Using CARRX models to study factors affecting the volatilities of Asian equity markets

North American Journal of Economics and Finance 26 (2013) 552–564

Contents lists available at ScienceDirect

North American Journal of Economics and Finance

Using CARRX models to study factors affecting the volatilities of Asian equity markets夽 Chor-Yiu (CY) Sin Department of Economics, National Tsing Hua University, Hsinchu 30013, Taiwan, ROC

a r t i c l e

i n f o

JEL classification: C5 C52 G11 Keywords: CARRX Daily price range European factor Parsimony principle Regional (Asian) factor U.S. factor Volatility spillover

a b s t r a c t The range of daily asset prices is often used as a measure of volatility. Using a CARRX (conditional autoregressive range with exogenous variables) model, and the parsimony principle, the paper investigates the factors affecting the volatilities of Asian equity markets. Since the beginning of the new Century, emerging Asian markets such as Taiwan and Shanghai have been undergoing various stages of financial globalization. The volatility of the equity market may not be explained solely by its own dynamics. In this paper, we examine volatility using the following factors: (i) lagged returns; (ii) lagged absolute returns; (iii) own trading volume; (iv) U.S. factors; (v) European factors; and (vi) regional (Asian) factors. Points (i) and (iii) are by and large significant, while (ii) is not. Controlling for (i), (ii) and (iii), we find evidence that the volatility of European markets has spillovers on to both the Taiwan and Tokyo markets, mild evidence that the volatility of the U.S. market has spillovers on to the Hong Kong market, but there are no spillovers from the European or U.S. markets on to the Shanghai market. © 2013 Elsevier Inc. All rights reserved.

1. Introduction Since the publication of Parkinson (1980), recent years have witnessed the use of daily high-low range (henceforth range) as a measure of volatility of asset return. In contrast to the high-frequency data, the high and low of asset price are often reported in daily news and thus easier to obtain. On the

夽 The author is most grateful for the helpful comments and suggestions of a referee and the guest editors. This research was partially supported by the National Science Council of Taiwan under grant NSC 101-2410-H-007-013.

E-mail address: [email protected] 1062-9408/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.najef.2013.02.021

C.-Y. Sin / North American Journal of Economics and Finance 26 (2013) 552–564

553

other hand, compared with the daily squared close-to-close return (henceforth squared return) or the daily absolute close-to-close return (henceforth absolute return), a dynamic range model similar to the prevalent GARCH model is more robust to outliers. In addition, there is some evidence that, compared with GARCH model, the dynamic range model is a good estimator of volatility (see, for instance, the results in Chou, 2005). Further, see also Gallant, Hsu, and Tauchen (1999), Alizadeh, Brandt, and Diebold (2001), Brandt and Diebold (2006) and Brandt and Jones (2006) for more applications of range in volatility forecast. Financial crises over the last thirty years, such as the 1987 stock market crash, the 1994 Mexican peso crisis, the 1997 Asian financial crisis in 1997, the 1998 Russian financial crisis, the 1999–2002 Argentine debt crisis in 2001, and the 2007 U.S. subprime crisis, have long suggested some information transmission from one equity market to the other or the others. Globalization results in spillovers over return and volatility among various markets. Focusing on the leading markets of the U.S., the U.K. and Japan, a number of papers find evidence of volatility spillovers. See, for instance, Hamao, Masulis, and Ng (1990), Martens and Poon (2001), Skintzi and Refenes (2006). As far as Asian markets are concerned, Bekaert and Harvey (1997) use time-series and crosssectional models and find that financial liberalizations increase return correlations between emerging markets and the leading markets, but the volatilities are not driven up. Similarly, Ng (2000) concludes that liberalization events such as capital market reform in the Pacific-Basin Markets, affect the relative importance of the world and regional market factors, where the world market factors are proxied by the U.S. shocks and the regional market factors are proxied by the Japan shocks. She also finds significant spillovers from the region to the majority of the Pacific-Basin markets. Constructing a bivariate EGARCH (exponential GARCH) model for Japan and other seven Asian markets with an exogenous U.S. factor, Miyakoshi (2003) finds that the volatility of the Asian market is by and large affected by Japan, but not the U.S. Further and rather surprisingly, the volatility of the Asian market has a negative impact on that of Japan. On the other hand, Worthington and Higgs (2004) investigate the volatility transmission among three developed Asian markets (Hong Kong, Japan and Singapore) and six Asian emerging markets (Indonesia, Korea, Malaysia, the Philippines, Taiwan and Thailand). Developing an MGARCH (multivariate GARCH) model, they find that own-volatility spillovers are generally higher than cross-volatility spillovers for all markets, but especially for the emerging markets. Using a panel of 10 emerging and industrial financial markets, crisis, Dungey, Fry, Gonzáalez-Hermosillo, and Martin (2007) showed contagion was spread all around international equity markets the during the LTCM crisis, but was spread selectively during the Russian crisis. Recently, Diebold and Yilmaz (2009) formulate and examine variants of measures of return spillovers and volatility spillovers based directly on the familiar notion of variance decompositions in vector autoregressions. Applying the framework to nineteen equity markets for the last twenty years, they find volatility spillovers display no trend but clear bursts associated with readily identified “crisis” events. Singh, Kumar, and Pandey (2010) examine volatility spillovers across North American, European and Asian equity markets. Using a VAR (Vector Autoregression)-GARCH of fifteen equity indices, they find that a greater regional influence among Asian and European equity markets. Examining the factors affecting the volatilities of the Asian equity markets, this paper differ from the existing literature in twofold: (i) we use range as a measure of volatility and employ a dynamic model for range, namely CARRX developed by Chou (2005); (ii) we include the trading volume as one of the exogenous variables. Trading volume is found in the literature one of the important exogenous variables. See, for instance, Karpoff (1987), Lamoureux and Lastrapes (1990), Fleming, Kirby, and Ostdiek (2006), and Asai and Brugal (in press). This paper considers two emerging markets (Taiwan and Shanghai)1 and two developed markets (Hong Kong and Japan) will be the object of investigation. Out of the emerging Asian markets, Shanghai and Taiwan have been undergoing various stages of financial globalization in the last decade, and it is interesting to see if there have been spillovers to these two markets. On the other hand, the other developed Asian markets such as Singapore are not

1 It is not straightforward to make an exact list of emerging or developed markets. That said, Taiwan and the Mainland of China are classified as emerging markets by groups of analysts such as FTSE, MSCI and S&P.

