Asymmetric responses of international stock markets to trading volume

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Physica A 360 (2006) 422–444 www.elsevier.com/locate/physa

Asymmetric responses of international stock markets to trading volume Richard Gerlacha, Cathy W.S. Chenb,, Doris S.Y. Linb, Ming-Hsiang Huangc a

School of Mathematical and Physical Sciences, University of Newcastle, Australia Graduate Institute of Statistics and Actuarial Science, Feng Chia University, Taichung 407, Taiwan c Department of Business Administration, National Changhua University of Education, Taiwan

b

Received 28 February 2005; received in revised form 30 May 2005 Available online 18 July 2005

Abstract The major goal of this paper is to examine the hypothesis that stock returns and return volatility are asymmetric, threshold nonlinear, functions of change in trading volume. A minor goal is to examine whether return spillover effects also display such asymmetry. Employing a double-threshold GARCH model with trading volume as a threshold variable, we find strong evidence supporting this hypothesis in five international market return series. Asymmetric causality tests lend further support to our trading volume threshold model and conclusions. Specifically, an increase in volume is positively associated, while decreasing volume is negatively associated, with the major price index in four of the five markets. The volatility of each series also displays an asymmetric reaction, four of the markets display higher volatility following increases in trading volume. Using posterior odds ratio, the proposed threshold model is strongly favored in three of the five markets, compared to a US news double threshold GARCH model and a symmetric GARCH model. We also find significant nonlinear asymmetric return spillover effects from the US market. r 2005 Elsevier B.V. All rights reserved. Keywords: Asymmetry; Double threshold GARCH; MCMC methods; Model selection; Trading volume change

Corresponding author. Tel.: +886 4 24517250x4412; fax: +886 4 24517092.

E-mail address: [email protected] (C.W.S. Chen). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.06.045

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1. Introduction As a result of globalization, de-regulation and advances in information technology, the modern theory of international markets has switched from the traditional view of market segmentation to the concept of market integration. A substantial amount of research emphasizes the co-movements of international stock markets and explores the dynamics of return co-variances and spillover effects between markets. For example, Jaffe and Westerfield [1] provide empirical evidence of a significant direct spillover effect among some national stock markets, while Eun and Shim [2] find return spillover effects among national markets and an influential role of the US market on the cross-country market index series. Ross [3] argues further that information from one stock market can be incorporated into the volatility process of other stock markets. Hamao et al. [4], Theodossiou and Lee [5], Chiang and Chiang [6] and Martens and Poon [7] subsequently find supporting evidence for volatility spillover among major stock markets. There is also substantial evidence in the literature that stock markets react asymmetrically to market news results. This phenomenon was first discovered by Black [8] and Christie [9] who discuss the leverage effect as the cause of higher volatility following negative stock returns; similar to the market over-reaction hypothesis discussed in Ref. [10]. There is also the volatility feedback hypothesis which says that higher volatility causes stock prices to fall. Many models have been developed to capture types of asymmetric behavior, most fit into the threshold GARCH framework. Glosten et al. (1993) employ a threshold GJR-GARCH model and find evidence that local negative market news causes increased market volatility [47]. This finding is confirmed in studies by Koutmos [11], Nam et al. [10] and Brooks [12], using double threshold models. These papers also find evidence of faster mean reversion dynamics following local market bad news. More recently, Chen et al. [13] employ a double-threshold GARCH model with a US market threshold variable, to explore the dynamics of daily stock-index returns for six international markets from 1985 to 2001. Their results provide strong evidence supporting an asymmetric nonlinear spillover effect from the US market to other markets in Europe and Asia. The US market news transmits asymmetrically, around a threshold value, to each of the national stock markets considered with average volatility in each national market much higher following bad US news. Further, Chen and So [14] explore a range of international markets to use as exogenous threshold values in a double threshold GARCH model. They find that the Japanese market has little spillover or threshold nonlinear effect on mean returns in Asian markets, in comparison with the US market. These results are supported by Wang and Firth [15], who find the emerging market of China does not exhibit significant spillover effects to other markets worldwide, including those in Asia. The US return has thus evolved as the preferred threshold variable in the examination of return spillover and nonlinear asymmetric effects. However, this previous work ignores the possible correlation between the stock price or return and trading volume. Numerous financial studies have documented this important relationship. Clark [16] and Epps and Epps [17] suggested that trading

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volume is a good proxy for information arrival from the capital market. The hypothesis has been further supported by empirical evidence; Lamoureux and Lastrapes [18], Kim and Kon [19], Andersen [20], Gallo and Pacini [21] found the same effect for the US stock market; Omran and McKenzie [22] observed this effect for the UK stock market; Bohl and Henke [23] reported similar evidence for the Polish stock market. Ying [24] was the first to provide strong empirical evidence supporting an asymmetric relation between trading volume and price-change. By investigating six series of daily data from NYSE, Ying made the following conclusions: a small trading volume is usually accompanied by a fall in price; a large volume is usually accompanied by a rise in price; and a large increase in volume is usually accompanied by either a large rise in price or a large fall in price. These propositions lay an important foundation for our nonlinear asymmetric hypothesis and illustrate that a linear relationship between price return and volume, and/or volatility and trading volume, may not be sufficient to capture the true relationship. This hypothesis is also documented by Karpoff [25] in an extensive survey of research into the relationship between stock–price change and trading volume. Karpoff suggests several reasons why the volume–price change relationship is important and provides evidence to support the asymmetric volume–price change hypothesis. His asymmetric hypothesis implies that the correlation between volume and price change is positive when the market trend is going up, but that this correlation is negative when the market trend is downwards. This is again important and highlights that we should not simply add a linear exogenous volume term to the mean equation in a GARCH model for stock returns. To capture the possible nonlinearity we will consider an asymmetric piecewise linear relationship between price (return) and volume, as can be captured by threshold models [26]. Departing from traditional work that focused on the contemporaneous relation between return and trading volume, Chordia and Swaminathan [27] examine the causal relationship and the predictive power of trading volume on the short-term stock return. Their empirical evidence suggests that volume plays a substantial role in the dissemination of national market-wide information. In a dynamic context, Lee and Rui [28] utilize the GARCH(1,1) model to investigate the relationship between stock returns and trading volume using the New York, Tokyo and London stock markets. Their empirical results suggest that US financial market variables, in particular US trading volume, have extensive predictive power in both the domestic and cross-country markets, after the 1987 market crash. Moosa et al. [29] employ a bivariate VAR model and find significant mean level asymmetry in the price–volume relationship for the future market in crude oil prices; they did not consider a heteroscedastic model and they enforced the threshold variable to be zero. The above findings further enforce our belief that a consideration of trading volume as a threshold variable might add to the understanding of capital market behavior in general. The major objective of this study is thus to investigate whether stock returns, volatility and international return spillover effects react in a threshold nonlinear fashion to changes in trading volume, in five international markets. As far

