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Shadows in DVARIt has been demonstrated that stock return can be decomposed into a couple of components, AA, and A W_{t}. A decompositionbased vector autoregressive (DVAR) model is proposed for a simultaneous modeling of ДА, and AW, to obtain a return forecast. In this chapter, we will further show that shadows in the candlestick charts are informative for forecasting ДА, and A W,. In Chapter 5, we show theoretically that stock return can be decomposed into a couple of components and propose a DVAR model for return modeling. The DVAR model establishes a new framework for analyzing the dynamics of stock return. In this chapter, we step further by showing that shadows in candlestick charts inform the forecasts of ДА, and A W_{n} and thus the forecasting of stock return. The main finding of this chapter is that both lower and upper shadows Granger cause ДА, and A W_{t}. This finding is important as it means that the information contained in the lower and upper shadows should be used if we model stock return using the DVAR model. This chapter is organized as follows: Section 1 presents simulation results. Section 2 presents the theoretical explanations. Empirical results are presented in Section 3. Summaries are given in Section 4. 6.1 Simulations In this section, we are going to argue through simulation that both upper and lower shadows Granger cause ДА, and A IT,. The simulating process proceeds as follows: Step 1: An i.i.d. sample of size 1,000,000 is generated from Щ0, er^{2}); Step 2: A geometric Brownian motion of size 1,000,000 is generated from the above data sample; Step 3: Divide the geometric Brownian motion into 1,000 periods; within each period there are 1,000 data observations which simulate how the price evolves within each period; Step 4: Within each period, the first data observation is used as the opening price, the last as the closing price, the maximum as the high price, and the minimum as the low price; Step 5: From 4, АЛ,, Д W„ Is, and us, are calculated. To be robust, we use different ers (a=0.01, 0.05, 0.1) to represent low, medium, and high volatility respectively. We repeat the above simulations for 1000 times for each a. Since the Granger causality test is very sensitive to the lag selection, different lags are used to consolidate the results. Feige and Pearce (1979), Christiano and Ljungqvist (1988), and Stock and Watson (1989) studied the sensitivity of Granger causality to lag selection. Tables 6.16.3 report the summary statistics of /statistic and the />value for the Granger causality test. Symbol "A t* В’ means that ‘A does not Granger causes B’. The statistics as a whole confirm that upper and lower shadows Granger cause AR, and Д however there is still something interesting. The results reported in Tables 6.16.3 reject the hypothesis that Is, and us, do not Granger cause ДЛ,_{+ 1} as the p values are reported to be 0, and the results are very robust, free from lags and volatility. The results on the hypothesis that Is, and us, do not Granger cause AW_{nl} are a little bit more complex: first, the hypothesis can not be consistently rejected that Is, and us, do not Granger cause AW_{ui} since the maximum p values are above 10% for all the simulation results; second, the mean p values increase with the lags. Table 6.1 Granger causality tests: a = 0.01
Shadows in DVAR 37 Tabic 6.2 Granger causality tests: cr = 0.05
Table 6.3 Granger causality tests: cr = 0.1
Figures 6.16.3 present the histograms of p values when a takes different values. The histograms demonstrate the distributions of p values center on the value of less than 5% and diverge with the increase of lags, which are consistent with the summary statistics. 6.2 Theoretical explanation We have demonstrated with simulations that upper and lower shadows Granger cause AII,. In this section, we are going to present a theoretical explanation. Before presenting the results, some preliminary knowledge on Taylor expansion is needed. Suppose function j(x) has continuous derivative of order n + 1. Thus fix) can be expanded at point x_{0} as follows:
where/^{4}' is the Mi derivative off(x), and в S (л<_{ь} x) if x > .v_{0} or в € (x, Xo) if Xq > x. With the fact that the Mi derivative of e^{x} is still e*, we can expand the high price H_{u} at the low price L_{u} i as follows:
where h_{t+1} = ln(H_{t+1}), l_{ul} = ln(L_{ul})_{y} £ € (l_{Hl}, h_{[+l}). Rearranging and taking logarithm on both sides of the above equality, we obtain
or
Through Eq. (6.3), we get In the same way, we can expand the low price L,_{+} at the high price H_{u}i as follows,
where jj £ (/,_{+b} h_{t+x}). Rearranging and taking logarithm on both sides yields or
By Eq. (6.6), we get
With the above preparations, we present the explanations to the Granger causality as follows:
Since /,+ [/, includes the lower shadow ls„ which thus makes lower shadow Granger cause Ai.
In this case, both Is, and b,+  h, include h,  l,, which indicates Is, gets the predictive information for forecasting AR_{f+1}. In other words, Is, contributes to forecasting AR_{u} j. In both cases, Is, Granger causes AR_{f+}i
Since b,_{+} i  h, includes the lower shadow us,, which thus makes upper shadow Granger cause AR,_{+} i. Although simulations are performed with the assumption that stock prices follow the random walk model, the theoretical explanations presented above actually require no specific assumptions for the data generating process (DGP) on the stock price. The reason why upper and lower shadows are Granger causality to the DVAR model is actually due to the information overlapping: AR_{r+}i overlaps with lower shadow (upper shadow). The information overlapping is well illustrated in Figure 6.4. To further consolidate these findings, it is of great necessity to perform empirical studies on real stock prices. Shadows in DVAR 43 Figure 6.4 Shadows in DVAR: Granger causality 6.3 Empirical evidence The following are the well documented facts: (1) there is no significant linear autocorrelation in stock returns; (2) the volatility of stock returns are clustering and highly persistent; (3) the distribution of stock returns are far from being normal. They are of high kurtosis, negative skewness, and so on. All these facts indicate that the real DGP of the stock prices is unknown. Thus, simulations based on the random walk hypothesis might produce biased results. To lower down the risk of potential bias, empirical studies performed on real stock prices are needed. The empirical studies performed in this section fulfil two Table 6.4 Empirical studies on S&P500
purposes: first, they are used to consolidate the simulations and the theoretical explanations; second, they are used to confirm that the Granger causality is due to information overlapping, free from the real DGP of the stock prices. We collected the daily, weekly, and monthly index data of the U.S. Standard and Poors 500 (S&P500) index data for the sample period from January, 1990 to December, 2011. For each frequency data, four pieces of price information, opening, high, low and closing, are reported. The data set is downloaded from the finance subdirectory of the website http://finance.yahoo.com. The observations for daily, weekly and monthly data are respectively 5547, 1147 and 264. The empirical results performed on the S&P500 index are reported in Table 6.4. For each frequency data observation, the p values of the Granger causality test are presented with different lags. For the sake of consistency, the lags are selected to be 2, 4, or 6. Highly consistent with the simulations, the hypothesis of no Granger causality from upper and lower shadows to AR, is rejected at a significance level of 5%. 6.4 Summary We have demonstrated in Chapter 5 that stock return can be modeled using the DVAR model. In this chapter, we show with both theoretical explanations and empirical evidence that the upper and lower shadows in the candlestick are informative for DVAR forecasting. The findings obtained in this chapter are of great importance, as they indicate that shadows in the candlestick should be considered when using the DVAR model to forecast asset returns. 
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