554

C.-Y. Sin / North American Journal of Economics and Finance 26 (2013) 552–564

considered since some of the variables are relatively difficult to obtain. See also Section 3. The DJAT (rather than the Tokyo Nikkei 225 index) is used as the Asian factor. The U.S. factor is the S&P500, while the European factor is one of the followings: DAX, FTSE100 and CAC40. This paper is organized as follows. The next section is on the methodology, while Section 3 is about the data description and the empirical model. Section 4 is devoted to the empirical results. Both in-sample estimation and the post-sample model comparisons can be found. Section 5 concludes. A code, which is modified upon the R-code acd.R written by Professor Ruey S. Tsay (http://faculty.chicagobooth.edu/ruey.tsay/teaching/bs41202/sp2012/), is available from the author. 2. A volatility spillover CARRX model Let Pt be the natural logarithm of (the close of) one of the equity indices. We confine our attention to the following definition of range. For each day t, Rt := 100 × (max{P } − min{P }),

 =t−1+

1 2 , t − 1 + , . . . , t, n n

(1)

where the parameter n is the number of intervals in a day. As the equity index is in natural logarithm, the close-to-close return is defined as: rt := 100 × (Pt − Pt−1 ),

(2)

The CARR (Conditional Autoregressive Range) model assumes the following dynamics of Rt : Rt =

t−1 t ,

(3)

where {t } is an i.i.d. series with E(t ) = 1; and conditional on an information set Ft−1 , expectation of Rt ,2 which is in turn specified as a GARCH-type model: t−1

=ω+

r 

˛i Rt−i +

s 

i=1

ˇj

+

t−1−j

j=1

q 

ık wt−1,k ,

t−1

is the

(4)

k=1

in which the wt−1,k ’s are the exogenous variables observable in day t − 1. In view of the possible presence of exogenous variables. this model is called CARRX model (CARR model with exogenous variables). See Chou (2005). Note both Rt and  t−1 are (assumed to be) non-negative, thus so is t . A natural choice for the distribution is the exponential (with unit expectation) as it has non-negative support. Similar to the ARCH model or the GARCH model, (4) specifies a dynamic structure which characterizes the persistence of shocks to the range Rt . Further, we may interprete t−1 as the standard deviation 2 (conditional on Ft−1 ), and t−1 as the conditional variance. The parameters (ω ; ˛i , i = 1, . . ., r ; ˇj , j = 1, . . ., s ; ık , k = 1, . . ., q) in (4) are often estimated with the QMLE (quasi-maximum likelihood estimation). Three distributions of t are considered, namely (standard) exponential, (standardized) Weibull and (standardized generalized) gamma, and the models are sometimes termed ECARRX, WCARRX and GCARRX respectively. See Tsay (2011) for the details (see also Allen, Ng, & Peiris, in press). More concretely, the density functions are respectively: Exponetial : f (Rt |Ft−1 ) =

t−1

 

Weibull : f (Rt |Ft−1 ) =  

Gamma : f (Rt |Ft−1 ) =



1

exp

1 1+ 

−

Rt t−1

 R−1

t  t−1



;

(5)

  exp

Rt−1  exp  () (G(, ) t−1 )

− 

 −

1 1+ 

t,

t−1



Rt G(, )

2 While the literature often denotes the conditional expectation, E(Rt |Ft−1 ), by it is realized in period t − 1. Similarly, for a general j, we denote E(Rt−j+1 |Ft−j ) by

 R 

t

t−1

,

we denote it by

t−j .

;

(6)

(7)

t−1

to signify the fact that

C.-Y. Sin / North American Journal of Economics and Finance 26 (2013) 552–564

555

where  (.) is the gamma function and G (, ) =  ()/ ( + 1/). It is not difficult to see that when the scale parameter  = 1, the (standardized generalized) gamma boils down to the (standardized) Weibull. Further when the shape parameter  = 1, the (standardized) Weibull boils down to the (standard) exponential. Further, E(21 ) = 2 (exponential), E(21 ) = ( (1 + (2/))/ 2 (1 + (1/))) (Weibull) and E(21 ) = ( () ( + (2/))/ 2 ( + (1/))) (gamma). By now it is well-known that the QMLE of the CARRX model is exactly the same as that of the ACD (autoregressive conditional duration) model for durations between trades. More precisely, denote as the vector of parameters (with the shape parameter if Weibull is used, and with the shape and the power parameters if gamma is used), and let the conditional density be ft−1 ( ) : = f(Rt |Ft−1 ). Given some initial values R−r+1 , R−r+2 , . . ., R0 ; −s , −s+1 , −1 , the QMLE of is obtained by maximizing the following log-likelihood function: L( , R1 , R2 , . . . , RT ) =

T 

ln[ft−1 ( )]

