Statistical properties
Closing price records the price level of a speculative asset, while price range measures the price variability. Despite the difference, these two indicators are highly related. It is straightforward that
where H_{tl} L_{t}, C, are, respectively, high, low and closing prices.
Note that (H,C,)/C,can be approximated by ln(H_{r}/C,), and {C_{t}L,)/C,can be approximated by ln(C_{t}/L_{t}). Thus Eq. (4.1) can be approximated by the following equation
Taking natural logarithmic operation on both sides of Equation (4.2) and rearranging, we obtain
Substituting respectively I'll, and PR, for H,L, and ln(H,)ln(L,)_{y} we get
where 111, and PR, are, respectively, the technical range and the Parkinson range. Eq. (4.4) shows that the closing price can be approximated by a linear combination of technical range and Parkinson range.
Under some mild assumptions, this chapter presents some interesting statistical properties of the candlestick. The remainder of this chapter is organized as follows. Section 1 presents the statistical properties of technical range. Sections 23 consolidate these properties with simulations and empirical evidence. Section 4 summarizes the contents of this chapter.
 20 Candlestick
 4.1 Propositions
To obtain the statistical properties of the technical range, the following two assumptions are needed:
Assumption 1: the natural logarithm of the closing price is a unit root process.
Assumption 2: the natural logarithm of the Parkinson range is a weakly stationary process.
The above mentioned two assumptions are not difficult to accept. For Assumption 1, it is accepted that the financial market is quite efficient and the asset price behaves almost like a random walk, which is definitely a unit root process. For Assumption 2, it is also accepted that the volatility is a stationary process. Since the Parkinson range is a volatility estimator, it thus could be accepted as a stationary process.
With these 2 assumptions, the properties of the technical range are presented as follows:
Proposition 1: log technical range is a unit root process.
Proposition 2: closing price and technical range are соintegrated of order (1,1) with the cointegration vector of (1, 1).
Proposition 3: two technical range series are соintegrated if their corresponding closing prices are cointegrated.
The proofs to Propositions 13 are presented as follows:
Proof to Proposition 1: Given the facts that the log closing price is a unit root process and the Parkinson range is a stationary process, it is self evident that the technical range must be a unit root process.
Proof to Proposition 2: Since ln(C,)  ln(TR,) can be approximated by In (PR,) (see Eq. (4.4)) which, by Assumption 2, is a stationary process, it is easy to obtain, by the definition of cointegration, that closing price and technical range must be соintegrated of order (1, 1).
Proof to Proposition 3: Suppose two closing price series C_{lt} and C_{2}, are соintegrated of order (1,1) with a cointegration vector of (ai, «2) By the definition of cointegration, there exists a linear combination
such that со, is stationary. Substituting Eq. (4.4) for C_{it} (i = 1, 2) in Eq. (4.5), one obtains
Rearranging Eq. (4.6), we get
Given that со, and ln(PR_{t}) are stationary, it is clear from Eq. (4.7) that ln(TR_{r}) and ln(TR2t) are also соintegrated of order (1, 1).
4.2 Simulations
This section is designed to evaluate these properties with simulations. Following the usual assumption, we assume stock price follows a geometric Brownian motion (GBM) with a drift:
where B, is the stock price, ц is the drift, W, is a Winner process, dW, is normally distributed with mean 0 and standard deviation dt, dW_{t} ~ N(0, dt). To generate discrete stock price, the following differential equation is used:
where z_{f+}i~N(0, 1).
When simulating, we assume f.i = 0.000001, a = 0.1. Suppose the initial price is 1, the open, high, low and closing prices are generated as follows:
 1 Generate a path of stock price with the length of 1000,000.
 2 Divide the 1000,000 stock prices into 1000 groups.
 3 Within each group, the first and the last stock prices are used as opening and closing prices. The maximum and minimum prices within each group are used as high and low prices.
With the simulated four pieces of price information, the technical range can be calculated.
To scrutinize the unit root property of the technical range, we employ the augmented DickeyFuller (ADF) test. The results are presented in Table 4.1. Consistent with Proposition 1, the ADF test shows that the null hypothesis that there is a unit root can not be rejected. Table 4.2 reports the cointegration test results.
Table 4.1 Unit root test results on simulated technical range
tStatistic 
Prob. 

Augmented DickeyFuller test statistic 
2.174 
0.216 
Test critical values: 1% level 
3.437 

5% level 
2.864 

10% level 
2.568 
Note: when performing unit root test on simulated technical range, the parameter fi is specified to be 0. The lag length determined by SIC is p = 2.
Unrestricted Cointegration Rank Test (Trace) 