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as the authors are aware, this is the first time that trading volume has been employed as a threshold variable in a dynamic heteroscedastic model of stock returns. To address the above issue, we follow Chen et al. [13] and employ a doublethreshold autoregressive GARCH model, using a Bayesian estimation approach through Markov chain Monte Carlo (MCMC) methods. This new model allows threshold nonlinearity in both the mean and volatility processes to be driven by lagged changes in trading volume. We compare this model to the US news threshold model in Ref. [13] and a symmetric GARCH model. We utilize the posterior odds ratio, the standard Bayesian model comparison technique, as in Ref. [30], to determine which is the most favored model in each market. We also employ the causality tests of Ref. [29] to motivate and validate our models. Building on the work of Ying [24], Karpoff [25] and Moosa et al. [29], our empirical results provide strong evidence supporting the return-volume nonlinear threshold relation. The stock return, volatility and international return spillover do react asymmetrically, around a threshold value of change in trading volume, in the five markets considered. Our findings shed new light on the application of trading volume in the market integration literature and develop a new avenue for asset pricing in a multi-market framework. The remainder of this study proceeds as follows. Section 2 describes the data used in this study and presents some statistical properties of the stock returns in a standard GARCH(1,1) specification. Section 3 discusses the Bayesian methods for estimation and present the double threshold models considered. Section 4 discusses the estimated results for each model, and compares the findings with the existing literature. Section 5 discusses the Bayesian model comparison methods and discusses the findings for each market. Section 6 contains concluding remarks.

2. Data and basic statistics The analysis undertaken in this article is based on daily closing prices and trading volume of five stock market indexes: the Korean Composite Price Index (South Korea), Thailand SET index (Thailand), the Taiwan Stock Exchange Weighted Stock Index (Taiwan), the CAC 40 (France), and the FTSE 100 (UK). We also employ the US Dow Jones Industrial Average as a benchmark. The data, obtained from Datastream International, run from January 1, 1994 to November 26, 2003. The market return at time t is defined by Rt ¼ ðlnðpt Þ
lnðpt
1 ÞÞ  100%, where pt is the price index at time t. Time series plots for each market return series are given in Fig. 1. We also consider the series of logarithm of trading volume over the same time period in each market. The variable change in volume at time t is defined as V t ¼ c  ðlnðvt Þ
lnðvt
1 ÞÞ  100%, where vt is the trading volume at time t. The multiple c is used here because the scales of the market return and percentage change in volume series are very different. We thus scale the change in volume series by a constant c, chosen to allow the series V to have a similar scale to the return series R. In order to form a statistically adequate model, the variables should first be checked as to what extent they can be considered stationary. The results of applying

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TWSI

SETI 10

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1000 DJIA

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-4 -6

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Fig. 1. Time series plots for the five market returns and the returns of US Dow Jones Industrial Average as a benchmark.

an augmented Dickey–Fuller (ADF) test to the logs of the price, returns, log-volume and volume changes are shown in Table 1. We conclude from the p-values in this table that the log-price and log-volume series contain unit roots, while the returns and volume change series appear stationary in mean. To provide a general understanding of the nature of each market’s returns, we summarize the daily return and scaled volume change statistics in Table 2. Perhaps due to the Asian financial crisis within this time period, all three Asian markets exhibit negative mean returns, while the UK, France and the US have positive mean returns over this period. All six return series exhibit the standard property of asset return data: they have fat-tailed distributions, as indicated by the positive coefficient of excess kurtosis. This characteristic is also shown by the highly significant Jarque–Bera normality test statistics, a joint test for the absence of skewness and kurtosis. All series are negatively skewed, except Thailand.

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Table 1 ADF tests for a unit root on log-prices, return, log-volume, and log-volume changes Index

Taiwan weighted

Thailand SET

South Korea SE composite

France CAC 40

UK FTSE 100

ADF statistic (log-price) ADF statistic (return) ADF statistic (log-volume) ADF statistic (volume changea)


1.8593


1.8327


2.0664


1.1472


1.4719


32.6736 ðo0:01Þ
3.1763


32.5623 ðo0:01Þ
2.4134


35.5427 ðo0:01Þ
1.6049


35.9939 ðo0:01Þ
2.2453


37.3928 ðo0:01Þ
2.1092


49.3402 ðo0:01Þ


23.3543 ðo0:01Þ


21.4808 ðo0:01Þ


43.8022 ðo0:01Þ


31.6039 ðo0:01Þ

a

V t ¼ c  ðlnðvt Þ
lnðvt
1 ÞÞ  100%, where vt is the trading volume at time t and c ¼ 0:1.

Table 2 Summary statistics: Stock Index Returns and volume change Return

Taiwan weighted

Thailand SET

South Korea SE composite

France CAC 40

UK FTSE 100

US DJI

Mean Std. Skewness Kurtosis Minimum Maximum Observations J–B testa


0.0038 1.7156
0.0824 1.9856
9.9360 8.5198 2408 398.3084 (o0.01)


0.0422 1.8673 0.4619 3.7963
10.0280 11.3495 2421 2039.7620 (o0.01)


0.0049 2.1918
0.0647 3.0240
12.8047 10.0238 2401 1298.5400 (o0.01)

0.0162 1.4830
0.0658 2.1205
7.6780 7.0023 2491 453.2870 (o0.01)

0.0103 1.1549
0.1574 2.3953
5.8853 5.9026 2498 714.1045 (o0.01)

0.0384 1.1220
0.2648 4.0236
7.4549 6.1547 2491 1957.0780 (o0.01)


0.0015 2.3952 0.1414 1.1420
12.8846 12.7297 2408 137.7552 (o0.01)

0.00093 3.7479 0.2297 1.3255
24.6453 18.1687 2421 197.0956 (o0.01)

0.0133 2.1108 0.3449 3.4734
9.7940 14.7527 2401 1247.7470 (o0.01)

0.0109 3.8009 0.0234 2.7646
25.9523 23.5100 2491 788.9356 (o0.01)

0.0078 2.9139 0.0523 4.7098
20.0322 19.7815 2498 2298.3250 (o0.01)

0.0045 2.4186 0.0889 5.6539
15.0145 18.3530 2491 3301.8310 (o0.01)

Volume changeb Mean Std. Skewness Kurtosis Minimum Maximum Observations J–B testa

a The Jarque–Bera normality test statistic and p-value are listed. The lower p-value indicates the null hypothesis of normality can be rejected. b V t ¼ c  ðlnðvt Þ
lnðvt
1 ÞÞ  100%, where vt is the trading volume at time t and c ¼ 0:1.