(8)

t=1

where T is the (in-)sample size and, depending on the choice of the distribution, ft−1 ( ) := f (Rt |Ft−1 ) is one of (5)–(7). Some caveats of the QMLE are in order: (i) We assume ω > 0, ˛i ≥ 0, i = 1, . . ., r; ˇj ≥ 0, j = 1, . . ., s. r s ˛ + ˇ < 1. (ii) For strict stationarity3 of t−1 , we assume i=1 i j=1 j (iii) For each exogenous variable, we standardize it such that its variance is the same as that of Rt . More precisely, for an exogenous variable xt−1,k , we consider wt−1,k = (xt−1,k − xk )

ˆ R ,

ˆ x

(9)

where xk denotes the sample mean of xt−1,k ; and ˆ R and ˆ x denote the sample standard deviations of Rt and xt−1,k respectively. Further, we set 0 ≤ ık < 1.

r ˆ + ˛ ˆ R + Denote the QMLE as ˆ and the predicted standard deviation as ˆ t−1 = ω i=1 i t−i q ˆ s ˆ ˆ + . The post-sample performance of a model can be evaluated by one of ˇ ı w k=1 k t−1,k j=1 j t−1−j the following criteria: T +N 

MAE R = N −1

|Rt − ˆ t−1 |;

(10)

t=T +1 T +N 

MAE ar = N −1

||rt | − ˆ t−1 |;

(11)

t=T +1

RMSE R =

T +N 

N −1

1/2 (Rt − ˆ t−1 )2

;

(12)

t=T +1

RMSE ar =

N

−1

T +N 

1/2 (|rt | − ˆ t−1 )

2

,

(13)

t=T +1

where N is the post-sample size and we recall that rt is the close-to-close return. See (3).

3 Strict stationarity, though may not be the weakest, is one of the usual conditions suffice for doing inference with asymptotic normality. See, for instance, p. 565 of Chou (2005).

556

C.-Y. Sin / North American Journal of Economics and Finance 26 (2013) 552–564

Table 1 Summary statistics of the raw data: TAIEX (Taiwan).

Nobs Minimum Maximum 1. Quartile 3. Quartile Mean Median Stdev Skewness Kurtosis

High

Low

Close

Volume

2433 4156.85 9859.65 6028.68 7933.00 7007.66 7093.47 1279.92 −0.15 −0.66

2433 3955.43 9735.43 5971.58 7843.05 6916.94 6998.13 1274.01 −0.16 −0.65

2433 4089.93 9809.88 5997.67 7886.34 6959.76 7040.90 1276.25 −0.16 −0.66

2433 1110800.00 11558200.00 2677000.00 4539800.00 3761157.58 3536200.00 1497882.95 1.11 1.93

High

Low

Close

Volume

2862 1019.92 6124.04 1525.24 2816.19 2277.65 2100.98 986.69 1.37 2.00

2862 998.23 6040.71 1505.04 2752.92 2230.79 2056.72 956.82 1.36 2.00

2862 1011.50 6092.06 1514.87 2790.41 2256.67 2078.74 973.96 1.37 2.00

2862 82083.00 27580120.00 1115101.50 8911063.25 5618449.44 4534924.50 4996758.09 0.91 0.16

Table 2 Summary statistics of the raw data: SSE (Shanghai).

Nobs Minimum Maximum 1. Quartile 3. Quartile Mean Median Stdev Skewness Kurtosis

3. Data and the empirical model We collect the daily index data from various sources such as Taiwan Economic Journal (http://www.tej.com.tw/twsite), Yahoo (http://www.tej.com.tw/twsite) and Google (http://www.google.com/finance). Four Asian equity indices are our objects of investigation, namely, TAIEX (Taiwan Capitalization Weighted Stock Index), SSE (Shanghai Stock Exchange Composite Index), HSI (Hang Seng Index) and Nikkei 225 (Nikkei Stock Average). These indices are chosen not only because they by and large represent the market trends of different Asian markets (emerging as well as developed), but also because the availability of the relevant variables noticeably trading volume. Tables 1–4 contain the summary statistics of the raw data (daily high, daily low, daily close and daily trading volume) of the four markets.

Table 3 Summary statistics of the raw data: HSI (Hong Kong).

Nobs Minimum Maximum 1. Quartile 3. Quartile Mean Median Stdev Skewness Kurtosis

High

Low

Close

Volume

2422 8430.62 31958.41 13923.22 21543.56 18027.56 18892.10 4733.13 0.01 −0.70

2422 8331.87 31361.91 13760.97 21230.53 17771.49 18626.04 4651.69 −0.01 −0.74

2422 8409.01 31638.22 13859.82 21375.49 17905.37 18783.29 4690.26 −0.00 −0.72

2422 102238400.00 9799120000.00 346230650.00 1890564200.00 1304308699.63 1253253800.00 1045172406.43 1.41 4.64

C.-Y. Sin / North American Journal of Economics and Finance 26 (2013) 552–564

557

Table 4 Summary statistics of the raw data: NIKKEI225 (Tokyo).

Nobs Minimum Maximum 1. Quartile 3. Quartile Mean Median Stdev Skewness Kurtosis

High

Low

Close

Volume

2406 7100.77 18300.39 9489.84 13675.50 11724.03 10817.11 2946.89 0.77 −0.69

2406 6994.90 18213.59 9359.98 13454.05 11568.75 10686.78 2922.89 0.77 −0.66

2406 7054.98 18261.98 9427.77 13600.38 11646.86 10747.65 2936.84 0.77 −0.67

2406 32000000.00 4158000000.00 920500000.00 1464000000.00 1212485209.31 1198000000.00 418271894.99 0.74 2.12

Denote the range of the index by Rt (see Table 5 for the summary statistics). Recall the CARRX model in (4) with r = s = 1, we have the following conditional expectation of Rt : t−1

= ω + ˛1 Rt−1 + ˇ1

t−2 +

q 

ık wt−1,k .

