2003 Kluwer Academic Publishers. Manufactured in The Netherlands.
Examining the Volatility of Taiwan Stock Index Returns
via a Three-Volatility-Regime Markov-Switching
ARCH Model
MING-YUAN LEON LI
Department of Banking and Finance, National Chi Nan University 1, University Rd., Puli, Nantou, Taiwan 545 Tel.: 886-49-291-0960 Ext. 4984, Fax: 886-49-291-4511
E-mail: [email protected]
HSIOU-WEI WILLIAM LIN
Department of International Business, National Taiwan University, Taiwan No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan 106 Tel.: 886-2-2363-0231 Ext. 3657, Fax: 886-2-2363-8399
E-mail: [email protected]
Abstract. This study adopts Hamilton and Susmel’s (1994) Markov-switching ARCH (hereafter SWARCH) model to examine the volatility of the valued-weighted Taiwan Stock Index (hereafter TAIEX) returns. We also conduct sensitivity tests on comparison observations of Dow Jones and Nikkei stock indices. Our empirical findings are consistent with the following notions. First, the SWARCH model appears to outperform the competing ARCH and GARCH models in estimating the volatilities of TAIEX. Second, the three-volatility-regime setting is descriptive for TAIEX and Nikkei. In contrast with Hamilton and Susmel (1994), the contemporaneous Dow Jones adopted in this paper has only two regimes. Our test results suggest that the optimal number of volatility regimes is sensitive to the choice of sample periods. Third, our empirical results also lend an explanation to such phenomenon: the probability that TAIEX directly moves from a low (high) volatility regime to the high (low) volatility regime approaches zero, whereas TAEIX happened to be in a low volatility regime during the pre-financial-crisis period from April, 1996 to July, 1997. These can explain why Taiwan was one of Asia’s few star performers compared with recession-hit neighbors during the first eighteen months of Asia’s financial crisis.
Key words: Markov-switching ARCH models, stock index returns, Asian financial crisis JEL Classification: C53, C22, G15
1. Introduction and institutional background
The Engle’s (1982) ARCH (auto-regressive conditional heteroskedasticity) or Bollerslev’s (1986) are the most commonly used methods to characterize the volatility of stock returns. However, while estimating financial and macroeconomic series, some economists found that both ARCH and GARCH would encounter high persistence in volatility and lower accuracy in the predicting performance.
Due to the above reasons, Bollerslev and Engle (1986) introduced the modified IGARCH (integrated GARCH) model. One could thus examine if a data series of interest is an IGARCH via testing whether conditional variance is a unit root series. Nevertheless, many prior studies such as French, Chou (1988), Schwert and Seguin (1990), as well as Bollerslev,
Chou and Kroner (1992) fail to reject the null hypothesis of unit roots. Diebold (1986) as well as Lamoureux and Lastrapes (1990) argued that the high persistence is caused by the structural changes in the volatility process during the estimation period.
By examining the figures of the TAIEX, Nikkei, and Dow Jones, one can easily find that the volatilities of stock index are substantially bigger (or smaller) than the average during some periods. Thus, one cannot take the overall sample period variance as a constant. Moreover, assigning dummy variables to partition the sample period as various phases to control the different volatility levels demands a subjective designation of cutoff dates, and cannot effectively predict the timing for structural changes. In contrast, if a model that can self-sufficiently partition different regimes and verify the timing of structural changes based on historical data, we believe that it can better control how the stock market behaves.
Following the above line of thought, Hamilton and Susmel (1994) established the SWARCH model incorporating Markov-switching and ARCH models. The characteristics of their SWARCH models are to use the Markov-switching setting to control the structural changes, represent changes in the regime as changes in the scale of the ARCH models, mit-igate the high persistence of variance in ARCH model, and therefore enhance the accuracy in predicting the future stage volatility. Cai (1994) also modeled a similar setting to analyze the volatility in Treasury bill yield in the United States. Moreover, Turner, Startz and Nelson (1989), Dueker (1997), and Schaller and Norden (1997) did the similar applications to stock market analysis. Ramchand and Susmel (1998) established the bivariate SWARCH model to analyze the correlations between the major stock markets and the American stock market. Hamilton (1988) modified the setting of Goldfeld and Quandt (1973) into the Markov-switching model, that was adopted by many time series studies regarding the influence of economic and political events on the macroeconomic and financial variables. These studies include: Hamilton (1988), Driffill (1992), Sola and Drifill (1994), Garcia and Perron (1996) as well as Gray (1996) on short-term interest rates; Hamilton (1989), Lam (1990, 1996), Ghysels (1994), Durland and McCurdy (1994), Filardo (1994), Diebold and Rudebusch (1996), Hamilton and Lin (1996) as well as Huange, Kuan and Lin (1998) on the aggregative output and the business cycles; Engel and Hamilton (1990) as well as Engle (1994) on the exchange rates; Bianchi and Zoega (1998) as well as Montgomery et al. (1998) on the unemployment rate; and Chow (1998) on the future markets.
