How Persistent is Volatility? An Answer with Stochastic Volatility Models with Markov Regime Switching State Equations

How Persistent is Volatility? An Answer with Stochastic Volatility Models with Markov Regime Switching State Equations Soosung Hwang Cass Business School, London, UK Steve E. Satchell Cambridge Univeristy, UK Pedro L. Valls Pereira Ibmec Business School, BRAZIL January 4 Abstract We introduce SV models with Markov regime changing state equation (SVMRS) to investigate the important properties of volatility, high persistence and smoothness. With the quasi-ml approach proposed in our study, we showed that volatility is far less persistent and smooth than the GARCH or SV models suggest. Keywords: Stochastic Volatility, Markov Switching, Persistence. JEL Codes: C3 Corresponding Author. Faculty of Finance, Cass Business School, 6 Bunhill Row, London, ECY 8TZ, UK. Tel: 44 () 74 9. Fax: 44 () 74 888. Email: s.hwang@city.ac.uk

Introduction One of the most common methods to obtain a proxy measure of volatility is to fit parametric econometric models such as GARCH or stochastic volatility (SV) models, and others include option implied volatility, the intra-day return volatility (Andersen and Bollerslev, 998) and range volatility (Parkinson, 98; Garman and Klass, 98; Alizadeh, Brandt, and Diebold, ). In most cases, the results from these parametric or nonparametric methods show that ex-post squared returns or absolute returns are too noisy and volatility is highly persistent and smooth. These results are consistent with the poor forecasting power of GARCH models on ex-post squared returns or absolute returns. However, we may ask if the persistence and smoothness represent the properties of true volatility. For example, GARCH conditional volatility reflects only lagged information and is not designed to take account of cross-sectional information. If asset returns follow linear factor models such as Fama and French (99), then there are multiple cross-sectional factors which are not explained by conditional volatility, but are sources of volatility. Campbell, Lettau, Malkiel, and Xu (), for example, using cross-sectional decomposition on equity volatility, showed that market and industry volatilities are important components for the explanation of individual asset volatility. Connor and Linton () and Hwang and Satchell (4) also suggested that there is common heteroskedasticity in asset-specific returns. Therefore, a significant amount of squared returns may not be noise but come from crosssectional heteroskedasticity in factors and factor loadings, which is not explained by conditional volatility. Another econometric question is that the persistence and smoothness obtained with well known volatility models such as GARCH or SV models may come from the restrictive nature of the models. For example, Lobato and Savin (998), Granger and Hyung (999) and Diebold and Inoue () suggested that structural breaks in the mean of volatility may be a source of persistence. As a second example,

Bollerslev s (986) nonnegativity constraints on coefficients on GARCH model may restrict autocorrelation structure of volatility. Nelson and Cao (99) showed that the Bollerslev s non-negativity conditions are too restrictive and in some cases negative estimates may be obtained in practice. He and Teräsvirta (999) further showed that allowing negative parameters in GARCH models can give us various autocorrelation structures of squared returns. In our study we use SV models with Markov regime changing state equations (SVMRS) to investigate the questions on persistence and smoothness of volatility. Existing models such as So, Lam, and Li (998), Kalimipalli and Susmel (), and Smith () provide such a structure, but our model is more general in that we allow volatility to have regime-dependent means, variances and autoregressive characteristics. SV models are useful for our purpose since they allow us to decompose squared returns into transitory noise and permanent innovation (volatility process). Note that the error (innovation) in the state equation matters over time through a process, whilst the error (noise) in observation equation does not, and it is the innovation term that captures the persistence of the model. In addition, in SV models we do not need nonnegativity restrictions on the parameters. Furthermore, by allowing regime changes in the parameters of the state equations in SV models, we can investigate properties of the volatility process. As is standard, we assume that the state equation in our SV model follows an AR(). However, the assumption of volatility process being AR() processes seems to be too restrictive, if there are structural breaks. Therefore, we allow the state equation in our SV model to Markov regime switch over time. We find that the squared returns are better specified with our SVMRS model. More importantly, we find that volatility is far less smooth and persistent. Our results suggest that the conventional SV and GARCH models may be too restrictive for squared returns. In addition, the large proportion of transitory noise in SV Using ARMA processes in the state equation does not change the persistence and smoothness of the volatility process. See Hwang and Satchell () for example. 3

models decrease significantly in the SVMRS model and many cases in our study show that there is little transitory noise in squared returns when the SVMRS model is used. We also show that when a AR() process follows regime changes, under certain conditions in the transition probabilities, autocorrelation coefficients may show more persistence than the AR parameter suggests. These results are consistent with those of Granger and Hyung (999) and Diebold and Inoue () for example. In the next section we introduce our model. We also derive the autocorrelation function of the regime switching AR() process. In section 3 using S&P5 and FTSE daily and weekly returns, we show estimates of our SVMRS model and compare the conventional SV model. Conclusions follow in section 4. Models. SV and Markov Regime Switching Models The stochastic volatility (variance) (SV) model was introduced by Taylor (986) and Hull and White (987) and has been further developed by Harvey and Shephard (993, 996) and Harvey, Ruiz and Shephard (994). In the SV model, the log of ε t, where ε t is typically asset returns (or residuals from a return process), is modelled as a stochastic process: ε t = ɛ t σ exp( h t) () h t = φh t + η t where ɛ t N(, ) and independent of η t N(,σ η), and σ is a positive scale factor. Squaring () and taking logs we have a process ln ε t = lnɛ t +lnσ + h t () = E[ln ɛ t ]+lnσ + h t +lnɛ t E[ln ɛ t ] = µ + h t + ϕ t 4