We choose to employ autoregressive GARCH-type models for analysis. It is of interest to examine whether the stock returns on price indexes in the advanced markets are affected by international market return spillover. Possible choices of exogenous factor here include the US, Japan and the emerging market in China.

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However, Chen and So [14] illustrate that the US market dominates Japan in terms of return spillover effects in Asian markets. China’s stock market has recently become more correlated with major international financial markets, but it is still an emerging market in nature. Previous literature suggests that return spillover effects are unidirectional, transmitted from advanced financial markets to lessdeveloped financial markets. This argument was further confirmed in recent work by Wang and Firth [15]. Their findings suggest the existence of only unidirectional return spillover effect from the US to China. Thus, for the time being it seems inappropriate to consider China as an exogenous factor. We thus consider only the US stock market for return spillover effects. It is important when doing this to note that stock markets in different countries operate in different time zones with subsequently different opening and closing times. To compare the realized returns for international markets in a given calendar day in different real-time periods is difficult. However, the stock trading at New York City (the Dow Jones Index in the US market) is the last one to close among the international stock exchanges under investigation. So the closing news in the US market at day ðt
1Þ will have sufficient time to transmit to the Southeast Asian markets and various European markets, see Ref. [13]. We employ standard Granger bi-directional causality tests to examine, using a bivariate VAR model between the US and each market separately and employing a regression F-statistic, whether each domestic market is significantly affecting the US (in mean) and/or vice versa. The results from these tests are presented in Tables 3. Clearly, the US is significantly affecting each market’s return in the mean, while each market has a much weaker and mostly insignificant effect on the US market return. For this reason, we restrict ourselves to a univariate, as opposed to a bivariate, model and include a 1-day lagged cross-asset return in the mean equation. We also assume the conditional variance is a

Table 3 Causality tests for stock returns in global markets Causal direction

Lag 1 F test

p-value

Lag 1–2 F test

p-value

US US US US US

171.99 228.45 142.99 76.72 112.36

0.0000 0.0000 0.0000 0.0000 0.0000

87.80 122.18 71.47 38.48 56.85

0.0000 0.0000 0.0000 0.0000 0.0000

! ! ! ! !

FR UK SK THAI TWN

FR ! US UK ! US SK ! US THAI ! US TWN ! US

3.70 2.712 0.037 0.187 0.309

0.054 0.100 0.847 0.666 0.578

3.134 2.561 2.945 0.212 0.517

0.044 0.077 0.053 0.809 0.596

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GARCH(1,1) process Model 1 Rit ¼ f0 þ f1 Rit
1 þ c1 Rjtþm
1 þ at , pffiffiffiffi at ¼ ht e t , ht ¼ a0 þ a1 a2t
1 þ b1 ht
1 ;

et tðnÞ ,

ð1Þ

where m is the time difference between i and j markets, Rit and Rjt are stock returns from countries i (i ¼ Taiwan, Thailand, South Korea, France and the UK) and j (the US), respectively; ht is the conditional variance; f0 , f1 , c1 , a0 , a1 and b1 are unknown parameters; and at is a random error term following a standardized tdistribution with n degrees of freedom, mean 0 and variance 1. As in Ref. [13] we estimate the degrees of freedom parameter n as part of the Bayesian estimation, see details in Appendix. 2.1. Bayesian estimation and preliminary results We employ Bayesian MCMC methods to estimate the model parameters, as in Ref. [13]; details are given in an Appendix. Briefly, MCMC is an iterative sampling scheme that, in turn, samples parameter values as follows:

  

Sample f0 ; f1 ; c1 jointly conditional on the data R and the other current parameter values. Sample a0 ; a1 ; b1 jointly conditional on the data R and the other current parameter values. Sample n conditional on the data R and the other current parameter values.

The Bayesian approach has the advantages over classical statistical methods of simultaneous inference, and incorporation of any prior information, on all model parameters. The priors used are described in Appendix, but are mostly chosen to be uninformative over the standard region ensuring stationarity and finite, positive variance in the model. Estimates of Model 1, for each of the five series, are presented in Table 4. This table contains the posterior means, together with standard errors, for the unknown parameters (f0 , f1 , c1 , a0 , a1 , b1 ). Here we see that, as expected, indeed there are significant spillover effects from the US market to each national stock index, as measured by the significantly positive estimates of c1 . We note the standard empirical result, high level of persistence in volatility in each market, with estimates of a1 þ b1 between 0.96 and 0.99 for each market. There is significant negative firstorder persistence in mean in the UK and French markets, but significant positive persistence in mean in the Thailand and South Korea markets, after accounting for the exogenous US mean effect. Finally, the degree of freedom parameter is especially low in the three Asian markets, indicating a higher level of kurtosis in the error distribution compared to the UK and France. This justifies the use of the tdistribution and indicates the tails of the conditional error distribution are significantly fatter than a normal distribution.

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Table 4 Bayesian estimates for Model 1

f0 f1 c1 a0 a1 b1 n

Taiwan

Thailand

South Korea

France

UK

0.0125 (0.0303) 0.0108 (0.0207) 0.3035 (0.0290)


0.0445 (0.0301) 0.1056 (0.0214) 0.2234 (0.0276)


0.0145 (0.0325) 0.0675 (0.0204) 0.3169 (0.0351)

0.0266 (0.0244)
0.0928 (0.0224) 0.3416 (0.0274)

0.0241 (0.0171)
0.0910 (0.0213) 0.2780 (0.0206)

0.1202 (0.0463) 0.0961 (0.0190) 0.8659 (0.0304)

0.0967 (0.0365) 0.1164 (0.0214) 0.8588 (0.0284)

0.0312 (0.0125) 0.0691 (0.0142) 0.9253 (0.0152)

0.0218 (0.0078) 0.0636 (0.0092) 0.9260 (0.0108)

0.0112 (0.0037) 0.0792 (0.0116) 0.9124 (0.0125)

6.5699 (0.9262)

7.1010 (1.0312)

9.6063 (1.6805)

20.4054 (8.4478)

20.7016 (8.7777)

Figures in brackets represent posterior standard errors for each parameter.