The primary purposes of this study are to search and discuss the adequate model to explore the volatility of the Taiwan stock market. We employ the SWARCH model of Hamilton and Susmel (1994) and the GARCH as well as ARCH models introduced by Engle (1982) and Bollerslev (1986). In addition, we also adopt the contemporary Dow Jones and Nikkei indices as our comparison observations to investigate the sensitivity of the test results related to TAIEX. Specifically, we are interested in the following issues. First, Hamilton and Susmel (1994) presented that the three-volatility-regime volatility setting could adequately picture the stock market in the United States. Does the three-volatility-regime volatility setting also exist in the Taiwan and Japan stock markets? Second, is the three-volatility-regime setting for the stock market consistent during different estimation periods? Specifically, are the numbers of the volatility state for the stock market unchangeable, if we adopt various estimation periods? Third, does the forecasting performance of the SWARCH models in estimating TAIEX also perform better than the GARCH and ARCH models? Moreover,
what are the characteristics of the volatility predicted by the competing models? Fourth, do the various regime periods identified by the SWARCH models match the historical events? Moreover, do the various regime settings for the volatility of the stock market picture the Asian financial crisis?
This study also contributes to practices in the finance fields. The research designs and test results add to the tasks of measuring value-at-risk (VaR), pricing derivative securities, as well as implementing risk management systems. First, the process of measuring VaR often starts with depicting historical returns of stocks, bonds or foreign currencies on a histogram. And yet VaR studies for TAIEX security returns without filtering out structural changes would find that the return series deviates from the normal distribution and follows the t distribution with a low degree of freedom. After filtering out the structural changes, it may generate a series that is normally distributed. Similarly, as documented in many prior studies such as Duffie and Pan (1997), the distributions of stock and bond market returns appear to have fat tails. Furthermore, there exist problems of twin peak, which are the results often found in the New Taiwan dollar value per U.S. dollar histograms. To handle these problems, one cannot ignore the issue of structural changes of financial variables. Second, in spite of the availability of derivative security pricing packages, the key to the pricing tasks is to determine how far one can go for historical data to estimate the variance. Statistically, an increase in sample size can eliminate measurement errors but inevitably gains exposure to structural changes. Third, the effectiveness of a firm’s risk management relies heavily on how accurately it could estimate the probabilities of extreme events. Nevertheless, transitional volatility in financial variables may dwarf most typical scenario analyses, stress tests, and risk limit guidelines.
The next section presents the Hamilton and Susmel’s (1994) SWARCH model specifica-tions. Section 3 presents the empirical results in estimating TAIEX, provides the test results regarding our comparison observations of Dow Jones and Nikkei indices, and discusses the Asian Financial crisis impacting on the stock markets. Section 4 concludes the study.
2. Model specification
Denoting the rate of return for the market index asyt, Hamilton and Susmel (1994) estab-lishes the following settings:
yt = φ0+ φ1yt−1+ · · · + φt−pyt−p+ et (2.1)
ut =
htvt, νt ∼ Gaussian or Student t distribution (2.2)
et =√gstut (2.3)
ht = a0+ a1u2t−1+ a2u2t−2+ · · · + aqu
2
t−q (2.4)
Here, st is an unobservable state variable with possible outcomes of 1, 2, . . . , k, and is assumed to follow a first-order Markov chain process:
p(st= j | st−1 = i, st−2 = k, . . . , yt−1, yt−2, . . .) = p(st = j | st−1= i) = pi j (2.5)
Also define the transition probability matrix: P = p11 p21 · · · pk1 p12 p22 · · · pk2 .. . ... · · · ... p1k p2k · · · pkk (2.6)
The sum of elements in each and every row in the above matrix should be equal to 1. According to Hamilton and Susmel (1994), Eqs. (2.1) to (2.7) are known as the k-state, q-th order Markov-switching ARCH models, denoted SWARCH(k, q). Moreover, Eqs. (2.3) and (2.4) show that ut is a typical ARCH(q) process, and Eq. (2.2) states that when st = 1, et is equivalent to ut in ARCH(q) multiplied by a constant√g1. Whereas,
when st = 2, et equals utin ARCH(q) multiplied by a constant√g2, . . . , etc. Without the
loss of generality, one can normalize g1, the coefficient for regime I, to be unity, whereas
gi ≥ 1, i = 2, 3, . . . , k, for the other regimes. Specifically, etis equal to utin an underlying fundamental ARCH(q) process multiplied by the square root of regime switching constant
gst. In a special case with g1= g2 = · · · = gk= 1, error term etin the model would follow
the fundamental ARCH(q) process.