(14)

k=1

The exogenous variables include some of the followings: the trading volume of the respective index; daily range of SP500 (Standard and Poors 500), daily close of VIX (Chicago Board Options Exchange Market Volatility Index); daily range of DAX (German stock index), daily range of FTSE100 (Financial Times and Stock Exchange 100), and daily range of CAC40 (French Stock Market Index); daily range of DJAT (Dow Jones Asian Titans 50 Index); negative of the lagged daily return; and the lagged absolute return. Each exogenous variable is standardized such that its (in-sample) sample standard deviation is the same as that of Rt . The whole sample goes from January 2, 2003 to October 31, 2012. The sample reflects the recent trends of globalization on the one hand, and contains enough information for analyses on the other hand. About 80% goes to the in-sample while about 20% goes to the post-sample, and thus the in-sample is chosen to end at December 31, 2010 and the post-sample is chosen to start from January 2, 2011. Deleting the missing data and considering only the sample that SP500, VIX, DAX, FTSE100, CAC40 and DJAT are also available, the in-sample size ranges from 1835 (Shanghai) to 1886 (Hong Kong); while the post-sample size ranges from 418 (Shanghai) to 430 (Hong Kong). The exact in-sample sizes (denoted by T) and the exact post-sample sizes (denoted by N) can be found in Tables 6–13. 4. Empirical results The empirical results for Taiwan, Shanghai, Hong Kong and Tokyo are presented, respectively, in Tables 6–8, and the discussions can be found in Sections 4.1–4.4. For each market, we first consider a basic CARR model (assuming exponential distribution), in which there is no exogenous variable, the Table 5 Summary statistics of the daily range: January 2, 2003–October 31, 2012.

Nobs Minimum Maximum 1. Quartile 3. Quartile Mean Median Stdev Skewness Kurtosis

TAIEX

SSE

HSI

NIKKEI225

2298 0.146 7.403 0.791 1.636 1.347 1.130 0.803 2.033 6.503

2257 0.499 10.170 1.190 2.511 2.034 1.682 1.229 1.959 5.562

2320 0.285 17.647 0.826 1.724 1.426 1.159 1.034 4.466 41.258

2277 0.236 13.763 0.827 1.659 1.382 1.153 0.982 4.362 33.756

558

C.-Y. Sin / North American Journal of Economics and Finance 26 (2013) 552–564

Table 6 Daily range of TAIEX (Taiwan), T = 1870. Estimate Exponential

(constant) Rt−1

Exponential

(constant) Rt−1

t−2

t−2

volume us europe asia −rt−1 |rt−1 | Weibull

(constant) Rt−1 t−2

volume us europe asia −rt−1 |rt−1 | Shape Gamma

(constant) Rt−1 t−2

volume us europe asia −rt−1 |rt−1 | power shape Exponential

(constant) Rt−1 t−2

volume europe −rt−1

t value

Pr(>|t|)

LLF

0.0167 0.1142 0.8733

Std. error 0.0122 0.0203 0.0223

1.3710 5.6265 39.1183

0.1704 0.0000 0.0000

−2382.61

0.0753 0.0758 0.8697 0.0183 0.0022 0.0164 0.0000 0.0659 0.0155

0.0365 0.0255 0.0263 0.0068 0.0211 0.0154 0.0137 0.0225 0.0317

2.0646 2.9765 33.0267 2.6676 0.1051 1.0665 0.0012 2.9244 0.4906

0.0390 0.0029 0.0000 0.0076 0.9163 0.2862 0.9990 0.0035 0.6237

−2374.71

0.0738 0.0547 0.8919 0.0169 0.0049 0.0156 0.0000 0.0676 0.0140 2.3114

0.0154 0.0099 0.0118 0.0025 0.0079 0.0058 0.0051 0.0087 0.0129 0.0375

4.7805 5.5242 75.2855 6.7074 0.6166 2.6717 0.0033 7.7710 1.0909 61.5827

0.0000 0.0000 0.0000 0.0000 0.5375 0.0075 0.9974 0.0000 0.2753 0.0000

−2375.22

0.0763 0.0881 0.8562 0.0190 0.0000 0.0171 0.0000 0.0625 0.0157 0.2388 100.0000

0.0160 0.0114 0.0118 0.0031 0.0095 0.0069 0.0062 0.0101 0.0138 0.0453 37.9385

4.7605 7.7197 72.7813 6.0641 0.0017 2.4650 0.0027 6.2031 1.1408 5.2740 2.6358

0.0000 0.0000 0.0000 0.0000 0.9986 0.0137 0.9979 0.0000 0.2540 0.0000 0.0084

−2374.87

0.0573 0.0803 0.8783 0.0182 0.0168 0.0644

0.0212 0.0186 0.0225 0.0064 0.0107 0.0213

2.7037 4.3199 38.9980 2.8291 1.5704 3.0220

0.0069 0.0000 0.0000 0.0047 0.1163 0.0025

−2374.86

results are presented in the upper panel of the respective table. The results of a general CARRX model, with all exogenous variables discussed around (14), are presented in the middle panel. Restricted model(s) will then be considered and the results are presented in the lower panel. For the CARRX models, the exponential, Weibull and gamma distributions will be considered, though to save space we will only present the results of the exponential distribution, should those of the other two distributions are similar. As one can see below, for all four markets, both trading volume and the negative of the lagged return are significant. The former is common in volatility models while the latter captures the leverage effect. The regional (Asian) factor is not significant though. For the U.S. factor, as SP500 (daily range) and VIX (daily close) give similar results, we only present that of SP500.4 Similarly, for the European factor, as DAX (daily range), FTSE100 (daily range) and CAC40 (daily range) give similar results, we only present that of DAX.