3. Double TAR-GARCH models and Bayesian methods In forming our models, we are motivated by the causality tests of Moosa et al. [29], useful for vector autoregressive models in order to determine directions of causality or dependence. While these tests make strict assumptions: the threshold variable is exactly zero, volatility is constant and errors are Gaussian; they do allow asymmetric relationships in the mean equation. They are thus useful as an initial guide in asymmetric model choice, while being much less general than the results in this paper. The results of these tests are presented in Table 5. Clearly, the results indicate that both positive and negative US market returns significantly affect returns in all five domestic markets. This motivates the choice of the US as both an exogenous variable in the mean equation and a threshold variable. The model we employ is a Double TAR-GARCH model, which generalizes the DT-ARCH model of Li and Li [31] and Chen [32]. This model is motivated by several nonlinear characteristics commonly observed in practice, such as asymmetry in declining and rising patterns of a process. It uses piecewise linear models to obtain a better approximation of the conditional mean and conditional volatility equations based on a threshold variable. We specify the first two-regime asymmetric heteroscedastic model as 8 ð1Þ ð1Þ j i < f0 þ fð1Þ Rjtþm
d pr1 ; 1 Rt
1 þ c1 Rtþm
1 þ at ; i Model 2 Rt ¼ : fð2Þ þ fð2Þ Ri þ cð2Þ Rj Rjtþm
d 4r1 ; t
1 0 1 1 tþm
1 þ at ;

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Table 5 Causality tests without assuming symmetry for stock returns in global markets Causal direction

Lag 1 F test

p-value

Lag 1–2 F test

p-value

USþ US
USþ US
USþ US
USþ US
USþ US


55.185 76.530 64.191 106.240 35.217 60.712 19.508 63.147 23.527 55.225

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

27.103 41.704 35.989 56.563 14.397 35.548 6.333 32.180 12.634 26.672

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.002 0.0000 0.0000 0.0000

! ! ! ! ! ! ! ! ! !

FR FR UK UK SK SK THAI THAI TWN TWN

is the return on the US market, are defined as The asymmetric variables, where RUS t ( ( RUS if RUS if RUS RUS t t X0; t t p0 ; þ
USt ¼ USt ¼ 0 otherwise; 0 otherwise :

at ¼ ht ¼

pffiffiffiffi ht et ; et tðnÞ , 8 ð1Þ ð1Þ 2 < a0 þ að1Þ 1 at
1 þ b1 ht
1 ;

Rjtþm
d pr1 ;

: að2Þ þ að2Þ a2

Rjtþm
d 4r1 ;

0

1

t
1

þ bð2Þ 1 ht
1 ;

ð2Þ

where m is the time different between i and j markets. The definition of Rit and Rjt is the same as Model 1. This is the model considered in Ref. [13] where the mean and volatility equations can react asymmetrically to good or bad return news from the US market. This model has the advantage that the threshold parameter, usually fixed at r1 ¼ 0, and the delay parameter, usually fixed at d ¼ 1, are not specified but can be estimated from the data simultaneously with the other model parameters, in a Bayesian framework. 3.1. Threshold trading volume DT-GARCH model Model 2 above ignores any potential relationship between trading volume and price movements on the stock market. Financial theory suggests that volume and stock return are strongly associated and many authors have found empirical results to back this up, see Refs. [27,28] among others. To motivate our model, we again employ the Moosa et al. [29] causality tests, without assuming symmetry, applied to each market’s volume-change and return series together in a bivariate VAR model. Results are presented in Table 6. Here we see, assuming a zero threshold, constant volatility and Gaussian errors (these assumptions are violated for these markets, see Table 8) that results are somewhat mixed, however, this test is meant as an initial guide only. Both positive and negative changes in trading volume significantly affect

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returns in South Korea (5% level), while Taiwan is affected strongly by only positive volume-change at 5% level. Also, at the 10% level, decreasing volume affects returns on the French and Taiwanese markets. However, the UK and Thailand seem not be significantly affected in the mean equation by trading volume-change. We also observed the standard causality tests, not presented here to save space, which illustrated that volume was not a significant exogenous factor in these markets. These results suggest that volume may potentially affect market returns asymmetrically and motivates the choice of volume as a threshold variable, but not as an exogenous variable in the mean equation. We thus develop a new model in this paper, motivated by the work of Ying [24] and Karpoff [25], attempting to capture the potentially asymmetric reaction of stock prices to changes in trading volume. We introduce a Double TAR-GARCH model using change in trading volume as the threshold variable. This is the first time that trading volume has been employed as a threshold variable in an asymmetric heteroscedastic dynamic model of stock returns. The model is posited as follows: 8 ð1Þ ð1Þ j i < f0 þ fð1Þ V it
d pr2 ; 1 Rt
1 þ c1 Rtþm
1 þ at ; i Model 3 Rt ¼ : fð2Þ þ fð2Þ Ri þ cð2Þ Rj V it
d 4r2 ; t
1 0 1 1 tþm
1 þ at ; pffiffiffiffi at ¼ ht et ; et tðnÞ , 8 ð1Þ ð1Þ 2 < a0 þ að1Þ V it
d pr2 ; 1 at
1 þ b1 ht
1 ; ð3Þ ht ¼ : að2Þ þ að2Þ a2 þ bð2Þ ht
1 ; V i 4r2 ; t
d 0 1 t
1 1 where m is the time difference between markets i and j. Here, V it ¼ c  ðlnðvit Þ
lnðvit
1 ÞÞ  100% is the change in trading volume, where vit is the trading volume of market i at time t and c ¼ 0:1. This model will allow us to test the major hypothesis about asymmetries between mean and volatility of returns in reaction to changes in trading volume, and will allow us to empirically confirm the findings and theories of Ying [24] and Karpoff [25] in a GARCH-type dynamic volatility setting. The model allows trading volume to drive or influence stock returns in a piecewise linear fashion, allowing different return behavior following increasing or decreasing volume. Again, the threshold change in volume level and delay lag parameter are estimated simultaneously with the unknown model parameters, a significant advantage of the Bayesian approach to estimation. We employ a similar MCMC method to that for Model 1, as in Ref. [13], to estimate both Models 2 and 3. See Appendix for some details of this approach. Briefly, this involves the same iterative sampling scheme as for Model 1 (the two steps above are performed separately for each regime) with the addition of the steps:

 

Sample rk conditional on the data R and the other current parameter values. Sample d from it’s discrete posterior distribution conditional on the data R and the other current parameter values.