It is worth noting that Hamilton and Susmel (1994) adopted a more generalized specifi-cation of Student t distribution with unit variance and a degree of freedom of v for the error term,vt.1Moreover, v, the degree of freedom is also regarded as an unknown parameter in the models. The conditional probability for etcan be expressed as:
f (et | st, st−1, . . . , st−q, et−1, . . . , et−q) = {(ν + 1)/2}√ (ν/2) (v − 2) −1/2σ−1 t × 1+ e 2 t σ2 t(ν − 2) −(ν+1)/2 (2.7) Given g1 = 1, k
j=1 pi j = 1, i = 1, 2, . . . , k and 0 ≤ pi j ≤ 1, one cansearch for
φ0, φ1, . . . , φp, a0, a1, . . . , aq, g2, g3, . . . , gk, p11, p12, . . . , pkk, v that maximize the foll-owing likelihood function:
=
T
t=1
ln f (yt | yt−1, yt−2, . . .) (2.8)
Although the regime variable stin the model is unobservable, one can still use the data to estimate the specific regime probabilities in time t . When the information set for estimation includes signals dated up to time t , the regime probability is a p(st | yt, yt−1, . . .), or filtering
probability. On the other hand, one could also use the overall sample period information set
to estimate the state at time t : p(st | yT, yT−1, . . .), or smoothing probability. In contrast, a predicting probability denotes the regime probability for an ex ante estimation, with the information set including signals dated up to the period t− 1: p(st | yt−1, yt−2, . . .).
3. Empirical results
This section adopts the weekly returns of TAIEX, Dow Jones and Nikkei, provided by the Taiwan Economic Journal (TEJ). The original data are obtained weekly from February 2, 1981 to September 25, 1998, which include 918 observations.
In this paper, we set the order of auto-regression setting of the variable, yt, to be unity, namely, p= 1, as well as the number of orders in ARCH to be two,2namely, q= 3. We use OPTIMUM, a package program from GAUSS, and the built-in BFGS3 algebra to get the negative minimum likelihood function value of all models. In the SWARCH models with the two-volatility-regime (three-volatility-regime) settings, we randomly generate 100 (20) sets of initial values. We then derive the ML function value for each of the 100 (20) sets of initial values, respectively. The mapped converged measure of the greatest ML function value then serves to estimate the parameter.
3.1. Searching the most adequate model for TAIEX
Table 1 presents the associated statistics for each of the competing models. We take the CV (constant variance) model as a benchmark and examine the ARCH, GARCH and SWARCH
Table 1. Summary statistics for the competing model in estimating TAIEX
Model Param.a L (θ)b AICc Schwarzd Freedome Persistencef
Gaussian CVg 3 −2687.8 −2690.8 −2694.6 – – Student t CV 4 −2593.3 −2597.3 −2620.6 3.1(0.35) – Gaussian GARCH(1,1)h 5 −2505.4 −2510.4 −2522.5 – 0.96 Student t GARCH(1,1) 6 −2492.0 −2498.0 −2512.5 8.0(1.93) 0.97 Gaussian ARCH(2)i 5 −2555.0 −2560.0 −2572.1 – 0.76 Gaussian SWARCH(2,2)j 8 −2492.0 −2500.0 −2519.3 – 0.5 Student t ARCH(2) 6 −2521.4 −2524.4 −2541.9 5.5(0.93) 0.8 Student t SWARCH(2,2) 9 −2485.9 −2494.9 −2516.6 9.1(2.43) 0.55 Student t SWARCH(3,2) 12 −2470.7 −2482.7 −2511.6 19.8(11.0) 0.29 1. The data sources are Taiwan Economic Journal (TEJ), and the time periods are from February 2, 1981 to September 25, 1998, includes 918 observations.
2. The values in the parentheses present the estimates for the standard deviations. 3.aDenotes the parameter numbers,
bDenotes the maximum value of likelihood function,
cAIC[]ML function value-N . N is the model parameter numbers,
dSchwarz value= ML function value-(N/2)x ln(T ). T is the numbers of sample, ePlease refer to the paper’s Eq. (2.7),
fPlease refer to Hamilton and Susme (1994) for the detail calculating, gConstant Variance (CV) model:σ2
t = α0, hARCH(2) model:σ2
t = α0+ α1e2t−1+ α2e2t−2,
iGARCH(1,1) model:σ2
t = α0+ α1e2t−1+ βσt2−1,
jPlease refer to the paper’s Eqs. (2.1) to (2.8) for the detail setting.
