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Technical range spilloverThe question of how changes on asset price and volatility are propagated across markets is of great interest in academic research. This question is highly related to the efficiency of financial markets. The efficient market hypothesis (EMH) interprets the interdependence of stock returns and volatility as an informational link across markets: news revealed in one country is decoded as informative to fundamentals of stock price in another country. This view can be attributed to real and financial linkage of economies.^{1} While the behavioral finance (BF) claims the spillover as market contagion: stock prices in one country are affected by changes in another country beyond what is conceivable by connections through economic fundamentals.^{2} This chapter will demonstrate how the statistical properties of the technical range can be used to investigate the spillover effect across markets. 9.1 Introduction Financial markets have witnessed in both developed and developing countries liberalized capital movements, financial reforms, advances in computer technology and information processing, which greatly reduce the isolation of domestic markets and increase their ability to react promptly to news and shocks originating from the rest of the world. This indicates that the linkages across stock markets around the world may have grown stronger. Modern portfolio theory says that gains from international portfolio diversification are inversely related to the correlation between equity returns. Investors hold various securities in the expectation of achieving a reduction in risk via diversification. In the mean variance framework, correlation is the measure of comovement in returns. Errunza (1977) demonstrated the advantage of international diversification based on low correlation between the equity markets. Therefore, understanding the interdependence in returns and volatility across different markets not only adds to insights related to diversification and hedging strategies but also contributes to devising policies related to capital inflow in the market. There is voluminous academic literature concerning information transmission across markets. Some of the literature focuses on the longterm interdependence and causality among stock markets, in order to find longterm or shortterm correlation among these markets (Eun and Shim, 1989; Nath and Verma, 2003; and Constantinou et al., 2005). Since information transmission across markets can take place through not only price changes but also price volatility, there is also growing literature concerning volatility spillover (Bekaert and Harvey, 1997; Ng, 2000; Baele, 2003; Christiansen, 2003; and Worthington and Higgs, 2004). This chapter is neither to investigate the mean spillover nor the volatility spillover, instead it is designed with an empirical study to show how the statistical properties of technical range can be used to scrutinize the information transmission across financial markets. An empirical study is performed on the German stock index (DAX) and the French stock index (CAC40). The results show persuasive evidence that there is information spillover from the German stock market to the French market, while not vice versa. This chapter is organized as follows: Section 2 presents the econometric method with some discussions. Section 3 presents the empirical results. In Section 4, we summarize. 9.2 Econometric method We have demonstrated in Chapter 4 that technical ranges are соintegrated if their corresponding closing prices are соintegrated. With this property in mind, we will present in the following that technical range spillover can be scrutinized through a vector error correction (VECM) model. Suppose two speculative assets, closing prices c_{u} and c_{2}, (in log form) are cointegrated of order (1, 1) with a cointegration vector of fi = (/J_{b} f)_{2})^{7}. By the definition of cointegration, there exists a linear combination
such that со, is 7(0). Given the facts that c_{it} « /«(TR_{it})  ln(PR_{it}) and that ln(PR_{it}) (i = 1, 2) is 7(0), we can get
where TR_{it}and PR_{it}arc the technical range (see Eq.(4.1)) and Parkinson range (see Eq.(4.2)). Since со, and filn(PR,) + fS_{2}ln(PRi,t) are stationary' processes, we can obtain that !5ln{TR _{t}) + p_{2}ln(TR_{2r}) is stationary'. Thus we get the result that log technical ranges, /«(TR, _{f}) (i  1, 2) are соintegrated so long as their closing prices are cointegrated. It is claimed that price changes are due to new information being reflected. Technical range as a measurement of price variation can thus be treated as an information proxy. In this chapter, the following vector error correction model (VECM) is proposed to investigate the information spillover effect across financial markets
where к is a lag structure, T, is a p x 1 vector of variables and is integrated of order one, /(1), ц is a p x 1 vector of constants, and e, is a p x 1 vector of white noise error terms. Tj is a p x p matrix that represents shortterm adjustments among variables across p equations at the /th lag. Ylj i ГДУ . and aft^{1}T_{t}_i are the vector autoregressive (VAR) component in first differences and errorcorrection components respectively, ft is a p x r matrix of соintegrating vectors, and a is a p x r matrix of speed of adjustment parameters. The соintegrating vector fi shows the longterm equilibrium relationship between the concerned variables while the adjustment factor a shows the speed of adjustment towards equilibrium in case there is any deviation. A larger a suggests a faster convergence toward longrun equilibrium in cases of shortrun deviations from the longrun equilibrium. In this chapter, Y_{r} = (/;/(TRi _{t}), ln(TR_{2 t}))^{r}.