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The prior distributions on model parameters here are described in Appendix, but are mostly chosen to be uninformative over the region ensuring stationarity and finite variance in each model.

4. Results for threshold models Results from Model 2, with a US news threshold parameter, are presented in Table 7. These illustrate a clear nonlinear asymmetric reaction to US return news in each of the five markets, similar to results found in Ref. [13]. Firstly in each market, the mean reversion is faster in the negative regime, following bad news from the US ð2Þ market, i.e., fð1Þ 1 of1 , with these parameters being significantly different to zero in all five markets following good or positive US return news. Again, this local persistence in return is positive in the three Asian markets, but negative in UK and France. In addition, the spillover effect from the US market is significantly greater ð2Þ following bad US news, i.e., cð1Þ 1 4c1 in each market. We note that the spillover effect from the US market is significantly stronger than the local return persistence in each of the five markets, as indicated by jc1 j4jf1 j and jc1 j4jf0 j in each regime. Also, the persistence in volatility, as measured by a1ðjÞ þ bðjÞ 1 is higher following bad news in each market. Table 9 highlights the asymmetric reactions in the five markets. Here, we see that volatility level increases by as little as 7.5 times and as much as 68 times following bad US news compared to that following good US news. Thailand is the exception, with volatility level appearing stable regardless of the news from the US. In fact, the asymmetric reaction in volatility appears to be much stronger in the UK and France, compared to the three Asian markets. We also see that the mean return estimate in each market has decreased from the positive to negative threshold, i.e., following bad news from the US. This mean return decrease is from positive return following good US news to negative return following bad US news, in three of the markets. We note that the threshold parameters in Table 7 are estimated to be negative and significant in four of the five markets. This illustrates that small negative US returns do not induce an asymmetric reaction and are not perceived as ‘bad news’ by these five national markets. However, significant negative US returns, that is below each threshold estimate, are needed to induce the asymmetric reaction in each market. Thailand is the exception to this rule, although the positive estimated threshold parameter is not significantly different to 0 in this case. We also note in Table 9 that the posterior mode for the delay parameter d is 1 in four of the five markets, but is two days for the Taiwan market. The degrees of freedom estimates are similar to those obtained for Model 1. Results for Model 3 are presented in Tables 8 and 9. Results here illustrate a clear nonlinear asymmetric reaction to change in trading volume in each of the five markets. Firstly in each market, the mean reversion is faster following negative ð2Þ changes or decreases in trading volume, i.e., fð1Þ 1 of1 in four of the five markets, excepting Taiwan. The international spillover effect from the US market is greater in

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ð1Þ the positive regime, following increases in trading volume, i.e., cð2Þ 1 4c1 in each market, except Thailand. We note that the spillover effect from the US market is again significantly stronger than the local return persistence in each of the five markets, as indicated by jc1 j4jf1 j and jc1 j4jf0 j in each regime. In this case, the persistence in volatility (a1 þ b1 ) is mostly higher following positive trading volume changes. Table 9 again highlights the asymmetric nonlinear reactions to trading volume change in the five markets. Here, we see that average volatility increases by as little as 1.5 times (South Korea) up to 100 times (UK, France) following increases in trading volume, compared to that following declining trading volume. This is consistent with the hypothesis of Ying [24] who suggested that large increases in volume lead to large increases or decreases in price and hence to volatility. Taiwan is the notable exception to this rule, with volatility level significantly decreasing as trading volume increases. This distinctive empirical result in the Taiwanese market is interesting in light of the under and over reaction hypothesis, in the modern theory of behavior finance, proposed by Daniel et al. [33] and Barberis et al. [34]. Their hypothesis states that individual investors tend to be over-confident and over-reliant on their own private information, in general, resulting in over-reaction to public information, on the basis that they perceive this to confirm their individual beliefs. Proposition 3 in Ref. [33] states that ‘overconfidence can increase or decrease volatility’ around public information events. The result for Taiwan thus agrees with this theory, as indeed the volatility has changed following news on trading volume, it just changes in a different way to that in the other four markets. The particular Taiwan market behavior has been well documented in previous studies of price–volume relationships with the most plausible explanation falling in the realm of market imperfection and investor irrationality [35,36]. The Taiwanese stock market’s prolonged behavioral characteristic is very short-run speculation, which may also contribute to the inherent political jitters in Taiwan, due to the Chinese Missiles incident, and economic crisis in 1997–1998. In addition, the majority (approximately over 92%) of investors are uninformed traders and individual investors. Thus, we suspect that this speculative behavior, incorporated with investor irrationality, caused the particular information flow pattern and price–volume relationship in the Taiwan stock market. On close examination of the mean and volatility of returns under Model 3 in Table 6, the evidence suggests that in regime 1 investors over-react, proxied by the relatively high volatility and lower mean return, to the negative volume change. On the other hand, the result in regime 2 suggests an under-reaction, with lower volatility, but in a bullish fashion, with increasing volume of trade. The estimates of average return in each market have increased following increases in trading volume, excepting Thailand. The greatest increases in mean return are in South Korea and Taiwan. The particular finding in the Thailand stock market is also of interest. The turmoil of the Asian currency crisis in July 1997 led the economy of Thailand into a well-documented seven-year stagnation, falling into a long sluggish bear market until June of 2003, except for a short rebound between August 1998 and June 1999. This phenomenon is reflected by the largest, in magnitude, negative mean return ð
0:0422Þ among our five sample markets in Table 2 and also a large negative

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Table 6 Causality tests without assuming symmetry for domestic returns and volume changes Causal direction

Lag 1 F test

p-value

Lag 1–2 F test

p-value

Vþ V
Vþ V
Vþ V
Vþ V
Vþ V


0.226 3.375 0.247 1.263 4.743 4.712 1.752 0.538 6.607 2.925

0.634 0.066 0.619 0.261 0.029 0.030 0.186 0.463 0.010 0.087

0.363 0.432 0.103 0.732 3.152 1.296 0.914 0.422 5.453 3.372

0.696 0.239 0.902 0.481 0.043 0.274 0.401 0.656 0.004 0.034

! ! ! ! ! ! ! ! ! !