where µ = E[ln ɛ t ]+lnσ and ϕ t =lnɛ t E[ln ɛ t ] is a martingale difference, but not normal. When we replace ln ε t with y t and µ + h t with x t, the SV model in equations () and () can be written as y t = x t + ϕ t (3) x t µ = φ(x t µ)+η t (4) SV models are useful to decompose log-squared returns into transitory noise and permanent innovation. This is because the innovation, η t, matters over time through the AR() process, whilst noise, ϕ t, does not. Using this concept, Hwang and Satchell () showed that squared daily index returns such as FTSE or S&P5 consists of 95% of noise and 5% of unobserved innovation (volatility). This result is asymptotically consistent with the poor forecasting power of GARCH models (see Andersen and Bollerslev (998)). Note that the AR() process is commonly used in the state equation (volatility process). However, the assumption of the volatility process being an AR() process seems to be too restrictive. We may use ARFIMA models to generalise the volatility process. In our study, we focus on structural breaks. Many studies such as Lobato and Savin (998), Granger and Hyung (999), and Diebold and Inoue () showed that there are structural breaks in the volatility process and the structural breaks have been blamed as a source of extreme persistence in volatility. Hwang (4) recently showed that in a mean zero AR() process, persistence (or the magnitude of estimated AR coefficient) is a function of structural breaks in the mean as well as in the AR parameter. It is clear that if there are structural breaks in volatility, the conventional GARCH or SV models are misspecified and we need a model for the structural breaks. In recent years economic time series have been modelled with the assumption that the distribution of the variables is known conditional on a regime or state occurring. The Markov regime switching models introduced by Hamilton (989) allow the unobserved regime to follow a first order Markov process. The models 5

have been used extensively in macroeconometrics as a means of capturing the different patterns of expected growth in output, see, for example, Filardo (994) and Goodwin (993). Suppose that there is a state variable, s t, which is unobservable. When we allow regime switching in the fundamental equation, a simple state equation is µ + η,t, when s t =, x t = µ + η,t, when s t =, where η i,t N(,σ η i ),i=,, and s t follows a Markov chain. Here x t does not allow persistence, and thus the simple state equations are not appropriate for volatility process. Note that we can easily allow x t to follow AR() processes, but this assumption can be generalised by the introduction of more lags in the AR component, i.e., AR(p) processes. For simplicity, throughout this study we assume that the state equation follows a regime switching AR() process; µ + φ (x t µ )+η,t, when s t =,s t =, µ + φ (x t µ )+η,t, when s t =,s t =, x t = µ + φ (x t µ )+η,t, when s t =,s t = µ + φ (x t µ )+η,t, when s t =,s t =. (5) Previous studies introduced regime switching state equations to investigate the effects of structural breaks on the persistence of volatility. See Diebold and Inoue () for example. However, they only allowed µ to have different values over time. Our model is more general in the sense that φ and σ η as well as µ are allowed to change.. Stochastic Volatility Models with Markov Regime Switching Equation It is interesting if we can combine (3) and (5), i.e., a stochastic volatility model with a Markov regime switching state equation (SVMRS). As pointed out in Andersen 6

and Bollerslev (998) among many others, if squared residuals are too noisy for a proxy volatility process, we need to take out noise from the squared residuals and then investigate the remainder to see if there are structural breaks or persistent. However, the decomposition of squared returns depends on which models are used for state equations. There are some previous attempts to model with stochastic volatility and regime switching models. For example, in the studies of So, Lam, Li (998) and Kalimipalli and Susmel (), µ is allowed to regime-change. So, Lam, Li (998), using weekly S&P5 index volatility, found that volatility is far less persistent than that of SV models. Kalimipalli and Susmel () applied their model to explain the behaviour of short-term interest rates. They found that their regime switching model performs better than the GARCH family of models and SV models. On the other hand, Smith () generalised these models and showed that Markov-regime switching or stochastic volatility models need to be improved to explain short-term interest rates. Our SVMRS model allows all three parameters in (5) to change and thus is a generalised version. That is, there are two regimes and the state equation is assumed to follow an AR() process; y t = x t + ϕ t (6) x t = µ i + φ i (x t µ j )+η i,t, when s t = i, s t = j, (7) where i =, andj =,, ϕ t N(,σ ϕ), η i,t N(,σ η i ), and the transition probabilities are given by: p (i,j) =Pr(s t = i s t = j) (8) and the transition matrix is given by: p = p(,) p (,) p (,) p (,) (9) We note that the above SVMRS model has two unobserved variables; x t which follows different processes according to an unobserved variable s t. We may allow 7

more states and lags, but the number of cases we should consider for x t increases rapidly, i.e., (number of states) lags+. The SVMRS model treats ϕ t as a transitory noise and η t as a permanent innovation (or volatility process) as suggested by Hwang and Satchell (). The treatment provides intuitively interesting perspective since we can decompose any process into just noise which is not explained by the state equation and innovation. In addition, when σ η = σ η, the volatility of volatility is unchanged regardless of states and when µ = µ, the unconditional level of volatility is the same across different states. Finally, when φ = φ,x t has the same persistence. The SVMRS model above is the generalised version of SV as well as the Hamilton s Markov regime switching model. By restricting parameters in appropriate ways, we can derive these models. Usually this can be achieved when we estimate the SVMRS model and investigate if parameters satisfy some conditions; If σ ϕ =, the SVMRS model becomes Hamilton s Markov regime switching model. If µ = µ,φ = φ,σ η = σ η, and σ ϕ, then we have the SV model. If µ = µ,φ = φ,σ η = σ η, and σ ϕ =, then we have the AR() model. If one of σ ηi is zero, then the unobserved process consists of a stochastic process and a deterministic process. Note that by restricting φ = φ,σ = σ, and σ ϕ = π under the assumption that ɛ t is standard normal in (), we have a model similar to Smith () which is y t = x t + ϕ t () c + φx t + η t, when s t =, x t = () c + φx t + η t, when s t = where ϕ t N(, π )andη t N(,σ η). The model can be easily generalised so that φ and σ η may have different values for different states. However, the model does not 8