4. As to the summaries of the statistics including AIC, Schwarze value and LR tests, we conclude that the student
models, respectively. As Table 1 shows, for criteria including the maximum value of like-lihood function, Akaike’s (1976) AIC and Schwarz’s (1978) Schwarz value, specifications under the Student t distribution turn out better than those under the Gaussian distribution. Therefore, we focus on discussing the specifications with the Student t distribution for the subsequent analyses.
First, compare the Student t C.V. and Student t ARCH(2) models, with the former serves as a special case for the latter with the constraints: a1 = a2 = 0. The associated
LR (likelihood ratio) statistic is 2(2593.3− 2521.4) = 143.8, and the P-Value equals 5.9 × 10−32. This result supports us to reject the null hypothesis of CV specifications.
Comparing the Student t ARCH(2) model and the Student t SWARCH(2,2) model, the former specification serves as a special case for the latter under the constraint of g1 =
g2 = 1. If the assumption of χ2 distribution still holds, LR statistic will be (2521.4−
2485.9) = 71 (P-Value = 2.6 × 10−15). With such an trivial P-Value, despite that the LR statistic under this condition no longer follows the standard χ2 distribution, we can
still confidently reject the specification that there is one-volatility-regime setting under null hypothesis.4The LR statistic between Student t SWARCH(3,2) and SWARCH(2,2) is equal
to 2(2485.9− 2470.7) = 30.4 (P-Value = 1.13 × 10−6). This result supports us to reject the null hypothesis of the two-volatility-regime specification.5
As for the comparison between SWARCH and GARCH models, we cannot use the LR statistic tests since these two models are not strictly nested.6 AS to the AIC statistics for relative performance, SWARCH(3,2) model appears to outperform the GARCH model. Also, in terms of Schwarz value statistics, SWARCH(3,2) also outperforms the GARCH model.7
The last column of Table 1 presents the volatility persistence of each of the competing models.8First, the persistence measure for the Student t GARCH(1,1) model is 0.97, which
is not very different from one. It demonstrates that the volatility in the GARCH models has the high persistence problems. This result is consistent with the finding of Bollerslev and Engle (1986) with IGARCH. Nevertheless, with SWARCH(2,2) and SWARCH(3,2) models, the persistence measures are 0.55 and 0.29, respectively, and are significantly less than the ARCH(2) models.
Therefore, we conjecture that Bollerslev and Engle’s IGARCH model is resulted from certain structural changes occurred during the estimation period. When one applies Markov-switching model to control the structural changes, the persistence of volatility in the under-lying ARCH component decreases significantly.
Let us attract attention on the more generalized setting in which error term follows a Student t distribution. Table 1 shows that the estimates of the degrees of freedom of the distribution measured via either ARCH or GARCH models are significantly greater than that via the CV model but less than that via SWARCH(2,2) model. Furthermore, the de-gree of freedom estimate via SWARCH(3,2) is even greater than that of SWARCH(2,2). This result supports the notion that with SWARCH(3,2) specification, the distribution of the error term measure would be the closest to normal distribution. Namely, the non-normality properties of the stock market returns will be significant dismissed when we use the Markov-switching models to control the structural changes of the return volatilities.
Table 2. One-period-ahead (one-week-ahead) forecast errors of the competing models in estimating TAIEX
Model MSE MAE [LE]2 |LE|
Student t CV 3025.8 26.87 9.59 2.30
Student t ARCH(2) 2759.7(0.09) 23.29(0.13) 7.26(0.24) 1.91(0.17) Student t GARCH(1,1) 2700.7(0.11) 22.77(0.15) 6.48(0.32) 1.80(0.22) Student t SWARCH(2,2) 2366.1(0.22) 21.48(0.20) 6.40(0.33) 1.79(0.22) Student t SWARCH(3,2) 2279.5(0.25) 20.60(0.23) 6.75(0.30) 1.78(0.23) 1. The numbers of observation, time period, and model setting are same with Table 1.
2. The four loss functions are defined as the follows:
MSE= T−1 T t=1 e2t − σt2 2 , MAE = T−1 T t=1 e2t − σt2, [LE]2 = T−1 T t=1 lne2t − lnσ2 t 2 , |LE| = T−1 T t=1 lnet2 − lnσ2 t
3. In the parenthesis, we present the percentage of reduction in forecast error each of the model achieves when used to substitute the CV model.
4. Summaries the forecasting performance, the student t SWARCH(3,2) model outperforms another models in estimating TAIEX.