4

For empirical comparisons between VIX and SP500 volatility, see Chang, Jimenez-Martin, McAleer, & Amaral (in press).

C.-Y. Sin / North American Journal of Economics and Finance 26 (2013) 552–564

559

Table 7 Daily range of SSE (Shanghai), T = 1835. Estimate Exponential

(constant) Rt−1

Exponential

(constant) Rt−1

t−2

t−2

volume us europe asia −rt−1 |rt−1 | Weibull

(constant) Rt−1 t−2

volume Us europe asia −rt−1 |rt−1 | shape Gamma

(constant) Rt−1 t−2

volume us europe asia −rt−1 |rt−1 | power shape Exponential

(constant) Rt−1 t−2

volume −rt−1

t value

Pr(>|t|)

LLF

0.0450 0.1584 0.8209

Std. error 0.0274 0.0308 0.0357

1.6403 5.1417 22.9716

0.1010 0.0000 0.0000

−3159.47

0.1373 0.1525 0.7837 0.0205 0.0000 0.0000 0.0028 0.0575 0.0234

0.0771 0.0503 0.0499 0.0129 0.0295 0.0172 0.0237 0.0283 0.0443

1.7815 3.0320 15.7144 1.5959 0.0006 0.0010 0.1203 2.0292 0.5285

0.0748 0.0024 0.0000 0.1105 0.9995 0.9992 0.9042 0.0424 0.5971

−3155.76

0.1286 0.1428 0.7978 0.0252 0.0000 0.0017 0.0000 0.0517 0.0158 2.2934

0.0341 0.0224 0.0239 0.0058 0.0127 0.0067 0.0099 0.0117 0.0190 0.0371

3.7661 6.3691 33.4368 4.3309 0.0013 0.2536 0.0017 4.4097 0.8337 61.8787

0.0002 0.0000 0.0000 0.0000 0.9989 0.7998 0.9987 0.0000 0.4044 0.0000

−3156.13

0.1420 0.1582 0.7751 0.0180 0.0000 0.0000 0.0051 0.0589 0.0264 0.2447 100.0000

0.0317 0.0208 0.0202 0.0053 0.0124 0.0074 0.0100 0.0119 0.0184 0.0350 28.7052

4.4853 7.6126 38.3335 3.4184 0.0013 0.0022 0.5160 4.9707 1.4355 6.9908 3.4837

0.0000 0.0000 0.0000 0.0006 0.9989 0.9982 0.6058 0.0000 0.1511 0.0000 0.0005

−3155.84

0.1057 0.1724 0.7793 0.0219 0.0576

0.0484 0.0361 0.0489 0.0119 0.0281

2.1818 4.7766 15.9474 1.8357 2.0518

0.0291 0.0000 0.0000 0.0664 0.0402

−3155.92

4.1. Taiwan in-sample results For all models in Table 6 (with in-sample size, T = 1870), the coefficient of Rt−1 is around 0.08 while ˆ 1 ≈ 0.93. On the other hand, judging from the LLF ˆ1 + ˇ that of t−2 is around 0.85. In other words, ˛ (log-likelihood function), both the general CARRX and the restricted CARRX beat the basic CARR. One can see from the middle panel, though the p-values vary from different distributional assumptions made,5 the variables “us”, “asia” and |rt−1 | are insignificant, while “volume”, “europe” and −rt−1 are by and large significant (or not that insignificant). We drop the former three variables and re-estimate the models. The results are summarized in the lower panel.6

5 Strictly speaking, one may consider some sort of variance–covariance matrix that is robust to distributional assumption. However, it remains to see how this type of matrices perform, especially when the Hessian may not be easier to invert in this QMLE. 6 For Taiwan and the other three markets, we only present the results for the exponential distribution, as those for the other two distributions are qualitatively the same.

560

C.-Y. Sin / North American Journal of Economics and Finance 26 (2013) 552–564

Table 8 Daily range of HSI (Hong Kong), T = 1886. t value

Pr(>|t|)