explain the two other cases, i.e., s t =ands t =, and s t =ands t =. If the frequency of inter-state changing is small, the effects of disregarding these two cases may be trivial, and this may be appropriate for most macroeconomic variables where the number of structural breaks is usually less than %. See Stock and Watson (996), Ben-David and Papell (998), McConnell and Perez-Quiros (), Hansen (), and Bai, Lumsdaine, and Stock (998) among many. However, as will be shown later, for volatility which has many structural breaks, Smith s () model may become restrictive. When we define ξ t as a random variable that is equal to unity when s t =and zero otherwise, the AR() representation of state is ξ,t =( p (,) )+( +p (,) + p (,) )ξ,t + v,t, () where v,t is a martingale difference sequence of state at time t. The unconditional probability that the process will be in regime at any time, p,is; p (,) p = E(ξ,t )= (3) p (,) p (,) Theorem The autocorrelation function with lag τ, ρ(τ), ofthestateequationin (7) which is equivalent to that of Markov regime switching process, is [ τ ] ρ(τ) =E (φ +(φ φ )s t s+ ). Proof. s= The state equation in (7) can be represented as x t [µ +(µ µ )s t ] = (φ +(φ φ )s t )[x t [µ +(µ µ )s t ]] (4) where ξ t N(, ). Note that +(σ η +(σ η σ η )s t )ξ t, E [x t [µ +(µ µ )s t ]] = E(x t ) E [µ +(µ µ )s t ] = E [ x t ξ,t + x t ( ξ,t ) ] E [µ +(µ µ )s t ] = µ p + µ ( p ) µ (µ µ )p =. 9

Substituting z t = x t [µ +(µ µ )s t ], we have the following mean zero AR() process; z t =(φ +(φ φ )s t )z t +(σ η +(σ η σ η )s t )ξ t, (5) and thus [ τ ] ρ(τ) =E (φ +(φ φ )s t s+ ). s= Note that when τ =,ρ() = φ +(φ φ )p, since E(s t )=p. However, when τ =, we have ρ() = E [(φ +(φ φ )s t )(φ +(φ φ )s t )] = E [ ] φ +(φ φ )φ s t +(φ φ )φ s t +(φ φ ) s t s t = φ +(φ φ )φ p +(φ φ ) E[s t s t ]. Since E[s t s t ] = E[ξ,t ξ,t ] = E[(( p (,) )+( +p (,) + p (,) )ξ,t + v,t )ξ,t ] = E[(( p (,) )ξ,t +( +p (,) + p (,) )ξ,t ] = ( p (,) )p +( +p (,) + p (,) )p = p (,) p, we have ρ() = φ +(φ φ )φ p +(φ φ ) p (,) p = [φ +(φ φ )p ] +(φ φ ) [p (,) p (p ) ] = ρ() +(φ φ ) p [p (,) p ]. Therefore, Theorem shows that generally ρ(τ) ρ() τ, unless either φ = φ or p =or, i.e., there is only one state. Therefore, the autocorrelations of the Markov regime changing AR() process may not show the persistence level that the value of ρ() suggests.

Remark Equation (3) suggests that when p (,) > p (,) and φ φ, we have [p (,) p ] > and thus ρ() >ρ(). Thus even though (5) has an AR() representation, the process does not show the same exponential decay rate as the conventional AR() process because of the probability of states. In addition, the difference between ρ() and ρ() is a positive function of persistence difference φ φ, p, and p (,) p..3 Estimation Procedure Harvey, Ruiz and Shephard (994) adopted a procedure based on the Kalman filter to estimate SV models in (3) and (4). However, since the distribution of ϕ t is not known, it is not possible to represent the likelihood function in closed form. However, quasi-maximum likelihood (QML) estimators of the parameters can be obtained using the Kalman filter by treating ϕ t and η t as normal. Harvey, Ruiz and Shephard (994) treated ϕ t as though it were N(,π /), and maximized the resulting quasi-likelihood function. Ruiz (994) suggested that for the kind of data typically encountered in empirical finance, the QML for the SV model has good finite sample properties. In this study we propose a Quasi-Maximum Likelihood (QML) estimation method using the Kalman filter. The basic concept is that both x t and conditional probability that are unobserved processes, can be obtained through predicting and updating which was proposed by Smith (). That is, we have (number of states) lags+ state equations, e.g., in our study 4 state equations, and each state equation is updated and predicted in the same way as for the standard Kalman filter. We use the method suggested in Hamilton (989) to update the conditional probability with a transition probability matrix. A detailed explanation on estimation and smoothing procedure can be found in the Appendix. We could use other methods such as We only show the cases of ρ() and ρ(). The autocorrelations with larger lags are complicted and we do not discuss them further in this study.

Markov Chain Monte Carlo (MCMC), the generalised method of moments (GMM), the efficient method of moments (EMM) to estimate the SVMRS. However, these methods are more complicated than the QML with the updating procedure proposed in the appendix. One problem of the Markov regime switching model is that the model is multimodal. To find out the global maximum of the log-likelihood, we try various starting values. However, the largest ML value does not always guarantee that the estimates are appropriate. Another criterion we use for the SVMRS model is the relative magnitude of σ ηi to σ ϕ (signal-to-noise ratio). If a model is well specified, then the proportion that is not explained by the model, i.e., the transitory noise in the SVMRS model, should be minimised. Since the true volatility process and thus the amount of transitory noise included in squared returns is not known, a model that explains squared returns as much as possible may be better than a model that does not. Thus, signal-to-noise ratio can be a criterion to differentiate different sets of converged estimates. 3 3 Empirical Tests We use two daily indices, i.e., S&P5 and FTSE, from 7 February 99 to 7 February. For the sample period, 548 and 66 log-returns are obtained for FTSE and S&P5 indices, respectively. We also use 5 weekly log-returns from 6 February 99 to 7 February for the two indices. To calculate residuals, we simply take the mean returns during the sample from the log-returns. 4 3 However, this criterion may be controversial. We may use the Bayesian analysis, but again we need some knowledge of the true volatility process. Empirical results in the next section show that there are not significantly differences in model selection between ML values and signal-to-noise ratio. 4 Since daily and weekly expected returns are very small, and taking the mean returns from the daily and weekly returns does not have significant effects. In the following we use log-squared returns for the logs of squared de-meaned returns.