Now we discuss the forecast performance of each of the competing models. Table 2 (Table 3) presents the four different functions of forecast errors of our one-period-ahead forecast (four-week- and eight-week-ahead forecasts.) We also adopt the CV model as a benchmark, exploring the extent of which each model helps reduce the forecast error when it replaces CV. Table 2 demonstrates that SWARCH(3,2) associates with the minimum forecast error for one-period ahead forecasts and is thus the best forecast model.9 The
runners-up are SWARCH(2,2), GARCH(1,1) and ARCH(2) models. Table 3 shows that on both one- and two-month ahead forecasts, SWARCH(3,2) also outperforms GARCH(1,1). Accordingly, the three-volatility-regime setting, specifically, Student t distribution SWARCH(3,2) model is statistical powerful in forecasting. We conclude that it is the most suitable model for picturing the volatilities of TAIEX.
Table 3. The percentage of four- and eight-week-ahead forecast error reductions relative to the CV model in
estimating TAIEX
Model MSE MAE [LE]2 |LE|
Four-week-ahead forecast Student t GARCH(1,1) 0.03 0.11 0.28 0.18 Student t SWARCH(3,2) 0.18 0.22 0.27 0.20 Eight-week-ahead forecast Student t GARCH(1,1) 0.01 0.10 0.23 0.14 Student t SWARCH(3,2) 0.17 0.20 0.25 0.19
1. The numbers of observation, time period, and model setting are same with Table 1. 2. The forecasting statistic settings are same with Table 2.
3. Summaries the performance in four- and eight-week-ahead forecasting, the student t SWARCH(3,2) model outperforms the Student t GARCH(1,1) in estimating TAIEX.
3.2. More discussion on the empirical results for TAIEX
As shown in the above section, in terms of statistical significance, AIC and Schwarz value conformance, and forecast accuracy, SWARCH(3,2) all appears to be the most descriptive model. We thus estimate the parameters of the model and have the following result. (With the estimated standard deviation in the parenthesis10):
yt = 0.24 + 0.10yt−1+ et (0.10) (0.04)
et =√gst · ut
where ut = ht· vtandvt follows a Student t distribution with its variance equals to unity and its degree of freedom equals 19.84(11.03)
h2t = 3.98 + 0.07u 2 t−1+ 0.06u 2 t−2 (0.43) (0.04) (0.05) g1 = 1, g2= 3.34(0.51), g3= 15.62(3.16) P = 0.9912 (0.0067) 0.0084 (0.0080) 0 0.0088 (0.0067) 0.9861 (0.0121) 0.0166 (0.0113) 0 0.0055 (0.0041) 0.9834 (0.0113)
Any element in the j -th row and the i -th column of the matrix P represents the probability that the series switches from regime i to regime j .
When we first estimate the SWARCH(3,2) model, we only impose 0 ≤ pi j ≤ 1 and k
j=1pi j = 1 constraints on the transition probabilities. Nevertheless, some elements of the switching probability matrix appear to approach zero. Specifically, p13, the probability
of a regime I to regime III switch, is equal to 3.3 × 10−7. Similarly, p31, the probability of
a regime III to regime I switch, equals 8.33 × 10−12. So we set the above two probabilities to be 0, taking this two parameters as known constants for the purpose of calculating the second derivatives of the log-likelihood and obtain the standard error.11 It is worth noting
that Hamilton and Susmel (1994) also found that the two different probabilities in the transition probability matrix are zero, specifically, the probability of regime II to regime I, and the probability of regime I to regime III.
In the SWARCH(3,2) model, we use gst, with st = 1, 2, 3, to represent market volatility under the three regimes. In our setting, we normalize g1to be unity. Examine the estimates
for g2 and g3 to be 3.34 and 15.62, respectively. The result suggests that the volatility of
three measures of volatility differ significantly from one another. Thus we take the regimes I, II and III as the low, medium, and high volatility regimes, respectively. The finding that
p31 = p13 = 0 suggests that TAIEX cannot enter a low volatility regime directly from
a high volatility regime without passing the medium volatility regime, and vice versa. In contrast, whereas Hamilton and Susmel (1994) also document the property of p13 = 0 in
the U.S. stock market, concluding that the U.S. market p31does not approach zero. Namely,
the U.S. stock market returns could directly switch from a high volatility regime to a low volatility regime.
Interestingly, the first-ordered auto-regression coefficientψ1significantly deviates from
zero. Namely, our adoption of the setting for probability distribution and the specification with heterogeneous conditional variance do not alter the conclusion that the weekly stock index returns are positively correlated.