FR FR UK UK SK SK THAI THAI TWN TWN

The asymmetric variables are defined, where V t is volume-change, as (  V t if V t X0; V t if V t p0 ; ¼ V
Vþ t t ¼ 0 otherwise; 0 otherwise :

volume threshold value of
1:5375 in Table 7. Financial theory suggests that trading volume tends to expand in the direction of the major market trend. If the observed stock market is up in trend, the volume should increase with price increase, and the volume should diminish as price decreases. Conversely, if the major trend of the stock market is down, the volume should diminish as price rallies up but expand as price dips down. The relatively large magnitude of negative mean return ð
0:0385Þ of regime 2 under Model 3 in Table 6 is consistent with these findings in the literature, a large increase in volume is usually accompanied by either a large rise in price or a large dip in price, the dip here associated with a period of poor market return performance in Thailand. These results suggest very clear and significant asymmetric and nonlinear reactions in each market to changes in trading volume and support the original work by Ying [24] and Karpoff [25]. The results suggest that the mean return in each market is higher (except Thailand) and the volatility level is much higher (except Taiwan) when trading volume increases: We note that the threshold parameters in Table 8 are estimated to be positive and significant in three of the five markets. This illustrates that small increases in trading volume do not induce an asymmetric reaction and are not perceived as significant by these five national markets. However, significant positive changes in volume, that is above each positive threshold estimate, are needed to induce the asymmetric reaction in Taiwan, South Korea and France. Thailand is again the exception to this rule, as explained above. We also note in Table 9 that the posterior mode for the delay parameter d is 1 day in all five markets. The degrees of freedom estimates are similar to those obtained for Model 2.

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Table 7 Bayesian estimates for Model 2

f0ð1Þ

Taiwan

Thailand


0.0051


0.0707

South Korea

France

UK

0.2828

0.0561

0.1072

f1ð1Þ

(0.0697) 0.0780

(0.0621) 0.0981

(0.1372) 0.0153

(0.0756)
0.1119

(0.0570)
0.1198

c1ð1Þ

(0.0377) 0.3030

(0.0298) 0.2497

(0.0418) 0.5464

(0.0394) 0.404

(0.0367) 0.3827

(0.0502)

(0.0537)

(0.1027)

(0.0691)

(0.0497)

f0ð2Þ

0.0864

0.083

0.0414

0.0091

f1ð2Þ

(0.0410) 0.1429

(0.1075) 0.1034

(0.0455) 0.0895

(0.0400)
0.0814

(0.0302)
0.0824

c1ð2Þ

(0.0257) 0.0890

(0.0366) 0.1808

(0.0244) 0.3224

(0.0269) 0.2855

(0.0279) 0.2623

(0.0392)

(0.0751)

(0.0550)

(0.0502)

(0.0365)


0.0435

a0ð1Þ

0.0869

0.0892

0.1614

0.1234

0.0514

a1ð1Þ

(0.0965) 0.1134

(0.0568) 0.1129

(0.0734) 0.1036

(0.0315) 0.0939

(0.0152) 0.1066

b1ð1Þ

(0.0284) 0.8608

(0.0282) 0.8639

(0.0248) 0.8811

(0.0197) 0.8985

(0.0231) 0.8871

(0.0400)

(0.0364)

(0.0296)

(0.0213)

(0.0240)

a0ð2Þ

0.1897

0.2018

0.019

0.0156

0.0051

a1ð2Þ

(0.0524) 0.1424

(0.1027) 0.1464

(0.0133) 0.0734

(0.0117) 0.043

(0.0040) 0.0542

b1ð2Þ

(0.0234) 0.8129

(0.0342) 0.801

(0.0197) 0.9149

(0.0121) 0.9114

(0.0159) 0.9117

(0.0374)

(0.0481)

(0.0219)

(0.0161)

(0.0192)


0.2410 (0.1332) 6.6773 (0.9553)

0.2697 (0.3178) 7.2158 (1.0223)


0.4208 (0.1197) 9.8639 (1.8114)


0.2269 (0.0697) 21.9762 (13.9071)


0.1735 (0.0722) 25.3133 (18.6993)

r1 n

Figures in brackets represent posterior standard errors for each parameter. r1 : threshold value for US returns.

5. Model comparison We use model selection to determine whether the threshold trading volume model, Model 3, adds any value over the simple and DTAR-GARCH models, Models 1 and 2. To compare models pairwise we use the posterior odds ratio which is the ratio of integrated likelihoods for each model, multiplied by the prior odds ratio. This is a Bayesian decision rule where for any two models A and B the posterior odds ratio, showing evidence in favor of A

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Table 8 Bayesian estimates for Model 3 Taiwan

Thailand

South Korea

France

UK

f0ð1Þ


0.0690


0.0369


0.2520

0.0193

0.0124

f1ð1Þ

(0.0380)
0.0027

(0.0549) 0.0160

(0.0432) 0.0275

(0.0303)
0.1293

(0.0247)
0.1447

c1ð1Þ

(0.0290) 0.2988

(0.0560) 0.2537

(0.0242) 0.2642

(0.0288) 0.2722

(0.0316) 0.2286

(0.0344)

(0.0481)

(0.0396)

(0.0360)

(0.0340)

f0ð2Þ

0.2261

0.6071

0.0441

0.0346

f1ð2Þ

(0.0667)
0.0431

(0.0367) 0.1309

(0.0786) 0.0910

(0.0411) 0.0207

(0.0245)
0.0266

c1ð2Þ

(0.0335) 0.3147

(0.0248) 0.2103

(0.0375) 0.3338

(0.0436) 0.4280

(0.0372) 0.3058

(0.0546)

(0.0343)

(0.0657)

(0.0529)

(0.0278)


0.0712

a0ð1Þ

0.1538

0.1810

0.0244

0.0082

0.0067

a1ð1Þ

(0.0562) 0.1348

(0.1169) 0.0905

(0.0153) 0.0785

(0.0085) 0.0633

(0.0069) 0.0938

b1ð1Þ

(0.0284) 0.8438

(0.0508) 0.8344

(0.0169) 0.9150

(0.0158) 0.8982

(0.0194) 0.8703

(0.0353)

(0.0539)

(0.0179)

(0.0186)