Table reports the property of two index returns. As expected, daily returns are negatively skewed and leptokurtic, suggesting non-normal. In addition, autocorrelation coefficients are not significant. For the two weekly returns, we also find similar property, but the magnitude of non-normality is much smaller than that of the daily returns. On the other hand, the log-squared residuals, as reported in many other studies, are negatively skewed and fat-tailed, and also are persistent. One noticeable difference between the daily and weekly log-squared residuals is that daily log-squared residuals are more persistent than weekly log-squared residuals. The temporal aggregation affects the level of persistence, i.e., autocorrelation structure. The large negative skewness in the log-squared residuals results from the so-called inlier problem in stochastic volatility models. For the daily returns used in this study, for example, the largest log-squared residuals of 3.559 for the FSTE and 3.935 for the S&P5 are within three standard deviations. However, the lowest log-squared residuals are -3.857 for the FTSE and -6.37 for the S&P5, respectively, and both of them are outside five standard deviations. These extremely small log-squared returns reflect returns close to zero. Various methods may be used for inlier adjustment for the squared residuals. Harvey and Shephard (993) set an arbitrary critical value and trim all values less than the critical value to the arbitrary critical value. These trimmed estimates are better behaved than the untrimmed estimates in their simulations. However, these kinds of inlier adjustments are criticised to be profoundly suspicious by Nelson (994). In this study, we use the following Breidt and Carriquiry (BC) (996) transformation as used in Harvey and Strieibel (996): ln ε t = ln(ε t + κσ ε ) κσ ε /(ε t + κσ ε ) (6) The idea behind the BC transformation is as follows. For the zero or extremely small ε t, ln(ε t + δ), where δ is a small increment, is evaluated. Then, the transformed ln(ε t ) can be obtained by the linear extrapolation from the point (ε t + δ, ln(ε t + δ)) 3

using the slop of the tangent line, (ε t + δ). In the above equation, δ is set to κσ ε. Table reports the property of log-volatility changes for the three different parameter values of κ, i.e.,.,.5,.. For different time series, different values of κ are required. For example, S&P5 daily log-squared residuals show that when κ =., we have the smallest Jarque and Bera statistic, whilst κ =. gives the smallest Jarque and Bera statistic for FTSE weekly log-squared residuals. Note that when κ is too large, then we may lose information included in the original data. For example, the autocorrelation coefficients increase as κ increases. On the other hand when κ is too small, we still have the inlier problem. In many case, the choice of κ is arbitrary and needs econometricians subjective decision. In this study, we choose κ =.5 to minimise the inlier problem in the stochastic volatility model. 3. SV and SVMRS Models We first estimate the SV model in (3) and (4) for daily data. As in many other previous studies, we find that the unobserved volatility process is highly persistent for both S&P5 and FTSE daily log-squared returns. This is a typical result of SV models; the extreme persistence in volatility process. However, the autocorrelation coefficients presented in table does not suggest such a high level of persistence. The difference between the two is usually attributed to high level of noise in squared returns (see Andersen and Bollerslev, 998). This is supported by the signal-to-noise (SN) ratios, σ η /σ ϕ, which are.8 and.49 for S&P5 and FTSE, respectively. This means that SV models (or asymptotically GARCH models) explain only a small proportion of squared residuals. Figures a and a show absolute values of residuals and smoothed standard deviation obtained from the SV model. As in most empirical results on SV models, they show the volatility is smooth. However, as discussed in the previous section, the smoothness in the SV model may be achieved by disregarding the structural breaks. The estimates of our SVMRS model for the daily data are reported in table. 4

We find that the ML values are larger than those of the SV model in both indices, suggesting that the SV model with the MRS state equation better specifies the logsquared residuals. This is also supported by large SN ratios. For S&P5, the SN ratios are 436 (s t =)and774(s t = ), whilst for FTSE, these ratios are.74 (s t = ) and.79 (s t = ). In particular, the transitory noise for S&P5 from the SVMRS is close to zero. Therefore, the large amount of transitory noise unexplained by the SV model is now explained by switching regimes. In addition, we also notice that state which is the lower level of volatility is more volatile; the SN ratios of state are larger than those of state. However, the large volatility of the lower level of log-volatility (s t = ) may not have significant meaning, because state represents lower level of volatility which is close to zero when transformed back to volatility using the exponential function. More importantly, the high persistence found with the SV model disappears in the SVMRS model. For example, the estimates of the AR parameter in the SV model are.999 and.99 for S&P5 and FTSE, respectively. However, the estimates of the SVMRS model show that the AR parameters are.36 (s t = ) and.6 (s t = ) for S&P5 and.899 (s t = ) and.6 (s t = ) for FTSE, respectively. These estimates are far from those of the SV model. The significantly different levels of means in states and suggest that structural breaks in mean may be a source of high persistence. Therefore, SV models without considering structural breaks can provide spurious persistence. These results are consistent with recent studies such as Lobato and Savin (998), Granger and Teräsvirta (999), Granger and Hyung (999), Diebold and Inoue (), and Hwang (4). Interestingly we find that the AR coefficients of ξ,t in equation () are all small negative and thus not persistent at all; i.e., the probability of state has the AR coefficient of +p (,) + p (,) which is close to zero. Therefore, structural breaks do not have memory and there are many structural breaks in the state process. These results seem to be inconsistent with Granger and Hyung (999) who found a small probability (less than %) of structural breaks in squared returns in a long 5

