standard errors. If the model is correctly specified, the asymptotic dis-tribution of nMt is the standard normal. For the SV model, the null hypothesis
of zero skewness is rejected at the 1% significance level. On the other hand, the MSSV model passes all diagnostic tests at the 5% significance level.
Figure 1 depicts the posterior probabilities of high-volatility state as inferred from the MSSV model. These probabilities can be calculated simply by averaging 10,000 draws of the state St sampled from its posterior distribution. We may
define period t as a turning point if the posterior probability P (St−1 = 1| yT) >
0.5 and P (St = 1 | yT) < 0.5 or if P (St−1 = 1 | yT) < 0.5 and P (St = 1 |
yT) > 0.5. Then, the low-volatility periods are: 04/09/1975–03/12/1986 and
03/30/1988–12/27/1989.
Figure 2 displays the posterior mean of volatility exp(ht/2) in each period
es-timated by the SV model and the MSSV model. The posterior means of volatility estimated by the MSSV model appear to be more volatile and less smoother than those by the SV model because the estimate of φ (ση) is much smaller (larger)
6. Conclusions
This article estimates the SV and MSSV models using a Bayesian MCMC method. Our estimation result confirms the finding by previous researchers that the estimate of the persistence parameter drops and the estimate of the error variance rises in the volatility equation of the MSSV model compared to those of the standard SV model. We also develop a particle filter for the MSSV model, which is used for evaluating the marginal likelihood and obtaining the variable for diagnostics. The model comparison based on the resulting marginal likelihood reveals that the MSSV model is favored over the SV model. We also find that the MSSV model passes all diagnostic tests while the SV model does not.
The MSSV model analyzed in this article is a simple one, so that several extensions are possible. It is worthwhile extending the model such that the number of states is more than two or other parameters may also shift. In stock markets, bull and bear are usually defined as a high-return stable state and low-return volatile state respectively (see Maheu and McCurdy (2000)). It is hence important to allowfor the shift in the mean of returns as well as log-volatility. It is straightforward to extend the particle filter constructed in this article for such extended models. We must, however, be careful in imposing identifiability constraints if more than two parameters are allowed to shift (see Fr¨uwirth-Schnatter (2001)).
Acknowledgements
The authors would like to thank the Editor and two anonymous referees for their useful comments which improved the quality of this paper. Thanks are also due to Yasuhiro Omori and the participants in the annual meetings of the Japan Statistical Society at Fuji University and the Japanese Economic Association at Okayama University in September 2004 for their valuable comments. This research is partially supported by the 21st Century COE grant and Grants-in-Aid for Scientific Research, both from the Ministry of Education, Culture, Sports, Science and Technology of the Japanese Government. The views expressed in this paper are those of the authors and do not represent the official views of the Bank of Japan or Institute for Monetary and Economic Studies. Remaining errors are ours alone.
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