3.3. Examining the volatility of Dow Jones and Nikkei via competing models
This section presents the results of sensitivity test for the comparison indices of Dow Jones and Nikkei Series. The result of our above empirical tests on TAIEX suggests that the Student
t SWARCH(3,2) is the most adequate setting for picturing the volatilities of TAIEX. One
would ask, nevertheless, whether such ranking would hold for different economies and for different sample periods. We compare the test groups, Dow Jones and Nikkei. For comparing the difference among the various stock markets, we also adopt the same test period from February 2, 1981 to September 25, 1998.
Tables 4 and 5 present the empirical results of the Likelihood function value, AIC and Schwarz values as well as persistence statistics for Dow Jones and Nikkei, respectively. Consistent with the findings for TAIEX, the Student t specification outperforms Gaussian distribution in the statistical tests including the AIC values and Schwarz criteria. Moreover, for both Dow Jones and Nikkei, the three-volatility-regime (single-regime) specification has the greatest (smallest) degree of freedom for the Student t distribution for the error
Table 4. Summary statistics for the competing model in estimating Dow Jones
Model Param. L (θ) AIC Schwarz Freedom Persistence
Gaussian CV 3 −1839.8 −1842.8 −1850.0 – – Student t CV 4 −1762.2 −1758.2 −1775.8 5.1(0.80) – Gaussian GARCH(1,1) 5 −1812.2 −1817.2 −1829.3 – 0.98 Student t GARCH(1,1) 6 −1746.9 −1752.9 −1767.4 6.3(1.30) 0.97 Gaussian ARCH(2) 5 −1766.1 −1771.1 −1783.2 – 0.41 Gaussian SWARCH(2,2) 8 −1745.6 −1753.6 −1772.9 – 0.26 Student t ARCH(2) 6 −1743.9 −1749.9 −1764.4 6.4(1.3) 0.34 Student t SWARCH(2,2) 9 −1734.0 −1743.0 −1764.7 7.3(1.6) 0.34 Student t SWARCH(3,2) 12 −1730.2 −1742.2 −1771.1 10.7(3.7) 0.17 1. The data sources, numbers of observation, time period, statistics, model settings and the other notations are same with Table 1.
2. As to the summaries of the statistics including AIC, Schwarze value and LR tests, we conclude that the student
Table 5. Summary statistics for the competing model in estimating Nikkei
Model Param. L (θ) AIC Schwarz Freedom Persistence
Gaussian CV 3 −1999.8 −2002.8 −2010.0 – – Student t CV 4 −1945.6 −1949.6 −1959.2 3.9(0.55) – Gaussian GARCH(1,1) 5 −1932.3 −1937.3 −1949.4 – 0.97 Student t GARCH(1,1) 6 −1911.9 −1917.9 −1932.4 5.3(1.04) 0.94 Gaussian ARCH(2) 5 −1967.3 −1972.3 −1984.4 – 0.34 Gaussian SWARCH(2,2) 8 −1892.7 −1900.7 −1928.0 – 0.24 Student t ARCH(2) 6 −1922.0 −1928.0 −1942.5 4.5(0.68) 0.46 Student t SWARCH(2,2) 9 −1888.5 −1897.5 −1919.2 10.8(4.0) 0.23 Student t SWARCH(3,2) 12 −1881.4 −1893.3 −1922.3 14.1(8.0) 0.24 1. The data sources, numbers of observation, time period, statistics, model settings, and the other notations are same with Table 1.
2. As to the summaries of the statistics including AIC, Schwarze value and LR tests, we conclude that the student
t SWARCH(3,2) model appears to outperform than another models in estimating Nikkei.
term. And the degree of freedom for the two-volatility-regime setting lies in between. We conclude that, when one uses the discrete state variable in the Markov-switching setting to filter out the structural changes in the volatility processes, he will significantly mitigate the non-normality problems for the stock market returns. We believe that the findings are robust, because they are consistent in TAIEX, Nikkei and Dow Jones.
We further discuss the most adequate numbers of volatility regime for the goodness of fitting Dow Jones and Nikkei. According to the statistical tests, our finding that a two- instead of three-volatility-regime setting is more adequate for U.S. stock market is inconsistent with the results of Hamilton and Susmel (1994), who adopted a three-volatility-regime setting. It is to be noted that Hamilton and Susmel’s (1994) sample period is from 1962 to 1987. In contrast, in this work the sample period is from 1981 to 1998, which does not cover the short-lived low-volatility regime during 1963–1965 identified by Hamilton and Susmel (1994). Our test results suggest that the optimal number of volatility regimes is sensitive to the choice of sample periods.12
According to the empirical results of Nikkei, with the statistical tests including the AIC, Schwarz value and LR ratio test, we conclude that the three-volatility-regime setting appears to be marginally superior alternative.13Interestingly, some characteristics of the transition
probability of three-volatility-regime setting in Nikkei significantly differ from those of TAIEX. Specifically, for Nikkei, the switching probability between the low- and the middle-volatility-regime approaches to zero ( p12= 1.35 × 10−8and p21 = 6.75 × 10−7), and are
quite different from TAIEX. This result suggests that one should take into account the special properties of each security market before imposing any preliminary constraints for the probability matrices while analyzing the market volatility for different economies.