memory volatility model. However, the difference between our approach and other previous studies including Granger and Hyung (999) is that in our model, all three components, i.e., the level of volatility, AR coefficient and the volatility of permanent error, are allowed to change. Note that the unconditional probability that s t =,E(s t =),is E (s t =)= p (,) p (,) p (,) from (3). Using this equation, we obtain the unconditional probability of s t = for the daily S&P index, p =.73 (and we have.759 for FTSE index). This means that around 7% of cases, volatility is in higher state with mean of -.6 and AR parameter of.6. The AR coefficient of the Markov regime switching state equation for the S&P5 is φ +( φ φ ) p =.96 from theorem. Using the same method we find that the estimated AR coefficient for the FTSE is.44. Thus when we remove the transitory noise and allow regimes, the AR coefficients estimated are much smaller than those with the SV model which show extreme persistence. In addition, the estimated transition probabilities in table (and weekly cases in table 3) show that p (,) < p (,) in all four cases, suggesting ρ() <ρ(). Therefore, at the second lag the autocorrelation coefficient decays faster than the ordinary AR() process whose autocorrelation coefficient at lag is equivalent to ρ(). Table 3 and Figure 3 report the results of weekly data. As in daily data, we find extreme persistence in volatility processes and small SN ratios from the estimates of the SV model; the estimated AR parameters are.99 and.969 and the SN ratios are.4 and.6 for the S&P5 and the FTSE, respectively. In addition, the AR coefficient of x t calculated from the estimates of the SVMRS model as in () are.9 and.4 for the S&P5 and the FTSE, respectively. Again there is clear difference in persistence between SV and SVMRS models. 6

We plot squared residuals, smoothed volatility and probability in figures to 3. Figures b, b and 3b clearly show that when we allow two different regimes for the mean, the AR parameter and the volatility of log-volatility, we have much more volatile smoothed volatility. For the S&P5, since the transitory noise is close to zero, most squared residuals are now explained by the two regimes. On the other hand, for the FTSE, we still have a significant portion of squared returns which is not explained by the SVMRS model. Tables and 3 and Figures to 3 suggest that the large amount of transitory noise unexplained by the SV model is now explained by either one of the two states. Asymptotically SV models are equivalent to GARCH models, and thus these results can also be applicable to GARCH models; when we allow structural breaks, the persistence level is reduced and the explanatory power of the model will increase. 3. Some Other Considerations in SVMRS Models The SVMRS model proposed in this study can be used to investigate various different cases. Here we investigate two cases; when the mean is regime changing and when the state equation follows () as in Smith (). The former is useful to investigate if structural breaks in mean are sources of persistence. 5 The results with daily and weekly data when φ = φ and σ η = σ η are reported in tables and 3. First of all, even though we allow regime changes in mean, the results still show extreme persistence; the estimated AR coefficients are all around from.97 to.99. Interestingly the ML values of this restricted model have larger ML values than the unrestricted model of SVMRS (except for weekly FTSE case). However, in all four cases, the transitory noise is much larger than the signal. This restricted model may not be appropriate if a good model should explain observed time series as much as possible. Figures c, c and 3c show an 5 See Diebold (986), Lamoureux and Lastrapes (99), Chu (995), Lobato and Savin (998) Lobato and Savin (998), Granger and Teräsvirta (999), Granger and Hyung (999), and Diebold and Inoue (999) for example. 7

interesting pattern. Since the permanent innovation is much less than the noise, the two states (the common high and low volatility process) move smoothly over time. However, because of frequently changing probability, the volatility becomes highly volatile; despite the high persistence, the volatility process is far less persistent or smooth. The figures for the restricted SVMRS model show that the lower volatility process is dominated by close to zero volatility. When we accept that volatility is a proxy measure of risk, we are less interested in small volatilities. In order to avoid econometric difficulties from inliers and to concentrate on large volatilities we use BC method in (6) with κ =.5. Table reports that we still have a similar high AR coefficient for the volatility process for the modified log-squared returns. However, there is significant reduction of transitory noise and the lower level of logvolatility is now significantly shifted upward. Figures d, d and 3d show that the volatility process is now more concentrated and the difference between higher and lower volatility is reduced. However, we still find that smoothed volatility is highly volatile because of the frequently changing state. For example, the AR coefficients of ξ,t, the probability of state at time t, are -. and -.3 for the S&P5 and FTSE daily modified log-squared residuals, respectively. The second case we consider in this study is the model proposed by Smith () as in (). The last two columns of tables and 3 report the estimates and their standard deviations. Interestingly we find that the estimated AR coefficients show that the volatility process is not persistent; all of them are less than.. Comparing the levels of persistence with the SV model, these estimates are significantly small. This may be further evidence that volatility may be far less persistent. Note that the estimated ML values and SN ratios of Smith s model are much larger than those of the SV model. However, as explained in the previous section, Smith s model does not consider the two other cases, i.e., s t =,s t =ands t =,s t =andthus is more restrictive than the SVMRS model. Figure 4 plots the four cases of Smith () model in () for the weekly S&P5; 8

i.e., SV model, SVMRS model, SVMRS model with φ = φ and σ η = σ η (see the last two columns of table 3a), and SVMRS model with φ = φ and σ η = σ η for BC modified log-squared returns with κ =.5. 6 As in figure 3, figure 4 confirms that the volatility is much more volatile and less persistent. 4 Conclusions This paper has presented a SV model with regime-dependent mean, variance, and autocorrelation that generalises existing SV regime-dependent models. We estimate our model using generalisations of the Kalman filter methods of Harvey, Ruiz, and Shephard (994). Our results show that squared returns are better specified by our SVMRS models. In addition, a broad pattern we have found seems to be that the regime-dependent estimates are less persistent (and more volatile) than single regime estimates provided by other authors. This suggests that ignoring regime switching increases the estimate of persistence. 6 The estimates that are not reported in the paper can be obtained from authors upon request. We also estimated these models using ten years monthly data. The results are similar and can be obtained from authors upon request. 9