(0.0258)

a0ð2Þ

0.1265

0.0733

0.0732

0.1263

0.0323

a1ð2Þ

(0.0941) 0.0534

(0.0416) 0.1171

(0.0452) 0.0806

(0.0385) 0.1020

(0.0109) 0.0825

b1ð2Þ

(0.0223) 0.8714

(0.0218) 0.8686

(0.0230) 0.9075

(0.0238) 0.8868

(0.0193) 0.9119

(0.0522)

(0.0253)

(0.0263)

(0.0270)

(0.0203)

1.1448 (0.3979) 6.6030 (0.9211)


1.5375 (0.9330) 7.2002 (1.0382)

1.0179 (0.1962) 8.5599 (1.3779)

1.0832 (0.6038) 22.3516 (12.9051)


0.1654 (0.4197) 21.4837 (10.0179)

r2 n

Figures in brackets represent posterior standard errors for each parameter. r2 : threshold value for volumechange.

over B is ORA;B ¼

pðAjRÞ pðRjAÞPrðAÞ ¼ . pðBjRÞ pðRjBÞPrðBÞ

The decision rule is to choose Model A if ORA;B 41, otherwise choose model B. See Ref. [37] for a complete discussion of Bayesian decision rules. We use the integrated or marginal likelihood function pðRjAÞ here, so that we may integrate over the possible values of unknown parameters based on the observed sample data, and thus

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Table 9 The estimates of means and unconditional variance for each regime Taiwan Model 1 m a0 1
a1
b1 Model 2 mð1Þ mð2Þ a0ð1Þ

Thailand

South Korea

France

UK

0.0243 3.1627


0.0402 3.9258


0.0024 5.6165

0.0380 2.0900

0.0337 1.3417


0.0768 0.0777 9.5368


0.2108 0.1824 3.2780


0.3954 0.1484 13.4063


0.2939 0.1985 19.0026


0.2101 0.1601 11.1588

1.3355

4.2612

1.4863

0.2911

0.1235

2

1

1

1

1
a1ð1Þ
b1ð1Þ a0ð2Þ 1
d1

a1ð2Þ




b1ð2Þ

Model 3 mð1Þ mð2Þ a0ð1Þ

1


0.0614 0.2279 7.6983


0.0318
0.0385 2.3874


0.2629 0.6763 4.1567

0.0223 0.0721 0.1696

0.0088 0.0568 0.1010

1.4935

5.7459

6.0943

14.7160

8.0672

1

1

1

1
a1ð1Þ
b1ð1Þ a0ð2Þ 1
d1

a1ð2Þ




b1ð2Þ 1

1

account for parameter uncertainty. We assume prior model ignorance so that the prior ratio is 1, equivalently PrðAÞ ¼ PrðBÞ ¼ 0:5. To estimate the integrated likelihood for each model can be quite difficult, with many Bayesian approaches suggested in the literature, see Refs. [38–40] for examples. We employ the method suggested in Ref. [30], employing importance sampling techniques. These methods are implemented in Ref. [14], who give a detailed exposition of this work applied to extended threshold GARCH models. See Appendix for details. Table 10 contains the logarithm of the odds ratio comparing each pair of Models 1, 2 and 3. The first listed model is preferred over the second listed model if the number in the table is positive (and hence the corresponding odds ratio is 41), otherwise the second model is preferred. The boxed number is for the most preferred model in each market, confirmed by the final line of the table. It is interesting to note that the model with trading volume as a threshold variable, Model 3, is strongly preferred over both Models 2 and 1 in the three Asian markets: Taiwan, Thailand and South Korea. Using the Bayesian odds ratio interpretation in Ref. [39], this is ‘very strong’ to ‘decisive’ evidence in favor of the volume threshold asymmetric model. In contrast, however, for the European markets of the UK and France, the asymmetric model with lagged US news as the threshold variable is preferred over

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Table 10 Logarithm of the posterior odds ratio

Model 2 vs Model 1 Model 3 vs Model 1 Model 3 vs Model 2 Preferred model

Taiwan

Thailand

South Korea

France

UK


2.1363 6:3943 8:5306 Model 3

3.4860 8:9871 5:5011 Model 3


8.4749 3:6360 12:1109 Model 3

1:7713
0.4313
2.2026 Model 2

1:5733
5.8874
7.4607 Model 2

Models 1 and 3. This represents ‘substantial’ evidence in favor of this model for these two markets. Clearly, the asymmetric effect from trading volume is comparatively strongest in the Asian markets considered, while the US trading news dominates trading volume news in the UK and France markets. Table 10 also shows that the symmetric GARCH model with leptokurtic innovations, Model 1, ranked second in four of the five markets. This result is not surprising, and confirms previous findings, such as those in Ref. [41], that a simple GARCH formulation with t-errors can sometimes outperform more complex models for real market returns. It is hard to make the usual good news–bad news financial argument with this trading volume threshold model, as asymmetric increases in volatility also coincide with increases in mean return in each market; following an increase in trading volume. However, this is a common finding in many financial studies; an increased level of return is associated with, and often caused by, an increased level of risk (volatility) [42]. The traditional bad news argument can only be applied in Taiwan, where investors seem to perceive decreases in trading volume as bad news, as this result is associated with lower negative mean returns and higher return volatility.

6. Conclusions In this paper, we have thoroughly examined the empirical dynamic relationship between stock return, return volatility, international return spillover and change in trading volume for five international stock markets. Conforming to well-established empirical results, stock returns and return volatility display a certain degree of persistence and consistent with a meteor-shower hypothesis, our empirical results suggest that the stock-return news developed from US is transmitted significantly to each of the five national stock markets. In particular, the US return-news is positively correlated with the stock price of each national market but this correlation is asymmetric around a threshold US return value and/or a value of trading volume change. This confirms the minor goal of this paper. The major issue of this study has been whether stock returns and return volatility react asymmetrically, in a threshold nonlinear fashion, to trading volume change. By employing a double-threshold GARCH model to capture the nature of market reaction to trading volume change, we find that a large increase in volume is accompanied by a large rise in average

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stock price, while a decrease in volume is associated with lower average returns, with the exception of Thailand. Moreover, the return volatility of all the stock-markets also displayed an asymmetric reaction to volume change, around a threshold level. With the exception of Taiwan, the average magnitude of return volatility following an increase in trading volume was much larger than that following decreases in volume. Our model comparison results revealed that the asymmetric nonlinear model, with change in volume as a threshold, was decisively favored in the three Asian markets: Taiwan, Thailand and South Korea. However, the DT-GARCH model of Chen et al. [13], with US return news as threshold variable, was favored in UK and France. In summary, our findings are consistent with, and add to, the arguments of Ying [24], Karpoff [25] and Gallant et al. [43]: that the joint study of stock price and trading volume, as an asymmetric nonlinear relationship, leads to a better understanding of capital market behavior. These findings shed new light on the application of trading volume in the market integration literature and develop a new avenue for asset pricing in a multi-market framework.