Hypothesized 
Trace 
0.05 

No. of CE(2) 
Eigenvalue 
Statistic 
Critical Value 
Prob**. 
None* 
0.054 
57.037 
12.321 
0.000 
At most 1 
0.002 
1.708 
4.130 
0.225 
Note: Trace test indicates 1 соintegrating equation at the 0.05 level; * denotes rejection of the hypothesis at the 0.05 level; Prob. is the MacKinnonHaugMichelis (1999) pvalue; ‘No. Of CE(s)’ means number of cointegration equation.
Consistent with Proposition 2, the result shows that the null hypothesis that there exists cointegration between closing price and technical range can not be rejected.
4.3 Empirical evidence
There is huge evidence that stock price is not a purely random walk. For example, the density' of stock return is found to be of high kurtosis and large skewness, and the volatility of stock return is clustering, etc. Therefore, the results obtained on the simulated random walk process are not very' convincing. Empirical studies performed on real asset prices are necessary' to further consolidate the above statistical properties of the technical range.
In this section, empirical studies are performed on different stock indices to see if the above listed theoretical properties are correct on real data.
We collect the monthly data of such stock indices as DAX of Germany, S&P500 of the United States, and SSEC index of China for the sample period over 1991.012013.12 with 276 observations. The collected data sets are typical, covering both developed financial markets (DAX, S&P500) and the largest developing financial market (SSEC). The data sets are downloaded from www.finance.yahoo.com. For each data set, four pieces of price information are reported, the opening, high, low and closing prices. The technical range is calculated on these prices.
Table 4.3 presents the augmented DickeyFuller (ADF) unit root test results for the closing price of each stock index. The null hypothesis of the ADF test is that there is a unit root. The results show that the null hypothesis for S&P500 and DAX can not be rejected at the significance level of 5%. However, the ADF test result shows that the null hypothesis of the unit root process for the Chinese stock market index, SSEC, is rejected at a significant level of 5%.^{1}
Table 4.4 reports the ADF test on the technical range. Consistent with the results in Table 4.3, ADF tests performed on the technical range also show that the hypothesis of unit root in S&P500 and DAX can not be rejected at a significance level of 5%. The unit root process in SSEC is rejected at a significance level of 1%.
Table 4.3 Unit root test results on closing price of S&P500, DAX and SSEC
Augmented DickeyFuller test statistic 
tStatistic 
Prob. 

S&P500 DAX 
1.700 
0.430 

1.422 
0.572 

SSEC 
3.340 
0.014 

Test critical values: 
1% level 
3.437 

5% level 
2.864 

10% level 
2.568 
Note: when performing the unit root test on the closing price, we specify there is a constant but no trend.
Table 4.4 Unit root test results on technical range of S&P500, DAX and SSEC
Augmented DickeyFuller test statistic 
2.174 
0.216 

Technical range ofS&P500 
2.108 
0.242 

Technical range of DAX 
2.570 
0.101 

Technical range of SSEC 
4.479 
0.000 

Test critical values: 
1% level 
3.437 

5% level 
2.864 

10% level 
2.568 
Note: when performing the unit root test on the closing price, we specify there is a constant but no trend.
Tabic 4.5 Cointegration rank test on S&P500 stock index
Unrestricted Cointegration Rank Test (Trace) 

Hypothesized 
Trace 
0.05 

No. of CE(2) 
Eigenvalue 
Statistic 
Critical Value 
Prob**. 
None* 
0.083 
25.335 
15.495 
0.001 
At most 1 
0.007 
1.950 
3.841 
0.163 
Note: Trace test indicates 1 соintegrating equation at the 0.05 level; * denotes rejection of the hypothesis at the 0.05 level; Prob. is the MacKinnonHaugMichelis (1999) pvalue; ‘No. of CE (s)’ means the number of the cointegration equation.
Cointegration tests are only performed on the S&P500 and DAX indices for the reason that the ADF test reports that SSEC is not a unit root process. Tables 4.5
 4.6 report the testing results. Consistent with Proposition 2, the null hypothesis of no cointegration is rejected at a significance level of 5%. The trace test indicates 1 cointegration equation between the closing price and the technical range.
 4.4 Summary
Technical range is an important component of the candlestick, and it has been widely used in technical analysis. However, its properties have never been, to our knowledge, scrutinized by academic research.
24 Candlestick
Table 4.6 Cointegration rank test on DAX stock index
Unrestricted Cointegration Rank Test (Trace) 

Hypothesized 
Trace 
0.05 

No. of CE(2) 
Eigenvalue 
Statistic 
Critical Value 
Prob**. 
None* 
0.086 
26.042 
15.495 
0.001 
At most 1 
0.006 
1.610 
3.841 
0.205 
Note: Trace test indicates 1 соintegrating equation at the 0.05 level; * denotes rejection of the hypothesis at the 0.05 level; Prob. is the MacKinnonHaugMichelis (1999) R value; ‘No. ofCE (s)’ means the number of the cointegration equation.
Under some mild assumptions, this chapter for the first time obtains the statistical properties of the technical range. Both simulations and empirical studies are performed to scrutinize these properties, and the results confirm the theoretical propositions.
The properties of the technical range obtained in this chapter are of great interest and importance to both academic researchers and investment practitioners. In the following chapters, we will show through empirical studies that these properties can be used in risk spillover investigation, technical range forecasting, and return forecasting. Ignorance of these properties will result in inefficient forecasts.
Note
1 The reason may be that the Chinese stock market is not informationally efficient. Jiang, et al. (2011) find significant predictability of the Chinese stock market, and the predictability can not be explained by an asset pricing model. Han et al. (2013) find there is a significant momentum effect in the Chinese stock market.