A. Estimation Procedure Appendix We need some notation for the prediction of the state vector and also for its variance depending upon which regime is being used in the conditional set. Prediction equation (x (i,j) (i,j) t t ), mean squared error associated with x(i,j) t t (M t t ), prediction error (v (i,j) t ), prediction variance (f (i,j) t ) and updating equations (x (i,j) t t,m (i,j) t t )can be obtained using the following procedure; For s t = i and s t = j, i, j =,, we have x (i,j) t t = E[x t s t = i, s t = j, I t ] (7) = (µ i µ j φ i )+φ i x (j) t t, M (i,j) t t = E[(x t x (i,j) t t ) s t = i, s t = j, I t ] (8) = φ i M (j) t t + σ i, v (i,j) t = y t x (i,j) t t (9) = x t x (i,j) t t + ϕ t f (i,j) t = M (i,j) t t + σ ϕ () x (i,j) t t = x (i,j) t t + M (i,j) t t M (i,j) t t = M (i,j) t t M (i,j) t t ( f (i,j) t ( f (i,j) t ) (i,j) v t () ) (i,j) M t t () Note that as in Hamilton (989), the filtered transition probability, Pr(s t = i, s t = j I t ), is updated with transition probability, Pr(s t = i s t = j), and conditional probability, Pr(s t = j I t ) as follows; Pr(s t = i, s t = j I t )=Pr(s t = i s t = j)pr(s t = j I t ). (3) The conditional probability updated with information at time t are given by: Pr(s t = i, s t = j I t )= f(y t,s t = i, s t = j I t ) f(y t I t ) f(y t s t = i, s t = j, I t ) Pr(s t = j I t ) =. f(y t s t = i, s t = j, I t ) Pr(s t = j I t ) i= j= (4)

by: and Assuming normality the density for y t conditional on s t,s t and I t is given Note that ( f(y t s t = i, s t = j, I t )= exp πf (i,j) t Pr(s t = i I t )= Pr(s t = i, s t = j I t ). j= v (i,j) t f (i,j) t Pr(s t = s t =)= Pr(s t = s t =) Pr(s t = s t =)= Pr(s t = s t =). ). (5) We also obtain x (i) t t = E[x t s t = i, I t ] Pr(s t = i, s t = j I t )x (i,j) t t j= =, Pr(s t = i I t ) M (i) t t = E[(x t x (j) t t ) s t = i, I t ] Pr(s t = i, s t = j I t )M (i,j) t t j= =. Pr(s t = i I t ) The above procedure should be repeated from t= to T to calculate the log likelihood: $(y θ) = T log[f(y t I t )] t= where θ = {µ,µ,φ,φ,σ,σ, Pr(s t = s t = ), Pr(s t = s t = )}. We choose different initial value sets of θ to find the global maximum likelihood value, since Markov regime switching models usually have many local maxima. The start-

ing values may be given by x (i) = µ i M (i) = σ i φ i P (s = I )=. B. Smoothing Procedure As before, in order to extract the volatility we need the smoother component. In order to get theses estimates we need some notation where and x (k,i) t T = E[x t s t+ = k, s t = i, I T ] (6) M (k,i) t T = E[(x t x (k,i) t T ) s t+ = k, s t = i, I T ] (7) Pr(s t+ = k, s t = i I T )=Pr(s t+ = k I T )Pr(s t = i I t ) Pr(s t+ = k s t = i),(8) Pr(s t+ = k I t ) Pr(s t+ = k I T )= Pr(s t = i I t )= Pr(s t+ = l, s t+ = k I T ), l= Pr(s t+ = k I t )= Pr(s t = i, s t = j I t ), j= Pr(s t+ = k, s t = i I t ) from the estimation procedure in Appendix A. The smoothing equations are i= where x (k,i) t T = x (i) t t + J (k,i) t M (k,i) t T = M (i) t t + J (k,i) t (x (k) t+ T x(k,i) t+ t ) (M (k) t+ T M (k,i) t+ t (k,i) )J t

Note that x (i) t T = M (i) t T = x t T = M t T = J (k,i) t = M (i) t t φ k(m (k,i) t+ t ). k= k= k= k= Pr[s t+ = k, s t = i I T ]x (k,i) t T Pr(s t = i I T ) Pr[s t+ = k, s t = i I T ]M (k,i) t T i= i= Pr(s t = i I T ) Pr[s t+ = k,s t = i I T ]x (k,i) t T Pr[s t+ = k,s t = i I T ]M (k,i) t T where t = T,T,...,. For t = T, we use the last values from estimation procedure; these are x (k,i) T T,x(i) T T,M(k,i) T T,M(i) T T, Pr(s T = k,s T = i I T ), Pr(s T = k I T ). 3