3.4. The regime identification
We depict TAIEX and its weekly returns during the test period in Figures 1 and 2; Figures 3– 5 represent the predicting and smoothing probabilities of the high, medium, and low
Figure 1. The weekly level value of TAIEX during the test period, 1981:02 to 1998:09.
Figure 3. Regime III smoothing and predicting probabilities with SWARCH(3,2) model during the test period,
1981:02 to 1998:09. ---: smoothing Probability, ---: predicting probability.
Figure 4. Regime II smoothing and predicting probabilities with SWARCH(3,2) model during the test period,
Figure 5. Regime I smoothing and predicting probabilities with SWARCH(3,2) model during the test period,
1981:02 to 1998:09. ---: smoothing probability, ---: predicting probability.
volatility regime, respectively. We find, just as its name imply, the smoothing probabili-ties are smoother than the predicting probabiliprobabili-ties. Because the predicting probabiliprobabili-ties are derived from prior period data, such algorithm demands less, if any, ex post signals. At the standpoint of T, at the end of the test period, one could more accurately measure the smooth-ing probability with more information. By the same logic, since we use the information from the current period in measuring the filtering probability, its accuracy and smoothness rank in between those of predicting and smoothing probabilities.
We conclude that the market will be in a high, medium or low volatility regime if the associated smoothing probability is greater than or equal to 0.5. As can be summarized form Table 6, the periods of the beginning of 1997 to February, 1983, June, 1983 to February, 1987 and April, 1996 to August, 1997 were of the low volatility regimes, whereas May, 1987 to January, 1989 and March, 1990 to March, 1991 were both high volatility periods. The remaining observations are of the medium volatility regime.
Let us first focus on the two periods, in which TAIEX was based on the high volatility regime. By examining the history of Taiwan, we find that both the periods of high volatility for TAIEX are accompanied by some political and economical events. Specifically, the critical events occurred from 1987 to 1989 which included (1) the strongly expecting NT dollars to appreciate, and hot money chasing speculative profit poured into TAIEX in 1987, (2) NYSE crash in October 1987 and (3) the department of Taiwan Treasury initiates capital gain taxes in September, 1998. Moreover, the second period for the high volatility in TAIEX was due to bubble burst.14
Now let us examine the impact of the East Asian financial crisis emerged in the middle of 1997. Interestingly, although Taiwan was not exempt from the big crisis, Taiwan’s economy
Table 6. The periods of the various volatility regimes for TAIEX and Nikkei
Low-volatility-regime Middle-volatility-regime High-volatility-regime
Nikkei 1980:02–1986:07 1986:11–1987:04 1986:08–1986:10 1988:01–1990:02 1991:02–1991:11 1987:05–1987:12 1992:12–1993:10 1990:02–1991:01 1994:04–1994:12 1991:12–1992:11 1995:10–1997:08 1993:11–1994:03 1995:01–1995:09 1997:09–1998:12 TAIEX 1980:02–1983:01 1983:02–1983:06 1997:05–1988:12 1983:07–1987:01 1987:01–1987:04 1990:02–1991:02 1996:04–1997:07 1989:01–1990:01 1991:02–1996:03 1997:08–1998:12
1. The data sources, numbers of observation, and time period are same with Table 1.
2. We conclude that the stock market was in a high, medium or low volatility regime if the associated smoothing probability is greater than or equal to 0.5.
3. TAIEX was at a middle volatility during the Asian financial crisis period occurred in the middle of 1997, in contrast, NIKKEI was at a high volatility regime during the same period.
was not as badly hurt as compared to the other East Asian countries. Specifically, TAIEX was at a middle volatility, and was in contrast with our finding with respect to NIKKEI, which was highly volatile during the East Asian financial crisis. Various concurrent theories aim to provide explanations to the difference in impacts. But we would like to rethink the issue from a different aspect.
