# 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 Htl Lt, C, are, respectively, high, low and closing prices.

Note that (H,-C,)/C,can be approximated by ln(Hr/C,), and {Ct-L,)/C,can be approximated by ln(Ct/Lt). 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 2-3 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 со-integrated of order (1,1) with the co-integration vector of (1, -1).

Proposition 3: two technical range series are со-integrated if their corresponding closing prices are co-integrated.

The proofs to Propositions 1-3 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 co-integration, that closing price and technical range must be со-integrated of order (1, 1).

Proof to Proposition 3: Suppose two closing price series Clt and C2, are со-integrated of order (1,1) with a co-integration vector of (ai, «2)- By the definition of co-integration, there exists a linear combination

such that со, is stationary. Substituting Eq. (4.4) for Cit (i = 1, 2) in Eq. (4.5), one obtains

Rearranging Eq. (4.6), we get

Given that со, and ln(PRt) are stationary, it is clear from Eq. (4.7) that ln(TRr) and ln(TR2t) are also со-integrated 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, dWt ~ N(0, dt). To generate discrete stock price, the following differential equation is used:

where zf+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 Dickey-Fuller (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 co-integration test results.

Table 4.1 Unit root test results on simulated technical range

 t-Statistic Prob. Augmented Dickey-Fuller 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 Co-integration 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 со-integrating equation at the 0.05 level; * denotes rejection of the hypothesis at the 0.05 level; Prob. is the MacKinnon-Haug-Michelis (1999) p-value; ‘No. Of CE(s)’ means number of co-integration equation.

Consistent with Proposition 2, the result shows that the null hypothesis that there exists co-integration 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 1991.01-2013.12 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 Dickey-Fuller (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 Dickey-Fuller test statistic t-Statistic 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 Dickey-Fuller 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 Co-integration rank test on S&P500 stock index

 Unrestricted Co-integration 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 со-integrating equation at the 0.05 level; * denotes rejection of the hypothesis at the 0.05 level; Prob. is the MacKinnon-Haug-Michelis (1999) p-value; ‘No. of CE (s)’ means the number of the co-integration equation.

Co-integration 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 co-integration is rejected at a significance level of 5%. The trace test indicates 1 co-integration 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.

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Table 4.6 Co-integration rank test on DAX stock index

 Unrestricted Co-integration 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 со-integrating equation at the 0.05 level; * denotes rejection of the hypothesis at the 0.05 level; Prob. is the MacKinnon-Haug-Michelis (1999) R value; ‘No. ofCE (s)’ means the number of the co-integration 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.

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