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Table Properties of Log-Squared Residuals A. S&P5 Index Volatility Returns (%) Log- Squared Residuals (Returns-Mean) Log- Squared Residuals Modified with Breidt and Carriquiry (996) Method (κ=.) Log- Squared Residuals Modified with Breidt and Carriquiry (996) Method (κ=.5) Log- Squared Residuals Modified with Breidt and Carriquiry (996) Method (κ=.) Daily Returns Mean.46 -.945 -.64 -.459 -.9 (A total number Standard Deviation.983.66.956.739.55 of 548 returns Skewness -.37 -. -.5.8.434 from 7 February Excess Kurtosis 4.937. -.868 -.867 -.78 99 to 7 February Jarque and Bera Statistics 687.36 4.458 8.96.34 36.93 ) Ljung and Box ().335 34.897 545.84 63.33 697.345 Ljung and Box (5) 95.94 37.533 38.8 66.69 87.545 Autocorrelation (Lag ).5.9.7.6.33 Autocorrelation (Lag 5) -.34.7.78.9.99 Autocorrelation (Lag ).43.33.7.83.9 Autocorrelation (Lag 5).6.99.37.43.47 Autocorrelation (Lag ) -.8.94..6.3 Autocorrelation (Lag 3) -..9.4.5.54 Autocorrelation (Lag 4)..84..3.38 Autocorrelation (Lag 5) -.3.88..4.5 Weekly Returns Mean.7 -..3.59.397 (A total number Standard Deviation.9.395.856.664.49 of 5 returns Skewness -.4 -.37 -.5..44 from 6 February Excess Kurtosis.743.7 -.7 -.89 -.786 99 to 7 February Jarque and Bera Statistics 8.463 39.5 6.66 4.957 8.6 ) Ljung and Box () 6.357 5.645 76.889 88. 98.54 Ljung and Box (5) 4.56 9.59 96.735 33.66 36.5 Autocorrelation (Lag ) -.94.99.7.3.8 Autocorrelation (Lag 5) -.5.35.67.83.98 Autocorrelation (Lag ).3.5.69.79.87 Autocorrelation (Lag 5).7.98..9.5 Autocorrelation (Lag ).43.64.8.46.58 Autocorrelation (Lag 3).3..3.8.3 Autocorrelation (Lag 4) -..6.37.47.55 Autocorrelation (Lag 5).45.97..6.3

B. FTSE Index Volatility Returns (%) Log- Squared Residuals (Returns-Mean) Log- Squared Residuals Modified with Breidt and Carriquiry (996) Method (κ=.) Log- Squared Residuals Modified with Breidt and Carriquiry (996) Method (κ=.5) Log- Squared Residuals Modified with Breidt and Carriquiry (996) Method (κ=.) Daily Returns Mean.4 -.556 -.339 -. -.74 (A total number Standard Deviation.3.3.85.666.496 of 66 returns Skewness -.5 -.84 -.34.6.36 from 7 February Excess Kurtosis.63.59 -.744 -.873 -.83 99 to 7 February Jarque and Bera Statistics 5.37 767.686 8.45 8.6 96.734 ) Ljung and Box () 4.9 83.55 389.479 443.857 5.353 Ljung and Box (5) 77.88 894.67 97.436 478.73 658.87 Autocorrelation (Lag ).59..6..6 Autocorrelation (Lag 5) -..3.36.4.48 Autocorrelation (Lag )..5.8.4.3 Autocorrelation (Lag 5) -.8.7...7 Autocorrelation (Lag ).6.89.7.3. Autocorrelation (Lag 3)..83.8.7.4 Autocorrelation (Lag 4) -.33.37.4.44.47 Autocorrelation (Lag 5).4.5.8.93. Weekly Returns Mean.99.47.93.47.548 (A total number Standard Deviation.47.49.854.66.488 of 5 returns Skewness.9 -.46 -.4 -.69.39 from 6 February Excess Kurtosis.5.7 -.73 -.947 -.99 99 to 7 February Jarque and Bera Statistics 9.57 335.635 6.5.989.438 ) Ljung and Box () 9.657 6.55 7.94 33.36 4. Ljung and Box (5) 59.8 64.69 95.459 9.86 4.7 Autocorrelation (Lag ) -.79.9.39.48.56 Autocorrelation (Lag 5) -.5.4.49.56.64 Autocorrelation (Lag ).67.5.69.8.9 Autocorrelation (Lag 5) -.53.4...8 Autocorrelation (Lag ) -.4.39.53.58.6 Autocorrelation (Lag 3) -.35 -.9 -. -..7 Autocorrelation (Lag 4) -.9.4.34.33.3 Autocorrelation (Lag 5).3.3.34.7.

Table Estimates of SV, SV with Regime Changing Mean and SV with MRS State Equation for Daily Volatility A. S&P5 Index Volatility SV Model SVMRS Model SVMRS Model with Restriction of φ = φ and σ η =σ η Smith's () Model Log-Squared Residuals Log-Squared Residuals Log-Squared Residuals BC Log-Volaitlity (κ=.5) Log-Squared Residuals Estimates STD Estimates STD Estimates STD Estimates STD Estimates STD µ -.976.455-4.6.78-9.4.676 -.86.65-6.546.9 µ -.6.99-3.796.64 -.334.68 -.75.64 φ.999..36.4.993..996.3.8.9 φ.6.36 σ η.44.6.87.8.5.6.66..89.5 σ η.548.4 σ ϕ.465.54..3.668.48.99... p (,).68.5.3..486..34.4 p (,).69.36.877..5..87.3 ML Values -67. -595.9-5854.5-4886.6-6.45 Notes: A total number of 66 returns from 7 February 99 to 7 February is used. B. FTSE Index Volatility SV Model SVMRS Model SVMRS Model with Restriction of φ = φ and σ η =σ η Smith's () Model Log-Squared Residuals Log-Squared Residuals Log-Squared Residuals BC Log-Volaitlity (κ=.5) Log-Squared Residuals Estimates STD Estimates STD Estimates STD Estimates STD Estimates STD µ -.549.5-4.84.537-7.69.467-3.383. -6.4.48 µ -.94.397 -.76.47 -.85.8 -.87.6 φ.99.5.899.4.985.5.986.5.79.8 φ.6.46 σ η.7.7.88.65.4.6..8.653.36 σ η.856.56 σ ϕ.96.4.84..49.36.99.9..46 p (,).7.53.57.3.38.9.35. p (,).75.5.89..595..897. ML Values -567.3-5494.9-543.9-4674.3-554.448 Notes: A total number of 548 returns from 7 February 99 to 7 February is used.