Let us review the TAIEX history before the occurrence of Asian financial crisis. The twice missile crisis and another bank runs occurred during the period between January and March of 1996. Bear market fueled by antagonism between Taiwan and PROC followed the presidential election. In March 1996, the TAIEX dropped further to 4,809 points. Thereafter, however, both the stock market and the foreign exchange market turned to be rather stable. Comparing our empirical results, TAIEX was at a low volatility period from April 1996 to August 1997, in contrast, NIKKEI was at a middle volatility regime during the same period. Moreover, our test results for TAIEX indicate that p13, the probability that TAIEX
directly moves from the low volatility regime to the high volatility regime, approaches zero. In contrast, the probability, p23, that moves from the middle volatility to the high volatility
regime for both TAIEX and NIKKEI were significantly different from zero, and larger than the probabilities p13.15 So, the Asian financial crisis that occurred in the middle of 1997
influenced Nikkei from the middle to the high volatility stage, but only made TAIEX from the low to the middle volatility stage.
These can explain why Taiwan was one of Asia’s few star performers compared with its recession-hit neighbors during the first eighteen months of Asia’s financial crisis.16We conjecture that if there were no missile crisis and no surge in foreign capital inflows into TAIEX in March 1996, Taiwan stock market would have been stayed at the medium volatility regime, and thus could have had the high volatility crashes when the East Asian financial crisis occurred. Nevertheless, because the missile crisis destroyed the last bubble of Taiwan
stock market and brought TAIEX to a low volatility stage and the almost impossibility for low-switching to high-volatility in TAIEX, the East Asia financial crisis shook but did not sink TAIEX in 1997. Our empirical result provides an alternative explanation for the scenario.
4. Concluding remarks
This study examines how competing models describe the volatility of TAIEX, and also collects Nikkei and Dow Jones for conducting sensitivity tests. The Markov-switching setting in the SWARCH models appears to significantly diminish the problem of high persistence in the volatility experienced by ARCH models. We also document that a three- rather than two-volatility-regime setting with the error term following a Student
t distribution outperforms the other specifications for TAIEX. Moreover, we provide a
potential explanation for the better performance in Taiwan stock market during the early stage of the East Asian financial crisis. Furthermore, we apply the competing models to U.S. and Japanese market returns in the sensitivity tests. Our findings indicate that the relative performance of each SWARCH specification varies with sample-period and economies.
Notes
1. Note that a Student t distribution would converge to a normal distribution as the degree of freedom approaches infinity. Therefore, the specification of Student t (normal) distribution is more (less) generalized.
2. Among all specifications, the third-order ARCH parameter estimate appears to be insignificantly different from zero. Therefore, for the specifications for ARCH, we only take into account the second order setting. 3. Boyden, Fletcher, Goldfarb, and Shanno (BFGS) algebra is effective for deriving the maximum value of the
non-linear likelihood functions. See Luenberger (1984).
4. With the constraint of g1 = g2 = 1, one would encounter an unidentified problem under null hypothesis
when testing the model. Hansen (1991, 1992) derive the limiting distribution for the statistic. And yet its implementation requires complicated computing work.
5. As to the four-regime SWARCH(4,2) specification, the empirical result is not robust, most of samples fall in some regimes, only rare outliers fall in another regime.
6. It is worth noting that we cannot combine the Markov-switching and GARCH model, except Gray (1996). Please refer to Hamilton and Susmel (1994) for a more detailed discussion.
7. Please refer to Schwarz (1978) for Schwarz value and Akaike (1976) for AIC.
8. Please refer to Hamilton and Susmel (1994) for the calculations for the volatility persistence of the competing models adopted in this paper.
9. As measured by LE squares, SWARCH(3,2) model slightly under-perform both GARCH(1,1) and SWARCH(2,2) model. Nevertheless, SWARCH(3,2) model outperforms the other models with all three other criteria. Thus we conclude that SWARCH(3,2) model is the optimal alternative in forecasting tasks. 10. We estimate the standard deviation via the second derivative of the likelihood function.
11. The estimate is economically and statistically insignificant. Thus the associated Hessian matrix is not positive definite.
12. Our empirical results also indicate that if one exclude the two prior high-volatility regimes and adopt a test period beginning in 1991, then he only needs a two-volatility-regime setting to describe the dynamics of TAIEX.
13. The LR statistic for SWARCH(3,2) versus SWARCH(2,2) equals 2× (1888.5 − 1881.4) = 14.2, with a significance level of 0.0026. Thus we can reject the specification of two-volatility-regime at 1%.
14. The TAIEX hit the new high of 12,495 points again in February 1990, and first exceed 10,000 points. Then the bubble burst, the index slid all the way to 2,655 in October 1990 and then rebounded to 6,000. It was not until March 1991 when TSE left the sickness stage.
15. According to our empirical results, in Nikkei, p23= 0.0343, p13= 0.0039; in TAIEX, p23= 0.0054, p13=
3.3 × 10−7.
16. One can also refers to “Taiwan Is Yet to Find Profit in Asia’s Woes,” A15, The Wall Street Journal, August 19, 1998.
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