LLF

0.0125 0.1165 0.8747

0.0104 0.0204 0.0215

1.1957 5.7213 40.5943

0.2318 0.0000 0.0000

−2429.48

0.0840 0.0884 0.8534 0.0086 0.0054 0.0033 0.0000 0.0284 0.0349

0.0377 0.0295 0.0301 0.0096 0.0204 0.0111 0.0149 0.0202 0.0302

2.2286 2.9934 28.3803 0.8965 0.2619 0.3021 0.0011 1.4066 1.1566

0.0258 0.0028 0.0000 0.3700 0.7934 0.7625 0.9991 0.1595 0.2474

−2425.72

0.1129 0.0746 0.8475 0.0089 0.0136 0.0001 0.0000 0.0322 0.0521 2.4635

0.0176 0.0129 0.0163 0.0043 0.0091 0.0046 0.0063 0.0081 0.0125 0.0398

6.4333 5.7721 52.0550 2.0588 1.4919 0.0209 0.0026 3.9543 4.1751 61.9586

0.0000 0.0000 0.0000 0.0395 0.1357 0.9833 0.9979 0.0001 0.0000 0.0000

−2426.14

0.0760 0.0964 0.8507 0.0088 0.0019 0.0050 0.0000 0.0255 0.0301 0.2558 100.0000

0.0145 0.0116 0.0111 0.0037 0.0080 0.0044 0.0059 0.0081 0.0118 0.0421 33.0059

5.2253 8.3133 76.5967 2.3529 0.2420 1.1209 0.0028 3.1604 2.5454 6.0687 3.0298

0.0000 0.0000 0.0000 0.0186 0.8088 0.2623 0.9977 0.0016 0.0109 0.0000 0.0024

−2425.81

0.0828 0.0892 0.8536 0.0077 0.0096 0.0275 0.0343

0.0367 0.0276 0.0295 0.0078 0.0137 0.0199 0.0294

2.2577 3.2311 28.8888 0.9899 0.7023 1.3768 1.1659

0.0240 0.0012 0.0000 0.3222 0.4825 0.1686 0.2437

−2425.77

Estimate Exponential

(constant) Rt−1

Exponential

(constant) Rt−1

t−2

t−2

volume us europe asia −rt−1 |rt−1 | Weibull

(constant) Rt−1 t−2

volume us europe asia −rt−1 |rt−1 | shape Gamma

(constant) Rt−1 t−2

volume us europe asia −rt−1 |rt−1 | power shape Exponential

(constant) Rt−1 t−2

volume us −rt−1 |rt−1 |

Std. error

The lower panel shows that, all variables are significant or marginally significant, which is in contrast to those in the middle panel. Thus, one may conclude that “volume”, “europe” and −rt−1 can explain the daily range of TAIEX. To see if DAX (the European factor) really has a spillover effect on TAIEX, we drop the variable “europe” (the detailed results are not presented in order to save space). Consider the twice of the difference in LLF, 2 × (−2374.86 + 2376.28) = 3.84, which p-value is around 5%. 4.2. Shanghai in-sample results For all models in Table 7 (with in-sample size, T = 1835), the coefficient of Rt−1 is around 0.15 while ˆ 1 ≈ 0.95. On the other hand, judging from the LLF, that of t−2 is around 0.80. In other words, ˛ ˆ1 + ˇ both the general CARRX and the restricted CARRX beat the basic CARR. One can see from the middle panel that, the variables “us”, “europe”, “asia” and |rt−1 | are insignificant, while “volume” and −rt−1

C.-Y. Sin / North American Journal of Economics and Finance 26 (2013) 552–564

561

Table 9 Daily range of NIKKEI225 (Tokyo), T = 1848. Std. error

Estimate

Exponential

(constant) Rt−1

Exponential

(constant) Rt−1

t−2

t−2

volume us europe asia −rt−1 |rt−1 | Weibull

(constant) Rt−1 t−2

volume us europe asia −rt−1 |rt−1 | shape Gamma

(constant) Rt−1 t−2

volume us europe asia −rt−1 |rt−1 | power shape Exponential

(constant) Rt−1 t−2

volume europe −rt−1

t value

Pr(>|t|)