Table 3 Estimates of SV, SV with Regime Changing Mean and SV with MRS State Equation for Weekly Volatility A. S&P5 Index Volatility SV Model SVMRS Model SVMRS Model with Restriction of φ = φ and σ η =σ η Smith's () Model Log-Squared Residuals Log-Squared Residuals Log-Squared Residuals BC Log-Volaitlity (κ=.5) Log-Squared Residuals Estimates STD Estimates STD Estimates STD Estimates STD Estimates STD µ -.5.396 -.77.474-5.973.46 -.93.447-4.87.67 µ.738.44 -.68.73.54.456.56.39 φ.99.7.46.53.995.4.994.9.9.64 φ.4.5 σ η.88.36.673.9.5.3.69.69.5.835 σ η.368.79 σ ϕ.8.9..5.56.6.5.56..867 p (,).56.86.3.3.39.47.5.47 p (,).78.57.886.47.65.53.886.3 ML Values -79.4-33.8-5.6-968. -45.73 Notes: A total number of 5 returns from 6 February 99 to 7 February is used. B. FTSE Index Volatility SV Model SVMRS Model SVMRS Model with Restriction of φ = φ and σ η =σ η Smith's () Model Log-Squared Residuals Log-Squared Residuals Log-Squared Residuals BC Log-Volaitlity (κ=.5) Log-Squared Residuals Estimates STD Estimates STD Estimates STD Estimates STD Estimates STD µ.5.7 -.4.68-5.88.46 -.658. -5.86.53 µ.75.94 -.7.6.9.8.654. φ.969.9.33.9.97..97.4.9.3 φ..38 σ η.4.58.795.7.64.39..3.646.8 σ η.9.56 σ ϕ.357.3..399.54.78.893.37..4 p (,).55.83.8..33.36..48 p (,).63.45.899.47.65.43.899. ML Values -98. -6.8-3.6-958.8-4.67 Notes: A total number of 5 returns from 6 February 99 to 7 February is used.

Figure Smoothed Volatility and State Probabillity for S&P5 Index Daily Volatility 5 a. SV Model 4 3 Absolute Values of Residuals Smoothed Fundamental Volatility 5 4 3 b. SVMRS Model Absolute Values of Residuals Smoothed Fundamental Volatility Probability of State - - -3-4 5 4 3 c. SVMRS Model with Restriction of φ=φ =φ and ση η=σ =ση Absolute Values of Residuals Smoothed Fundamental Volatility Probability of State - - -3-4 5 4 3 d. SVMRS Model with Restriction of φ=φ =φ and ση η=σ =ση for BC Modified Log-Volatility Absolute Values of Residuals Smoothed Fundamental Volatility Probability of State - - -3-4 Notes: A total number of 66 returns from 7 February 99 to 7 February is used.

Figure Smoothed Volatility and State Probabillity for FTSE Index Daily Volatility 5 a. SV Model 4 3 Absolute Values of Residuals Smoothed Fundamental Volatility 5 4 3 b. SVMRS Absolute Values of Residuals Smoothed Fundamental Volatility Probability of State - - -3-4 5 4 3 c. SVMRS Model with Restriction of φ=φ =φ and ση η=σ =ση Absolute Values of Residuals Smoothed Fundamental Volatility Probability of State - - -3-4 5 4 3 d. SVMRS Model with Restriction of φ=φ =φ and ση η=σ =ση for BC Modified Log-Volatility Absolute Values of Residuals Smoothed Fundamental Volatility Probability of State - - -3-4 Notes: A total number of 548 returns from 7 February 99 to 7 February is used.

Figure 3 Smoothed Volatility and State Probabillity for S&P5 Index Weekly Volatility 5 3a. SV Model 4 3 Absolute Values of Residuals Smoothed Fundamental Volatility 5 4 3 3b. SVMRS Model Absolute Values of Residuals Smoothed Fundamental Volatility Probability of State - - -3-4 5 4 3 3c. SVMRS Model with Restriction of φ=φ =φ and ση η=σ =ση Absolute Values of Residuals Smoothed Fundamental Volatility Probability of State - - -3-4 5 4 3 3d. SVMRS Model with Restriction of φ=φ =φ and ση η=σ =ση for BC Modified Log-Volatility Absolute Values of Residuals Smoothed Fundamental Volatility Probability of State - - -3-4 Notes: A total number of 5 returns from 6 February 99 to 7 February is used.

Figure 4 Smoothed Volatility and State Probabillity for S&P5 Index Weekly Volatility with Smith () Model 5 4a. SV Model 4 3 Absolute Values of Residuals Smoothed Fundamental Volatility 5 4 3 4b. SVMRS Model Absolute Values of Residuals Smoothed Fundamental Volatility Probability of State - - -3-4 5 4 3 4c. SVMRS Model with Restriction of φ=φ =φ and ση η=σ =ση in Smith () Model Absolute Values of Residuals Smoothed Fundamental Volatility Probability of State - - -3-4 5 4 3 4d. SVMRS Model with Restriction of φ=φ =φ and ση η=σ =ση for BC Modified Log-Volatility in Smith () Model Absolute Values of Residuals Smoothed Fundamental Volatility Probability of State - - -3-4 Notes: A total number of 5 returns from 6 February 99 to 7 February is used.