Acknowledgements The authors thank two anonymous referees and the Editor, H. Eugene Stanley, whose comments improved the paper. C.W.S. Chen is supported by National Science Council (NSC) of Taiwan grants NSC93-2118-M-035-003. R. Gerlach was supported by the Mathematical Research Promotion Center of the NSC of Taiwan, Feng Chia University and the University of Newcastle, via an ECA networking grant and the School of MAPS.

Appendix A This appendix gives some details for the Bayesian MCMC sampling scheme used to estimate the GARCH models in this paper and of the method used to estimate the marginal likelihood required in the posterior odds ratio used for model comparison. For further details of MCMC sampling see Ref. [44] or [14]. A.1. MCMC methods Firstly, let

  

ðjÞ ðjÞ /ðjÞ ¼ ðfðjÞ j ¼ 1; 2, 0 ; f1 ; c1 Þ; ðjÞ aðjÞ ¼ ða0ðjÞ ; aðjÞ ; b Þ; j ¼ 1; 2, 1 1 ð1Þ ð2Þ ð1Þ ð2Þ H ¼ ð/ ; / ; a ; a ; rk ; dÞ.

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The likelihood function pðRjHÞ is given by (  
ðnþ1Þ=2 ) n 2 X Y Gððn þ 1Þ=2Þ 1 ðRt
mt Þ2 2;n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 1 þ pðR jHÞ ¼ I jt , ðn
2Þht ht t¼2 j¼1 Gðn=2Þ ðn
2Þp where I jt is the indicator variable IðRjtþm
d pr1 Þ, Gð:Þ is the Gamma function and n is the degrees of freedom. For Bayesian inference, we need to set prior distributions on ð1Þ ð1Þ ð2Þ all model parameters. We use the prior pðHÞ / Iðað1Þ 0 40; a1 þ b1 o1ÞIða0 40; ð2Þ ð2Þ a1 þ b1 o1ÞIðq1 ork oq3 ÞIðd 2 1; 2; 3ÞIðt 2 ½0; 0:25Þ, where IðÞ is an indicator function, t ¼ 1=n and q1 and q3 are the first and third quantiles, respectively, of the required threshold variable. In threshold modelling, it is important to set a minimum amount of observations in each regime, so there is sufficient sample size to generate meaningful inference results, driving our prior choice for rk here. We restrict n44 so that the variance of
t is finite, while ensuring that the kurtosis is greater than 3. Finally, for the mean equation parameters we assume the normal prior /j Nð/j0 ; V j Þ, where usually we set /j0 ¼ 0 and V
1 to be a large number, j such as 1, to ensure a reasonably flat prior. Incorporating the likelihood above and the priors, using Bayes rule, leads to the conditional posterior distributions for the parameter groupings we use in the MCMC sampling scheme. We use the following iterative sampling scheme to construct the desired posterior sample: 1. 2. 3. 4. 5.

Draw the vector /ðjÞ jR; H
fj for j ¼ 1; 2, using the random walk MH algorithm. Draw the vector aðjÞ jR; H
aj for j ¼ 1; 2 using the random walk MH algorithm. Draw the parameter rk jR; H
r using the random walk MH algorithm. Draw djR; H
d by noting that d is discrete valued and using it’s discrete posterior. Draw t using the random walk MH algorithm.

These posterior distributions are in general not of a standard form and require us to employ techniques such as the Metropolis random walk method [45,46] to achieve the desired sample, as detailed now. Consider a general parameter vector q, a subset of H. The posterior distribution for this parameter vector is evaluated by pðqjR2;n ; H
q Þ / pðR2;n jHÞpðqÞ . Details of the random walk MH algorithm are: Step 1: Generate initial values q½0 from the prior distribution for this parameter vector Step 2: At iteration i, generate a point q from the kernel density, q Nðq½i
1 ; aOÞ , where q½i
1 is the ði
1Þth iterate of q. Step 3: Accept q as q½i with the probability p ¼ minf1; pðq jR2;n ; H
r Þ= pðq½i
1 jR2;n ; H
r Þg. Otherwise, set q½i ¼ q½i
1 . To yield good convergence properties, the choices of O and a for each parameter vector are important. These choices can be made to ensure good coverage for each

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conditional posterior distribution, with O usually chosen with sufficiently large diagonal terms, while the coverage and sampling efficiency can also be tuned using the acceptance rate, in the burn-in period of the MCMC sample; see Ref. [14] for details. A suitable value of a, with good convergence properties, can usually be selected by having an acceptance probability of between 25% and 50% in the burnin period. Parameter estimates of any functions of parameters are obtained as posterior means by averaging the function over all sample iterates after the burn-in period. A.2. Marginal likelihood estimation We follow Gerlach et al. [30] in estimating the marginal likelihood term for each Q model pðRi Þ ¼ nt¼1 pðRit jR1;t
1 Þ by estimating the respective marginal term at each time t using PD pðRi jR1;t
1 ; Yk½i Þ=pðRt;k jR1;t
1 ; Y½i 1;t
1 i kÞ ^ t jR , Þ ¼ i¼1 PtD pðR ½i t;k 1;t
1 ; Yk Þ i¼1 1=pðR jR n Y ^ iÞ ¼ ^ it jR1;t
1 Þ , pðR pðR t¼1

Y½i k

where is the ith MCMC iterate from the posterior distribution of the unknown parameter vector YjR1;k , conditional on the first k observations only. Following Gerlach et al. [30], we choose a maximum value for k
t ¼ 300 and allow ^ it jR1;t
1 Þ for t ¼ k
k ¼ 300; 600; . . . ; n. For each k, we simultaneously compute pðR 300; . . . ; k. This means we need to run the MCMC sampling scheme multiple times (in fact n=300 integer rounded times) for each model. While this process can be more time consuming than 1 single MCMC run, we consider the gain from including parameter uncertainty in model comparison is sufficient, from a Bayesian viewpoint, to justify this procedure.

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