LLF

0.0267 0.1551 0.8260

0.0174 0.0298 0.0340

1.5341 5.2111 24.3129

0.1250 0.0000 0.0000

−2412.14

0.0857 0.1221 0.8187 0.0045 0.0139 0.0139 0.0000 0.0732 0.0055

0.0488 0.0374 0.0453 0.0077 0.0244 0.0181 0.0190 0.0251 0.0337

1.7572 3.2652 18.0646 0.5818 0.5722 0.7657 0.0009 2.9131 0.1630

0.0789 0.0011 0.0000 0.5607 0.5672 0.4438 0.9993 0.0036 0.8705

−2404.58

0.0755 0.1076 0.8401 0.0042 0.0127 0.0077 0.0000 0.0670 0.0110 2.4203

0.0199 0.0151 0.0190 0.0029 0.0088 0.0067 0.0078 0.0097 0.0135 0.0400

3.7984 7.1241 44.2146 1.4440 1.4358 1.1437 0.0021 6.9318 0.8149 60.4604

0.0001 0.0000 0.0000 0.1487 0.1510 0.2527 0.9983 0.0000 0.4151 0.0000

−2404.76

0.0923 0.1330 0.8031 0.0043 0.0147 0.0176 0.0000 0.0763 0.0029 0.2480 100.0000

0.0204 0.0158 0.0192 0.0033 0.0106 0.0079 0.0080 0.0105 0.0140 0.0462 37.2620

4.5204 8.4119 41.7814 1.2878 1.3863 2.2346 0.0021 7.2515 0.2078 5.3690 2.6837

0.0000 0.0000 0.0000 0.1978 0.1657 0.0254 0.9983 0.0000 0.8354 0.0000 0.0073

−2404.64

0.0705 0.1240 0.8271 0.0061 0.0195 0.0764

0.0313 0.0282 0.0372 0.0062 0.0142 0.0234

2.2567 4.4065 22.2107 0.9889 1.3790 3.2639

0.0240 0.0000 0.0000 0.3227 0.1679 0.0011

−2404.82

Table 10 TAIEX (Taiwan): post-sample comparison, N = 424. MAER

MAEar

RMSER

RMSEar

Table 6: Upper

Exponential

0.4044

0.7063

0.5965

0.8642

Table 6: Middle

Exponential Weibull Gamma

0.3916 0.3932 0.3901

0.6803 0.6814 0.6796

0.5766 0.5779 0.5763

0.8461 0.8459 0.8460

Table 6: Lower

Exponential

0.3905

0.6750

0.5789

0.8413

562

C.-Y. Sin / North American Journal of Economics and Finance 26 (2013) 552–564

Table 11 SSE (Shanghai): post-sample comparison, N = 418. MAER

MAEar

RMSER

RMSEar

Table 7: Upper

Exponential

0.4826

0.8954

0.6607

1.0254

Table 7: Middle

Exponential Weibull Gamma

0.5461 0.5633 0.5376

0.9982 1.0171 0.9881

0.6929 0.7042 0.6876

1.1259 1.1437 1.1164

Table 7: Lower

Exponential

0.5465

0.9964

0.6932

1.1236

Table 12 HSI (Hong Kong): post-sample comparison, N = 430. MAER

MAEar

RMSER

RMSEar

Table 8: Upper

Exponential

0.4139

0.7211

0.5865

0.9290

Table 8: Middle

Exponential Weibull Gamma

0.4307 0.4331 0.4296

0.7525 0.7578 0.7498

0.5823 0.5820 0.5827

0.9496 0.9542 0.9480

Table 8: Lower

Exponential

0.4277

0.7485

0.5801

0.9466

are by and large significant. We drop the former four variables and re-estimate the models. The results are summarized in the lower panel. The lower panel shows that, all variables are significant or marginally significant, which is a little in contrast to those in the middle panel. Thus, one may conclude that “volume” and −rt−1 can explain the daily range of SSE. Consider the twice of the difference in LLF, 2 × (−3155.76 + 3155.92) = 0.12, which is highly insignificant.

4.3. Hong Kong in-sample results For all models in Table 8 (with in-sample size, T = 1886), the coefficient of Rt−1 is around 0.09 ˆ 1 ≈ 0.94. On the other hand, judging from the ˆ1 + ˇ while that of t−2 is around 0.85. In other words, ˛ LLF, both the general CARRX and the restricted CARRX beat the basic CARR. The middle panel shows that, the variables “europe” and “asia” are insignificant, and the significances of other variables are mixed. We drop the two variables “europe” and “asia”, and re-estimate the models. The results are summarized in lower panel. To see if the remaining variable “us” has really an impact on the daily range of HSI, we further drop this variable and re-estimate the models (the detailed results are not presented in order to save space). Consider the twice of the difference in LLF, 2 × (−2425.77 + 2426.03) = 0.52, which is highly insignificant. In other words, the spillover effect from the U.S. is at most mild.

Table 13 NIKKEI225 (Tokyo): post-sample comparison, N = 425. MAER

MAEar

RMSER

RMSEar

Table 9: Upper

Exponential

0.3958

0.6348

0.7803

0.9191

Table 9: Middle

Exponential Weibull Gamma

0.4360 0.4323 0.4361

0.6529 0.6512 0.6523

0.7784 0.7782 0.7774

0.9101 0.9110 0.9085

Table 9: Lower

Exponential

0.4366

0.6554

0.7799

0.9138

C.-Y. Sin / North American Journal of Economics and Finance 26 (2013) 552–564

563

4.4. Tokyo in-sample results For all models in Table 9 (with in-sample size, T = 1848), the coefficient of Rt−1 is around 0.12 while ˆ 1 ≈ 0.95. On the other hand, judging from the LLF, ˆ1 + ˇ that of t−2 is around 0.83. In other words, ˛ both the general CARRX and the restricted CARRX beat the basic CARR. The middle panel shows that, the variables “us”, “asia” and |rt−1 | insignificant; the variable −rt−1 is significant; while the significances of “volume” and “europe” are a kind of mixed. We drop the first three variables and re-estimate the models. The results are summarized in the lower panel. The lower panel shows that, unlike the middle panel, “europe” and “volume” are significant (or not that insignificant). To see if the variable “europe” has really an impact on the daily range of NIKKEI225, we further drop this variable and re-estimate the models (the detailed results are not presented in order to save space). Consider the twice of the difference in LLF, for the exponential distribution, 2 × (−2404.82 + 2406.02) = 2.40, which is at most marginally significant. In other words, we may conclude that there is some evidence of spillover from Europe to Tokyo, but the evidence is not as strong as that to Taiwan.

4.5. Post-sample model comparisons We evaluate the post-sample forecast performance, based on the criteria mean absolute error and root mean squared errors with various definitions of volatility, MAER , MAEar RMSER and RMSEar . See (10)–(13). The results are summarized in Tables 10–13, in which N stands for the post-sample size. As one can see from Tables 10-13, the post-sample performance is by and large consistent with the in-sample results in Tables 6–9. This is especially the case for RMSER or RMSEar . One striking result is, except for TAIEX, as far as MAER and MAEar are concerned, both the general CARRX model and the restricted CARRX model(s) do not beat the basic CARR model. Whether this is due to the stability of models, or due to the evaluation criteria which are different from the loss function in the QMLE, is an open issue. We leave this issue to further study.

5. Concluding remarks Using daily range, this paper employs a CARRX (conditional autoregressive range with exogenous variables) model to investigate the factors affecting the volatilities of four major Asian equity markets, namely, Taiwan, Shanghai, Hong Kong and Tokyo. First proposed by Chou (2005), the CARRX model is easier to estimate and the data requirement is less stringent. The following factors are under considered: (i) lagged return; (ii) lagged absolute return, (iii) own trading volume, (iv) the U.S. factor, (v) the European factor, and (vi) the regional (Asian) factor. In-sample results, which are supported by the post-sample forecast results, suggest that Points (i) and (iii) are by and large significant, while Point (ii) is not. Being an emerging market, Taiwan behaves closer to Tokyo, which is a developed market. In sum, there is evidence that the lagged daily range of DAX (German market) has an impact on the daily range of TAIEX (Taiwan market) and that of NIKKEI225 (Tokyo market). The effect on the Taiwan market is stronger though. Further, we find mild evidence that the lagged daily range of SP500 (the U.S. market) has an impact on the daily range of HSI (Hong Kong). No spillover (neither from Europe nor from the U.S.) to the daily range of SSE (Shanghai market) though. Understanding the volatility spillover is essential for asset pricing, for global asset allocation and global hedging strategies, as well as for policy evaluation regarding international capital flows. As one can see in Table 6 and 9 (the middle panel), one standard deviation increase in the Europe factor will result in 0.0156 to 0.0171 standard deviation increase in the daily range of Taiwan; and 0.0077–0.0195 standard deviation increase in that of Tokyo. On the other hand, from Table 8, one standard deviation increase in the U.S. factor will result in 0.0019–0.0136 standard deviation increase in the daily range of Hong Kong. The magnitude is not that substantial, compared to the other factors such as lagged trading volume and lagged absolute return, but the impact may be huge, should there be a big increase in the daily range of equity indices such as DAX or SP500.