We collected the monthly data of the German stock index DAX and the French stock index CAC40 for the sample period 1994.012014.12 with 252 observations. For each stock index, four pieces of price information, opening, high, low and closing prices were reported. The data sets were downloaded from http:// finance.yahoo.com. We do not use the data after 2007 for the reason that there seems to be a structure change in time series data. Figure 9.1 presents the time series plot of log closing prices of DAX and CAC40 over 1994.012014.12. From the plot, it can be observed that the deviation between DAX and CAC40 sharply increases after 2007. Table 9.1 presents the summary statistics of the log technical range and the differenced log technical range. The LjungBox Q statistics show high persistence of log technical range. The LjungBox ^statistics of differenced log technical range, though not so much larger as the technical range, are still significant at a level of 1%. The ADF testing results show: (1) at the significance level of 5% the null hypothesis of unit root process can not be rejected for the log technical range of DAX. The null hypothesis of unit root process can be rejected for the the differenced log technical range of DAX; (2) at the significance level of 5%, the null hypothesis of unit root process is rejected for the technical range of CAC40. However, the ADF testing results performed on the log closing prices of both DAX and CAC40 show that the null hypothesis of unit root process Figure 9.1 Time series plot of log closing prices, DAX and CAC40: 1994.01 2014.12 Table 9.1 Summary statistics of technical range
We use ***, **, and * to mean significance at the level of 1%, 5%, 10%. can not be rejected. By Proposition 1 in Chapter 4 (p.20), technical range is a unit root conditional on the closing price being a unit root. Therefore in this chapter, we still take the technical range of CAC40 as a unit root process. The Johansen Trace test performed on the technical ranges of DAX and CAC40 shows that there is a cointegration equation at the level of 5%. Figure 9.2 presents the time series plot of technical ranges of DAX and CAC40 over 1994.01 2007.12. The cointegration relation is clear in Figure 9.2. Figure 9.2 Time series plot of log technical range, DAX and CAC40: 1994.01 2014.12 9.3.2 Estimation The VECM model estimates are reported in Table 9.2. The lags of the VECM model are determined by SIC criteria. In this VECM model, the lagged returns on the closing prices of DAX and CAC40 are used as exogenous variables. The reason is that closing price is exogenous to technical range and is informative for predicting technical range (see the empirical results in Chapter 8). The cointegration vector in the technical range for DAX and CAC40, is given by P = (1.000, 0.933, 1.605)^{T}. The cointegration relationship is presented as follows:
Since all the variables are in logarithmic function form, the coefficients in pi^{1 }can be interpreted as longterm elasticities: technical range on DAX increases 1.324 percent with respect to 1 percent increase of technical range on CAC40. The matrix of speed of adjustment parameters a for DAX and CAC40, is given by a = (0.042, 0.520). The fact that a {a_{liax}  0.042) in the technical range on DAX is small and insignificant suggests that technical range on DAX is exogenous to the technical range on CAC40. The large and significant a (a_{cac} = 0.520) in the technical range on CAC40 means that technical range on CAC40 responds quickly to the changes in the technical range on DAX. Tabic 9.2 Vector error correction estimates on DAX and CAC40
In econometrics and other applications of multivariate time series analysis, a variance decomposition or forecast error variance decomposition is used to aid in the interpretation of a vector autoregressive (VAR) model once it has been fitted. The variance decomposition indicates the amount of information each variable contributes to the other variables in the autoregressive. It determines how much of the forecast error variance of each of die variables can be explained by exogenous shocks to the other variables. Figure 9.3 presents the plots of variance decomposition. The results show that the forecast error variance of technical range on DAX due to exogenous shocks to CAC40 is almost zero, most of its forecast error variance is accounted by exogenous shocks to DAX itself. The forecast error variance of technical range on CAC40 due to exogenous shocks to CAC40 itself decreases very fast with the lags: after about 20 lags the exogenous shocks to CAC40 itself contributes no more than 20% of the forecast error variance. While the forecast error variance of technical range on CAC40 due to exogenous shocks to DAX increases sharply with the lags: after about 20 lags contributes more than 80%. All these findings indicate that, in terms of technical range, the DAX index is exogenous to the CAC40 index, and there is significant technical range spillover from DAX to CAC40, not vice versa. 9.4 Summary This chapter is designed to show how the statistical properties of the technical range can be used to investigate the information spillover etfect. We scrutinize the range spillover etfect between the German stock market and the French stock market with the vector error correction model. Figure 9.3 Plots of variance decomposition of technical range, DAX and CAC40 An empirical study performed on the DAX (Germany) and CAC40 (France) stock index shows that there is range spillover effect from the German stock market to the French market while not vice versa. The German stock market is exogenous to the French stock market. Of course, this chapter only serves as an illustration that the statistical properties of the technical range can be used to investigate the information spillover effect. Notes

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