564

C.-Y. Sin / North American Journal of Economics and Finance 26 (2013) 552–564

Two interesting issues arise. The first one is technical. Following the lines in the literature, we did compute the Ljung–Box i . i . d . test for the residual Rt / ˆ t−1 . Taking 12 lags, depending on the model, the Q(12) test statistic ranges from 9.00 to 17.00 and it is unclear if the statistic is significant or not. It is because, following the practice in the ARMA literature that, one may want to adjust for the degrees of freedom of the number of parameters estimated. The second issue is conceptual. This paper uses DJAT as the regional (Asian) factor (and find no significance at all), but it may be the case that one market (such as Shanghai) has an direct impact on the other one (such as Taiwan). If this is the case, a simultaneous model needs to be specified. We leave these interesting issues to further study. References Alizadeh, S., Brandt, M., & Diebold, F. X. (2001). Range-based estimation of stochastic volatility models or exchange rate dynamics are more interesting than you think. Journal of Finance, 57, 1047–1092. Allen, D., Ng, K.-H., & Peiris, S. Estimating and simulating Weibull models of risk or price durations: An application to ACD models. North American Journal of Economics and Finance, in press. Asai, M., & Brugal, I. Forecasting volatility via stock return, range, trading volume and spillover effects: The case of Brazil. North American Journal of Economics and Finance, in press. Bekaert, G., & Harvey, C. R. (1997). Emerging equity market volatility. Journal of Financial Economics, 43, 29–77. Brandt, M. W., & Diebold, F. X. (2006). A no-arbitrage approach to range-based estimation of return covariances and correlations. Journal of Business, 79, 61–73. Brandt, M. W., & Jones, C. S. (2006). Volatility forecasting with range-based EGARCH models. Journal of Business and Economic Statistics, 24, 470–486. Chang, C.-L., Jimenez-Martin, J.-A., McAleer, M., & Amaral, T. P. The rise and fall of S&P500 variance futures. North American Journal of Economics and Finance, in press. Chou, R. Y. (2005). Forecasting financial volatilities with extreme values: The conditional autoregressive range (CARR) model. Journal of Money, Credit and Banking, 37, 561–582. Diebold, F. X., & Yilmaz, K. (2009). Measuring financial asset return and volatility spillovers, with application to global equity markets. Economic Journal, 119, 158–171. Dungey, M., Fry, R., Gonzáalez-Hermosillo, B., & Martin, V. L. (2007). Contagion in global equity markets in 1998: The effects of the Russian and LTCM crises. North American Journal of Economics and Finance, 18, 155–174. Fleming, J., Kirby, C., & Ostdiek, B. (2006). Stochastic volatility, trading volume, and the daily flow of information. Journal of Business, 79, 1551–1590. Gallant, A., Hsu, C., & Tauchen, G. (1999). Using daily range data to calibrate volatility diffusions and extract the forward integrated variance. The Review of Economic and Statistics, 81, 617–631. Hamao, Y., Masulis, R. W., & Ng, V. (1990). Correlations in price changes and volatility across international stock markets. Review of Financial Studies, 3, 281–307. Karpoff, J. (1987). The relation between price change and trading volume: A survey. Journal of Financial and Quantitative Analysis, 22, 109–126. Lamoureux, C., & Lastrapes, W. (1990). Heteroskedasticity in stock return data: Volume versus GARCH effect. Journal of Finance, 45, 221–229. Martens, M., & Poon, S. H. (2001). Returns synchronization and daily correlation dynamics between international stock markets. Journal of Banking and Finance, 25, 1805–1827. Miyakoshi, T. (2003). Spillover of stock return volatility to Asian equity markets from Japan and the US. Journal of International Financial Markets. Institutions and Money, 13, 383–399. Ng, A. (2000). Volatility spillover effects from Japan and the US to the Pacific-Basin. Journal of International Money and Finance, 19, 207–233. Parkinson, M. (1980). The extreme value method for estimating the variance of the rate of return. Journal of Business, 53, 61–65. Singh, P., Kumar, B., & Pandey, A. (2010). Price and volatility spillover across North American, European and Asian stock markets. International Review of Financial Analysis, 19, 55–64. Skintzi, V. D., & Refenes, A. N. (2006). Volatility spillover and dynamic correlation in European bond markets. Journal of International Financial Markets, Institutions and Money, 16, 23–40. Tsay, R. (2011). Autoregressive conditional duration models. In: T. C. Mills, & K. Patterson (Eds.), Palgrave handbook of econometrics, Volume 2: Applied econometrics (pp. 1004–1024) (Chapter 21). Worthington, A., & Higgs, H. (2004). Transmission of equity returns and volatility in Asian developed and emerging markets: A multivariate GARCH analysis. International Journal of Finance and Economics, 9, 71–80.